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first commit
authorA-Larsen <austin.larsen@mailfence.com>
Wed, 14 Oct 2026 00:33:47 +0000 (20:33 -0400)
committerA-Larsen <austin.larsen@mailfence.com>
Wed, 14 Oct 2026 00:33:47 +0000 (20:33 -0400)
128 files changed:
Algrebra/Angles.adoc [new file with mode: 0644] patch | blob
Algrebra/Angles.html [new file with mode: 0644] patch | blob
Algrebra/Division.adoc [new file with mode: 0644] patch | blob
Algrebra/Division.html [new file with mode: 0644] patch | blob
Algrebra/Exponents.adoc [new file with mode: 0644] patch | blob
Algrebra/Exponents.html [new file with mode: 0644] patch | blob
Algrebra/Fractions.adoc [new file with mode: 0644] patch | blob
Algrebra/Fractions.html [new file with mode: 0644] patch | blob
Algrebra/Logarithms.adoc [new file with mode: 0644] patch | blob
Algrebra/Logarithms.html [new file with mode: 0644] patch | blob
Algrebra/Polynomials.adoc [new file with mode: 0644] patch | blob
Algrebra/Polynomials.html [new file with mode: 0644] patch | blob
Algrebra/README [new file with mode: 0644] patch | blob
Algrebra/Radicals.adoc [new file with mode: 0644] patch | blob
Algrebra/Radicals.html [new file with mode: 0644] patch | blob
Algrebra/Sequences.adoc [new file with mode: 0644] patch | blob
Algrebra/Sequences.html [new file with mode: 0644] patch | blob
Algrebra/Terminology.adoc [new file with mode: 0644] patch | blob
Algrebra/Terminology.html [new file with mode: 0644] patch | blob
Algrebra/Triangulation.adoc [new file with mode: 0644] patch | blob
Algrebra/Triangulation.html [new file with mode: 0644] patch | blob
Algrebra/complex_numbers.adoc [new file with mode: 0644] patch | blob
Algrebra/complex_numbers.html [new file with mode: 0644] patch | blob
Algrebra/index.adoc [new file with mode: 0644] patch | blob
Algrebra/index.html [new file with mode: 0644] patch | blob
Algrebra/linear_inequalities.adoc [new file with mode: 0644] patch | blob
Algrebra/linear_inequalities.html [new file with mode: 0644] patch | blob
Algrebra/rules.adoc [new file with mode: 0644] patch | blob
Algrebra/rules.html [new file with mode: 0644] patch | blob
Calculus/Asymptotes.adoc [new file with mode: 0644] patch | blob
Calculus/Asymptotes.html [new file with mode: 0644] patch | blob
Calculus/Derivatives.adoc [new file with mode: 0644] patch | blob
Calculus/Derivatives.html [new file with mode: 0644] patch | blob
Calculus/Integrals.adoc [new file with mode: 0644] patch | blob
Calculus/Integrals.html [new file with mode: 0644] patch | blob
Calculus/Intermediate-Value-Theorem.adoc [new file with mode: 0644] patch | blob
Calculus/Intermediate-Value-Theorem.html [new file with mode: 0644] patch | blob
Calculus/Limits.adoc [new file with mode: 0644] patch | blob
Calculus/Limits.html [new file with mode: 0644] patch | blob
Calculus/Mean-Value-Theorem.adoc [new file with mode: 0644] patch | blob
Calculus/Mean-Value-Theorem.html [new file with mode: 0644] patch | blob
Calculus/Spherical-Coordinates.adoc [new file with mode: 0644] patch | blob
Calculus/Spherical-Coordinates.html [new file with mode: 0644] patch | blob
Calculus/index.adoc [new file with mode: 0644] patch | blob
Calculus/index.html [new file with mode: 0644] patch | blob
Calculus/scarp [new file with mode: 0644] patch | blob
Geometry/Angles-And-Sides.adoc [new file with mode: 0644] patch | blob
Geometry/Angles-And-Sides.html [new file with mode: 0644] patch | blob
Geometry/Circle-Theorems.adoc [new file with mode: 0644] patch | blob
Geometry/Circle-Theorems.html [new file with mode: 0644] patch | blob
Geometry/Geometric-Proportions.adoc [new file with mode: 0644] patch | blob
Geometry/Geometric-Proportions.html [new file with mode: 0644] patch | blob
Geometry/Lines.adoc [new file with mode: 0644] patch | blob
Geometry/Lines.html [new file with mode: 0644] patch | blob
Geometry/Triangles.adoc [new file with mode: 0644] patch | blob
Geometry/Triangles.html [new file with mode: 0644] patch | blob
Geometry/index.adoc [new file with mode: 0644] patch | blob
Geometry/index.html [new file with mode: 0644] patch | blob
Graphics/3D-Objects.adoc [new file with mode: 0644] patch | blob
Graphics/3D-Objects.html [new file with mode: 0644] patch | blob
Graphics/Quanternions.adoc [new file with mode: 0644] patch | blob
Graphics/Quanternions.html [new file with mode: 0644] patch | blob
Graphics/Rasterization.adoc [new file with mode: 0644] patch | blob
Graphics/Rasterization.html [new file with mode: 0644] patch | blob
Graphics/index.adoc [new file with mode: 0644] patch | blob
Graphics/index.html [new file with mode: 0644] patch | blob
Graphing/Parent-Functions.adoc [new file with mode: 0644] patch | blob
Graphing/Parent-Functions.html [new file with mode: 0644] patch | blob
Graphing/index.adoc [new file with mode: 0644] patch | blob
Graphing/index.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Graphing.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Graphing.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Math.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Math.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Operations.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Operations.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Types.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Matrix-Types.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Properties.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Properties.html [new file with mode: 0644] patch | blob
Linear-Algrebra/SCRAP [new file with mode: 0644] patch | blob
Linear-Algrebra/System-of-Equations.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/System-of-Equations.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Terminology.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Terminology.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Vector-Math.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Vector-Math.html [new file with mode: 0644] patch | blob
Linear-Algrebra/Vector-Rules.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/Vector-Rules.html [new file with mode: 0644] patch | blob
Linear-Algrebra/index.adoc [new file with mode: 0644] patch | blob
Linear-Algrebra/index.html [new file with mode: 0644] patch | blob
Makehtml [new file with mode: 0755] patch | blob
Physics/Ball-Bouncing.adoc [new file with mode: 0644] patch | blob
Physics/Ball-Bouncing.html [new file with mode: 0644] patch | blob
Physics/Ball-Dynamic-Collision.adoc [new file with mode: 0644] patch | blob
Physics/Ball-Dynamic-Collision.html [new file with mode: 0644] patch | blob
Physics/Ball-Static-Collision.adoc [new file with mode: 0644] patch | blob
Physics/Ball-Static-Collision.html [new file with mode: 0644] patch | blob
Physics/Basics.adoc [new file with mode: 0644] patch | blob
Physics/Basics.html [new file with mode: 0644] patch | blob
Physics/Collision.adoc [new file with mode: 0644] patch | blob
Physics/Collision.html [new file with mode: 0644] patch | blob
Physics/index.adoc [new file with mode: 0644] patch | blob
Physics/index.html [new file with mode: 0644] patch | blob
Probability/Pascals-Triangle.adoc [new file with mode: 0644] patch | blob
Probability/Pascals-Triangle.html [new file with mode: 0644] patch | blob
Probability/Permutations-and-Combinations.adoc [new file with mode: 0644] patch | blob
Probability/Permutations-and-Combinations.html [new file with mode: 0644] patch | blob
Probability/index.adoc [new file with mode: 0644] patch | blob
Probability/index.html [new file with mode: 0644] patch | blob
README.adoc [new file with mode: 0644] patch | blob
Triginometry/Basics.adoc [new file with mode: 0644] patch | blob
Triginometry/Basics.html [new file with mode: 0644] patch | blob
Triginometry/Centers.adoc [new file with mode: 0644] patch | blob
Triginometry/Centers.html [new file with mode: 0644] patch | blob
Triginometry/Identities.adoc [new file with mode: 0644] patch | blob
Triginometry/Identities.html [new file with mode: 0644] patch | blob
Triginometry/index.adoc [new file with mode: 0644] patch | blob
Triginometry/index.html [new file with mode: 0644] patch | blob
diagram/diag-ditaa-md5-748c8d76a4b154138a91a12caf43c78a.png.cache [new file with mode: 0644] patch | blob
diagram/index.adoc [new file with mode: 0644] patch | blob
diagram/index.html [new file with mode: 0644] patch | blob
docinfo.conf [new file with mode: 0644] patch | blob
favicon.png [new file with mode: 0644] patch | blob
index.adoc [new file with mode: 0644] patch | blob
index.html [new file with mode: 0644] patch | blob
notes.adoc [new file with mode: 0644] patch | blob
rendered.html [new file with mode: 0644] patch | blob

diff --git a/Algrebra/Angles.adoc b/Algrebra/Angles.adoc
new file mode 100644 (file)
index 0000000..6f6e7a8
--- /dev/null
+++ b/Algrebra/Angles.adoc
@@ -0,0 +1,23 @@
+= Algrebra | Complex numbers
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+
+== Perpendicular Lines
+
+You can tell if two rays are perpendicular to one another, through a few ways. 
+
+=== Negative reciprocals
+
+The first way is if the two rays of slopes that are negative reciprocals of each
+other.
+
+=== Multiplying Slopes
+
+Similar to the above method, if you multiplied both of these slopes and got
+stem:[1] you would know the rays are perpendicular
+
diff --git a/Algrebra/Angles.html b/Algrebra/Angles.html
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index 0000000..c3d8bf7
--- /dev/null
+++ b/Algrebra/Angles.html
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Complex numbers</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_perpendicular_lines">Perpendicular Lines</a>
+<ul class="sectlevel2">
+<li><a href="#_negative_reciprocals">Negative reciprocals</a></li>
+<li><a href="#_multiplying_slopes">Multiplying Slopes</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_perpendicular_lines">Perpendicular Lines</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>You can tell if two rays are perpendicular to one another, through a few ways.</p>
+</div>
+<div class="sect2">
+<h3 id="_negative_reciprocals">Negative reciprocals</h3>
+<div class="paragraph">
+<p>The first way is if the two rays of slopes that are negative reciprocals of each
+other.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_multiplying_slopes">Multiplying Slopes</h3>
+<div class="paragraph">
+<p>Similar to the above method, if you multiplied both of these slopes and got
+\(1\) you would know the rays are perpendicular</p>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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\ No newline at end of file
diff --git a/Algrebra/Division.adoc b/Algrebra/Division.adoc
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+++ b/Algrebra/Division.adoc
@@ -0,0 +1,30 @@
+= Algrebra | Division
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Division Rules
+
+[%header, cols="1,1"]
+|===
+| Divisor      | Rule Description
+| 1            | Every integer is divisible by 1.
+| 2            | A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
+| 3            | A number is divisible by 3 if the sum of its digits is divisible by 3.
+| 4            | A number is divisible by 4 if the last two digits form a number that is divisible by 4.
+| 5            | A number is divisible by 5 if its last digit is 0 or 5.
+| 6            | A number is divisible by 6 if it is divisible by both 2 and 3.
+| 7            | To check divisibility by 7, double the last digit, subtract it from the rest of the number, and see if the result is divisible by 7.
+| 8            | A number is divisible by 8 if the last three digits form a number that is divisible by 8.
+| 9            | A number is divisible by 9 if the sum of its digits is divisible by 9.
+| 10       | A number is divisible by 10 if it ends in 0.
+|===
+
+== Long Division
+* https://www.youtube.com/watch?v=_FSXJmESFmQ[The Organic Chemestry Tutor | Long division with polynomials]
+* https://www.youtube.com/watch?v=FxHWoUOq2iQ[The Organic Chemestry Tutor | Synthetic division of polynomials]
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+.quoteblock blockquote,.quoteblock p{color:rgba(0,0,0,.85);font-size:1.15rem;line-height:1.75;word-spacing:.1em;letter-spacing:0;font-style:italic;text-align:justify}
+.quoteblock blockquote{margin:0;padding:0;border:0}
+.quoteblock blockquote::before{content:"\201c";float:left;font-size:2.75em;font-weight:bold;line-height:.6em;margin-left:-.6em;color:#7a2518;text-shadow:0 1px 2px rgba(0,0,0,.1)}
+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
+.verseblock{margin:0 1em 1.25em}
+.verseblock pre{font-family:"Open Sans","DejaVu Sans",sans-serif;font-size:1.15rem;color:rgba(0,0,0,.85);font-weight:300;text-rendering:optimizeLegibility}
+.verseblock pre strong{font-weight:400}
+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Division</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_division_rules">Division Rules</a></li>
+<li><a href="#_long_division">Long Division</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_division_rules">Division Rules</h2>
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Divisor</th>
+<th class="tableblock halign-left valign-top">Rule Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Every integer is divisible by 1.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 3 if the sum of its digits is divisible by 3.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 4 if the last two digits form a number that is divisible by 4.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 5 if its last digit is 0 or 5.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 6 if it is divisible by both 2 and 3.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">To check divisibility by 7, double the last digit, subtract it from the rest of the number, and see if the result is divisible by 7.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 8 if the last three digits form a number that is divisible by 8.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 9 if the sum of its digits is divisible by 9.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">A number is divisible by 10 if it ends in 0.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_long_division">Long Division</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=_FSXJmESFmQ">The Organic Chemestry Tutor | Long division with polynomials</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=FxHWoUOq2iQ">The Organic Chemestry Tutor | Synthetic division of polynomials</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
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\ No newline at end of file
diff --git a/Algrebra/Exponents.adoc b/Algrebra/Exponents.adoc
new file mode 100644 (file)
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--- /dev/null
+++ b/Algrebra/Exponents.adoc
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+= Algrebra | Exponents
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Exponent Rules
+
+[cols="^.^1a, ^.^1a, <.^2a", frame="none", grid="rows", options="header"]
+|===
+| Rule Name | Math (LaTeX) | Details
+
+| Product Rule
+| 
+[stem]
+++++
+a^m \cdot a^n = a^{m+n}
+++++
+| [%collapsible]
+====
+When multiplying two powers with the same base, keep the base and add the exponents.
+====
+
+| Quotient Rule
+| 
+[stem]
+++++
+\frac{a^m}{a^n} = a^{m - n}
+++++
+| [%collapsible]
+====
+When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
+====
+
+| Power of a Power
+| 
+[stem]
+++++
+(a^m)^n = a^{m \cdot n}
+++++
+| [%collapsible]
+====
+To raise a power to a power, multiply the exponents together.
+====
+
+| Power of a Product
+| 
+[stem]
+++++
+(ab)^m = a^m \cdot b^m
+++++
+| [%collapsible]
+====
+Each factor inside the parentheses is raised to the power.
+====
+
+| Power of a Quotient
+| 
+[stem]
+++++
+\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}
+++++
+| [%collapsible]
+====
+Both the numerator and the denominator are raised to the power. (Note: $b \neq 0$).
+====
+
+| Zero Exponent
+| 
+[stem]
+++++
+a^0 = 1
+++++
+| [%collapsible]
+====
+Any non-zero base raised to the power of zero is exactly 1.
+====
+
+| Negative Exponent
+| 
+[stem]
+++++
+a^{-m} = \frac{1}{a^m}
+++++
+| [%collapsible]
+====
+A negative exponent represents the reciprocal of the base raised to the positive power.
+====
+|===
+
+== Odd vs Even exponents and negative coefficients
+
+=== Even Exponents
+[stem]
+++++
+(-x)^4
+++++
+
+In this case this has a coefficients of negative one. so it is equavilant to
+
+[stem]
+++++
+x^4
+++++
+
+=== Odd Exponents
+
+[stem]
+++++
+3(-x)^3
+++++
+
+This is equivalant to
+
+[stem]
+++++
+-3x^3
+++++
+
+
+== Multiplying Negative Exponents
+
+=== When the bases are the same
+
+[stem]
+++++
+a^{-n} \cdot a^{-m} = a^{-(n+m)} = 1 \div a^{n+m}
+++++
+
+=== When the bases are different and the negative powers are the same
+
+[stem]
+++++
+a^{-n} \cdot b^{-n} = (a \cdot b)^{-n} = 1 \div (a \cdot b)^n
+++++
+
+=== When the bases and the negative powers are different
+
+[stem]
+++++
+a^{-n} \cdot b^{-m} = (a^{-n}) \cdot (b^{-m})
+++++
+
+=== Exponents of exponents
+
+== Links
+- https://www.cuemath.com/numbers/multiplying-exponents/[cuemath | multiplying exponents]
+- https://www.cuemath.com/algebra/exponent-rules/[cuemeth | exponent rules]
diff --git a/Algrebra/Exponents.html b/Algrebra/Exponents.html
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+++ b/Algrebra/Exponents.html
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+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Exponents</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_exponent_rules">Exponent Rules</a></li>
+<li><a href="#_odd_vs_even_exponents_and_negative_coefficients">Odd vs Even exponents and negative coefficients</a>
+<ul class="sectlevel2">
+<li><a href="#_even_exponents">Even Exponents</a></li>
+<li><a href="#_odd_exponents">Odd Exponents</a></li>
+</ul>
+</li>
+<li><a href="#_multiplying_negative_exponents">Multiplying Negative Exponents</a>
+<ul class="sectlevel2">
+<li><a href="#_when_the_bases_are_the_same">When the bases are the same</a></li>
+<li><a href="#_when_the_bases_are_different_and_the_negative_powers_are_the_same">When the bases are different and the negative powers are the same</a></li>
+<li><a href="#_when_the_bases_and_the_negative_powers_are_different">When the bases and the negative powers are different</a></li>
+<li><a href="#_exponents_of_exponents">Exponents of exponents</a></li>
+</ul>
+</li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_exponent_rules">Exponent Rules</h2>
+<div class="sectionbody">
+<table class="tableblock frame-none grid-rows stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-center valign-middle">Rule Name</th>
+<th class="tableblock halign-center valign-middle">Math (LaTeX)</th>
+<th class="tableblock halign-left valign-middle">Details</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Product Rule</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[a^m \cdot a^n = a^{m+n}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>When multiplying two powers with the same base, keep the base and add the exponents.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Quotient Rule</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\frac{a^m}{a^n} = a^{m - n}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Power of a Power</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[(a^m)^n = a^{m \cdot n}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>To raise a power to a power, multiply the exponents together.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Power of a Product</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[(ab)^m = a^m \cdot b^m\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>Each factor inside the parentheses is raised to the power.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Power of a Quotient</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>Both the numerator and the denominator are raised to the power. (Note: $b \neq 0$).</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Zero Exponent</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[a^0 = 1\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>Any non-zero base raised to the power of zero is exactly 1.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Negative Exponent</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[a^{-m} = \frac{1}{a^m}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>A negative exponent represents the reciprocal of the base raised to the positive power.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_odd_vs_even_exponents_and_negative_coefficients">Odd vs Even exponents and negative coefficients</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_even_exponents">Even Exponents</h3>
+<div class="stemblock">
+<div class="content">
+\[(-x)^4\]
+</div>
+</div>
+<div class="paragraph">
+<p>In this case this has a coefficients of negative one. so it is equavilant to</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[x^4\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_odd_exponents">Odd Exponents</h3>
+<div class="stemblock">
+<div class="content">
+\[3(-x)^3\]
+</div>
+</div>
+<div class="paragraph">
+<p>This is equivalant to</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[-3x^3\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_multiplying_negative_exponents">Multiplying Negative Exponents</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_when_the_bases_are_the_same">When the bases are the same</h3>
+<div class="stemblock">
+<div class="content">
+\[a^{-n} \cdot a^{-m} = a^{-(n+m)} = 1 \div a^{n+m}\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_when_the_bases_are_different_and_the_negative_powers_are_the_same">When the bases are different and the negative powers are the same</h3>
+<div class="stemblock">
+<div class="content">
+\[a^{-n} \cdot b^{-n} = (a \cdot b)^{-n} = 1 \div (a \cdot b)^n\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_when_the_bases_and_the_negative_powers_are_different">When the bases and the negative powers are different</h3>
+<div class="stemblock">
+<div class="content">
+\[a^{-n} \cdot b^{-m} = (a^{-n}) \cdot (b^{-m})\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_exponents_of_exponents">Exponents of exponents</h3>
+
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.cuemath.com/numbers/multiplying-exponents/">cuemath | multiplying exponents</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/algebra/exponent-rules/">cuemeth | exponent rules</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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\ No newline at end of file
diff --git a/Algrebra/Fractions.adoc b/Algrebra/Fractions.adoc
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+= Algrebra | Fractions
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Dividing Fractions
+When dividing two fractions, you can flip the second fraction and solve as if
+they were being multiplied to get your answer
+
+[stem]
+++++
+\frac{1}{2} \div \frac{2}{1} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
+++++
+
+== Rationalizing the denominator
+Rationalizing is the process of multiplying a surd with another similar surd, to
+result in a rational number. The surd that is used to multiply is called the
+rationalizing factor (RF).
+
+In math, a conjugate refers to a pair of binomials that have the same terms but
+opposite signs between them.
+
+Here we can multiply the top and bottom of the fraction by the conjugate to
+simplify the denomitor
+[stem]
+++++
+\frac{10}{6+\sqrt{5}} * frac{6-\sqrt{5}}{6-\sqrt{5}} =
+\frac{10(6-\sqrt{5})}{(6+\sqrt{5})(6-\sqrt{4})} =
+\frac{10(6-\sqrt{5})}{36-\sqrt{25}} =
+\frac{10(6-\sqrt{5})}{36-5} =
+\frac{10(6-\sqrt{5})}{31}
+++++
+
+== Canceling in fractions
+
+=== Example 1
+[stem]
+++++
+\frac{x^5y^2}{x^3} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot z}
+++++
+
+[stem]
+++++
+\frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot y \cdot y}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot z}
+++++
+
+[stem]
+++++
+\frac{x^2y^2}{z}
+++++
+
+=== Example 2
+
+[stem]
+++++
+\frac{a^2b^3(xy - pq)^4(c + d)}{ab^4z(xy - pq)^3}
+++++
+
+[stem]
+++++
+\frac{a(xy - pq)(c + d)}{bz}
+++++
+
+=== When you can't cancel
+
+The stem:[+ 1] makes you unable to cancel.
+
+[stem]
+++++
+\frac{a^2b^3(xy - pq)^4(c + d) + 1}{ab^4z(xy - pq)^3}
+++++
+
+=== Links
+* https://www.youtube.com/watch?v=5j8a75aHaSE[The Oganic Chemestry Tutor | Rationalizing the denominator]
+* https://www.cuemath.com/numbers/rationalize-the-denominator/[cue math | rationalizing the denominator]
+* https://www.mathsisfun.com/algebra/conjugate.html[mathisfun | conjugate]
diff --git a/Algrebra/Fractions.html b/Algrebra/Fractions.html
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+<html lang="en">
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+<title>Algrebra | Fractions</title>
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+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Fractions</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_dividing_fractions">Dividing Fractions</a></li>
+<li><a href="#_rationalizing_the_denominator">Rationalizing the denominator</a></li>
+<li><a href="#_canceling_in_fractions">Canceling in fractions</a>
+<ul class="sectlevel2">
+<li><a href="#_example_1">Example 1</a></li>
+<li><a href="#_example_2">Example 2</a></li>
+<li><a href="#_when_you_cant_cancel">When you can&#8217;t cancel</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_dividing_fractions">Dividing Fractions</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When dividing two fractions, you can flip the second fraction and solve as if
+they were being multiplied to get your answer</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{1}{2} \div \frac{2}{1} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\$
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_rationalizing_the_denominator">Rationalizing the denominator</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Rationalizing is the process of multiplying a surd with another similar surd, to
+result in a rational number. The surd that is used to multiply is called the
+rationalizing factor (RF).</p>
+</div>
+<div class="paragraph">
+<p>In math, a conjugate refers to a pair of binomials that have the same terms but
+opposite signs between them.</p>
+</div>
+<div class="paragraph">
+<p>Here we can multiply the top and bottom of the fraction by the conjugate to
+simplify the denomitor</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{10}{6+\sqrt{5}} * frac{6-\sqrt{5}}{6-\sqrt{5}} =
+\frac{10(6-\sqrt{5})}{(6+\sqrt{5})(6-\sqrt{4})} =
+\frac{10(6-\sqrt{5})}{36-\sqrt{25}} =
+\frac{10(6-\sqrt{5})}{36-5} =
+\frac{10(6-\sqrt{5})}{31}\$
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_canceling_in_fractions">Canceling in fractions</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_example_1">Example 1</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{x^5y^2}{x^3} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot z}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot y \cdot y}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot z}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{x^2y^2}{z}\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_example_2">Example 2</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{a^2b^3(xy - pq)^4(c + d)}{ab^4z(xy - pq)^3}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{a(xy - pq)(c + d)}{bz}\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_when_you_cant_cancel">When you can&#8217;t cancel</h3>
+<div class="paragraph">
+<p>The \$+ 1\$ makes you unable to cancel.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\frac{a^2b^3(xy - pq)^4(c + d) + 1}{ab^4z(xy - pq)^3}\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_links">Links</h3>
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=5j8a75aHaSE">The Oganic Chemestry Tutor | Rationalizing the denominator</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/numbers/rationalize-the-denominator/">cue math | rationalizing the denominator</a></p>
+</li>
+<li>
+<p><a href="https://www.mathsisfun.com/algebra/conjugate.html">mathisfun | conjugate</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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+</html>
\ No newline at end of file
diff --git a/Algrebra/Logarithms.adoc b/Algrebra/Logarithms.adoc
new file mode 100644 (file)
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+= Algrebra | Logarithms
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== What is a Logarithm?
+
+[stem]
+++++
+2^3 = 8
+++++
+
+Is the same as
+
+[stem]
+++++
+\log_{3}8 = 3
+++++
+
+Can be an inverse of an exponential function
+
+[.center]
+image::logorithm-exponent-inverse-graph.png[]
+
+== Properties
+
+[%header, cols="1,2a", options="header"]
+|===
+| Property Name | Formula
+
+| **Logarithm of 1**
+| [stem]
+++++
+\log_c 1 = 0
+++++
+
+| **Logarithm of the Base**
+| [stem]
+++++
+\log_c c = 1
+++++
+
+| **Product Property**
+| [stem]
+++++
+\log_c(ab) = \log_c a + \log_c b
+++++
+
+| **Quotient Property**
+| [stem]
+++++
+\log_c\left(\frac{a}{b}\right) = \log_c a - \log_c b
+++++
+
+| **Power Property**
+| [stem]
+++++
+\log_c a^b = b \log_c a
+++++
+
+| **Change of Base Formula**
+| [stem]
+++++
+\log_a b = \frac{\log_c b}{\log_c a}
+++++
+
+| **Inverse Property (Base to Log)**
+| [stem]
+++++
+\log_a a^b = b
+++++
+
+| **Inverse Property (Log to Base)**
+| [stem]
+++++
+a^{\log_a b} = b
+++++
+|===
+
+== Common Logarithms (Base 10)
+
+The **Common Logarithm** is a logarithm with base 10. It is frequently used in
+science and engineering (for example, in the pH scale or Richter scale
+calculations).
+
+When a logarithm is written without an explicit base, such as stem:[\log x], it
+is generally assumed to be the common logarithm (base 10).
+
+=== Standard Notation
+[stem]
+++++
+\log x = \log_{10} x
+++++
+
+=== Common Values
+[cols="1,2a", options="header"]
+|===
+| Expression | Result
+| stem:[\log 10] | 1
+| stem:[\log 100] | 2
+| stem:[\log 1] | 0
+| stem:[\log 0.1] | -1
+|===
+
+NOTE: In some advanced mathematics and computer science contexts, stem:[\log x]
+      may refer to the natural logarithm (base stem:[e]). Always check your
+      specific field's conventions!
+
+== Natural Logarithms (Base stem:[e])
+
+The **Natural Logarithm** is a logarithm with the base stem:[e], where stem:[e \approx 2.71828]. It is widely used in physics, biology, and economics to describe growth and decay.
+
+Instead of writing stem:[\log_e], we use the specific notation **stem:[\ln]**.
+
+=== Standard Notation
+[stem]
+++++
+\ln x = \log_{e} x
+++++
+
+=== Fundamental Properties
+[cols="1,2a", options="header"]
+|===
+| Expression | Result
+| stem:[\ln e] | 1
+| stem:[\ln 1] | 0
+| stem:[\ln e^x] | stem:[x]
+| stem:[e^{\ln x}] | stem:[x]
+|===
+
+[IMPORTANT]
+====
+The relationship between the exponential function and the natural log is inverse:
+[stem]
+++++
+y = e^x \iff x = \ln y
+++++
+====
+
+=== Properties of the Natural Logarithm (\ln)
+
+[cols="1,2a", options="header"]
+|===
+| Name/Type | Property & Condition
+
+2+^| *Fundamental Values*
+
+| Log of 1
+|
+[stem]
+++++
+\ln 1 = 0
+++++
+
+| Log of base e
+|
+[stem]
+++++
+\ln e = 1
+++++
+
+2+^| *Algebraic Properties*
+
+| Product Property
+|
+[stem]
+++++
+\ln(xy) = \ln x + \ln y \quad (x, y > 0)
+++++
+
+| Quotient Property
+|
+[stem]
+++++
+\ln(x/y) = \ln x - \ln y \quad (x, y > 0)
+++++
+
+| Power Property
+|
+[stem]
+++++
+\ln(x^y) = y \ln x \quad (x > 0)
+++++
+
+| Root Property
+|
+[stem]
+++++
+\ln(\sqrt[y]{x}) = \frac{\ln x}{y} \quad (x > 0, y \neq 0)
+++++
+
+2+^| *Comparison and Limits*
+
+| Monotonicity
+|
+[stem]
+++++
+\ln x < \ln y \quad (0 < x < y)
+++++
+
+| Limits
+|
+[stem]
+++++
+\lim_{x \to 0^{+}} \ln x = -\infty, \quad \lim_{x \to \infty} \ln x = \infty
+++++
+
+2+^| *Important Specific Limits*
+
+| At x=0
+|
+[stem]
+++++
+\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1
+++++
+
+| Limit Form
+|
+[stem]
+++++
+\lim_{\alpha \to 0} \frac{x^{\alpha} - 1}{\alpha} = \ln x \quad (x > 0)
+++++
+
+2+^| *Inequalities*
+
+| Standard
+|
+[stem]
+++++
+\frac{x - 1}{x} \le \ln x \le x - 1 \quad (x > 0)
+++++
+
+| Alternate
+|
+[stem]
+++++
+\ln(1 + x^\alpha) \le \alpha x \quad (x \ge 0, \alpha \ge 1)
+++++
+|===
diff --git a/Algrebra/Logarithms.html b/Algrebra/Logarithms.html
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+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Logarithms</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_what_is_a_logarithm">What is a Logarithm?</a></li>
+<li><a href="#_properties">Properties</a></li>
+<li><a href="#_common_logarithms_base_10">Common Logarithms (Base 10)</a>
+<ul class="sectlevel2">
+<li><a href="#_standard_notation">Standard Notation</a></li>
+<li><a href="#_common_values">Common Values</a></li>
+</ul>
+</li>
+<li><a href="#_natural_logarithms_base_e">Natural Logarithms (Base \(e\))</a>
+<ul class="sectlevel2">
+<li><a href="#_standard_notation_2">Standard Notation</a></li>
+<li><a href="#_fundamental_properties">Fundamental Properties</a></li>
+<li><a href="#_properties_of_the_natural_logarithm_ln">Properties of the Natural Logarithm (\ln)</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_what_is_a_logarithm">What is a Logarithm?</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[2^3 = 8\]
+</div>
+</div>
+<div class="paragraph">
+<p>Is the same as</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\log_{3}8 = 3\]
+</div>
+</div>
+<div class="paragraph">
+<p>Can be an inverse of an exponential function</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/logorithm-exponent-inverse-graph.png" alt="logorithm exponent inverse graph">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_properties">Properties</h2>
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Property Name</th>
+<th class="tableblock halign-left valign-top">Formula</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Logarithm of 1</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_c 1 = 0\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Logarithm of the Base</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_c c = 1\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Product Property</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_c(ab) = \log_c a + \log_c b\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Quotient Property</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_c\left(\frac{a}{b}\right) = \log_c a - \log_c b\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Power Property</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_c a^b = b \log_c a\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Change of Base Formula</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_a b = \frac{\log_c b}{\log_c a}\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Inverse Property (Base to Log)</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\log_a a^b = b\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Inverse Property (Log to Base)</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[a^{\log_a b} = b\]
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_common_logarithms_base_10">Common Logarithms (Base 10)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The <strong>Common Logarithm</strong> is a logarithm with base 10. It is frequently used in
+science and engineering (for example, in the pH scale or Richter scale
+calculations).</p>
+</div>
+<div class="paragraph">
+<p>When a logarithm is written without an explicit base, such as \(\log x\), it
+is generally assumed to be the common logarithm (base 10).</p>
+</div>
+<div class="sect2">
+<h3 id="_standard_notation">Standard Notation</h3>
+<div class="stemblock">
+<div class="content">
+\[\log x = \log_{10} x\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_common_values">Common Values</h3>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Expression</th>
+<th class="tableblock halign-left valign-top">Result</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\log 10\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>1</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\log 100\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>2</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\log 1\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>0</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\log 0.1\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>-1</p>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+In some advanced mathematics and computer science contexts, \(\log x\)
+      may refer to the natural logarithm (base \(e\)). Always check your
+      specific field&#8217;s conventions!
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_natural_logarithms_base_e">Natural Logarithms (Base \(e\))</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The <strong>Natural Logarithm</strong> is a logarithm with the base \(e\), where \(e \approx 2.71828\). It is widely used in physics, biology, and economics to describe growth and decay.</p>
+</div>
+<div class="paragraph">
+<p>Instead of writing \(\log_e\), we use the specific notation <strong>\(\ln\)</strong>.</p>
+</div>
+<div class="sect2">
+<h3 id="_standard_notation_2">Standard Notation</h3>
+<div class="stemblock">
+<div class="content">
+\[\ln x = \log_{e} x\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_fundamental_properties">Fundamental Properties</h3>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Expression</th>
+<th class="tableblock halign-left valign-top">Result</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\ln e\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>1</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\ln 1\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>0</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(\ln e^x\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>\(x\)</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">\(e^{\ln x}\)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>\(x\)</p>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<div class="admonitionblock important">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-important" title="Important"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>The relationship between the exponential function and the natural log is inverse:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[y = e^x \iff x = \ln y\]
+</div>
+</div>
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_properties_of_the_natural_logarithm_ln">Properties of the Natural Logarithm (\ln)</h3>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Name/Type</th>
+<th class="tableblock halign-left valign-top">Property &amp; Condition</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-center valign-top" colspan="2"><p class="tableblock"><strong>Fundamental Values</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Log of 1</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln 1 = 0\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Log of base e</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln e = 1\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-top" colspan="2"><p class="tableblock"><strong>Algebraic Properties</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Product Property</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(xy) = \ln x + \ln y \quad (x, y &gt; 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Quotient Property</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(x/y) = \ln x - \ln y \quad (x, y &gt; 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Power Property</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(x^y) = y \ln x \quad (x &gt; 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Root Property</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(\sqrt[y]{x}) = \frac{\ln x}{y} \quad (x &gt; 0, y \neq 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-top" colspan="2"><p class="tableblock"><strong>Comparison and Limits</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Monotonicity</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln x &lt; \ln y \quad (0 &lt; x &lt; y)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Limits</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\lim_{x \to 0^{+}} \ln x = -\infty, \quad \lim_{x \to \infty} \ln x = \infty\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-top" colspan="2"><p class="tableblock"><strong>Important Specific Limits</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">At x=0</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Limit Form</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\lim_{\alpha \to 0} \frac{x^{\alpha} - 1}{\alpha} = \ln x \quad (x &gt; 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-top" colspan="2"><p class="tableblock"><strong>Inequalities</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Standard</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\frac{x - 1}{x} \le \ln x \le x - 1 \quad (x &gt; 0)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Alternate</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(1 + x^\alpha) \le \alpha x \quad (x \ge 0, \alpha \ge 1)\]
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
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+  TeX: { equationNumbers: { autoNumber: "none" } }
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+</html>
\ No newline at end of file
diff --git a/Algrebra/Polynomials.adoc b/Algrebra/Polynomials.adoc
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+= Algrebra | Polynomials
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Standard Form of a Polynomial
+[stem]
+++++
+P(x) = a_{n}X^n + a_{n-1}X^(n-1) + a_{n-2}X^(n-2) + ... + a_{1}X + a_{0}
+++++
+
+[highlight]#stem:[a_{n},a_{n-1},a_{n-2}]# are coefficients of 
+[highlight]#stem:[x^n,x^(n-1),x^(n-2)]#  and constant term respectively and it
+should belong to real number (stem:[\exists\mathbb{R}]).
+
+== Types of Polynomials
+=== Binomial
+[stem]
+++++
+ax^m+bx^n
+++++
+=== Quadradic Equation
+
+.Standard Form
+[stem]
+++++
+ax^2 + bx + c
+++++
+
+.Vertex Form
+[stem]
+++++
+y = a(x-h)^2 + k
+++++
+
+.Intercept form
+[stem]
+++++
+y = a(x-p)(x - q)
+++++
+
+=== Line Slope-Intercept Form
+[stem]
+++++
+y = mx + b 
+++++
+m:: slope
+b:: y-intercept
+
+=== Line https://www.youtube.com/watch?v=yoHs1h5qtuQ[Point-Slope Form]
+I you have one point of the line and a slope you can figure out more points on
+the line with this formula
+
+[stem]
+++++
+(y - y_{1}) = m(x - x_{1})
+++++
+m:: slope
+stem:[x_{1},y_{1}]:: any pont on the line
+
+
+== Zeros of Polynomial (Roots of the equation)
+Values of the variable such that the polynomial equals 0. Often represented as
+stem:[\alpha], stem:[\beta], and y respectively.
+
+[.center]
+image::polynomial-roots.png[]
+
+WARNING: This image is not related to the below equation
+
+[.inline]
+* *Positive real root*: is a positive number (like stem:[1], stem:[5], or
+  stem:[\sqrt{2}]) that, when plugged in for stem:[x], makes stem:[P(x) = 0].
+
+[.inline]
+* *Negative real root*: is a negative number (like stem:[-1], stem:[-3], or
+stem:[-\pi]) that makes stem:[P(x) = 0].
+
+[.inline]
+* *Imaginary roots* (often part of Complex roots, written as stem:[a \pm bi])
+  are solutions that involve the imaginary unit stem:[i] (stem:[\sqrt{-1}]). You
+  cannot plot an imaginary number on a standard Cartesian stem:[(x,y)] grap
+
+.example equation
+[stem]
+++++
+f(x) = x^3 - 2x^2 - x + 2
+++++
+
+=== Finding the number of total real zeros
+The degree of the polynomial represents the total number of real zeros.
+
+=== Finding the number of positive real zeros
+Descrates Rule of Signs::
+    The number of positive real zeros is either equal to the number of sign
+    changes, or less a positive even integer
+
+According To *Descartes Rule of signs* The numnber of positive real zeros could
+be 2, because there were to sign changes, or 0 because 2 - 2 is zero and 2 is a
+positive even integer.
+
+=== Finding the number of negative real zeros
+First look at the equation as if you added a negative symbol to x.
+
+[stem]
+++++
+f(-x) = (-x)^3 - 2(-x)^2 - (-x) + 2 \
+\ = -z^3 - 2x^2 + x + 2
+++++
+
+The number of negative real zeros, is either equal to the number of sign
+changes, or less than a positive even integer in the equation.
+
+There is one sign change so the answer is one. One is the only answer because
+one minus two would be negative one and you cannot have a negative number of
+negative real zeros.
+
+=== Finding the number of imaginary real zeros
+In this case if we look at the chart, we know that there has to be three total
+real zeros. If we fall into the situation where there are 0 positive real zeros
+than there would have to be 2 imaginary real zeros.
+
+[%header,cols="1,1,1,4"]
+|===
+|Pos | Negative | Imaginary | Total
+|2
+|1
+|0
+|3
+
+|0
+|1
+|2
+|3
+|===
+
+== Factoring Trinomials
+=== Leading Coeficient Of One
+1. Find two numbers that when multiplied together equal the constant term
+   but when added together equal the first degree term.
+
+2. Take the two numbers from step one and put it in the multiplication of two
+   binomials in the form of stem:[(x+a)(x+b)], where `a` is the first number
+   number and `b` is the second term.
+
+Example:
+[stem]
+++++
+x^2 + 5x + 6 = 0
+++++
+
+answer:: stem:[(x+3)(x+2)]
+
+=== Leading Coeficient Other Than One
+1. Multiply the leading coeficient by the constant.
+
+2. Find two numbers that when multiplied equal the number from step 1, but when
+   added equals the first degree term.
+
+3. Replace the first degree term with the two numbers from step 2 added
+   together, and make sure they are multiplied by the same variable of the first
+   degree term.
+
+4. Factor until you get two binimals mulitiplied together
+
+Example:
+[stem]
+++++
+2x^2 - 3x - 2
+++++
+
+1. stem:[2 * -2 = 4]
+
+2. stem:[-4 + 1 = -3\ and\ -4 * 1 = -4]
+
+3. stem:[2x^2 - 4x + 1x -2]
+
+4. stem:[2x(x-2) + 1(x - 2)] -> stem:[(x-2)(2x+1)]
+
+== Quadradic Trinomial Formula
+[stem]
+++++
+x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}
+++++
+
+== Discriminant
+The part under the square root of the Quadradic Formula.
+
+[stem]
+++++
+D = b^2-4ac
+++++
+
+stem:[D\gt0]:: 2 real solutions, factorable if perfect square
+stem:[D=0]:: 1 real solution
+stem:[D<0]:: 2 imaginary solutions
+
+== Important Factoring
+=== Difference of Squares
+[stem]
+++++
+a^2-b^2 = (a+b)(a-b)
+++++
+
+=== Factoring Perfact cubes
+[stem]
+++++
+A^3 + B^3 = (A+B)(A^2-AB+B^2)
+++++
+
+[stem]
+++++
+A^3 - B^3 = (A-B)(A^2+AB+B^2)
+++++
+
+== Standard Form to Vertex Form
+1. Take the first degree term and take half of it squared added to both sides
+2. factor
+
+example:
+[stem]
+++++
+y = x^2 + 6x - 5
+++++
+
+1. stem:[y+3^2=x^2+6x+3^2-5]
+2. stem:[y + 9 = (x + 3)^2-5] -> -9 from both sides -> stem:[y=(x+3)^2-14]
+
+== Perfect Square Trinomials
+[stem]
+++++
+a^2 + 2ab + b^2 
+\ 
+or
+\ 
+a^2 - 2ab + b^2 
+++++
+
+[.center]
+image::Perfect-Square-Trinomial.png[]
+
+== Links
+- https://www.desmos.com/calculator[Desmos Graphing]
+- https://www.cuemath.com/algebra/perfect-square-trinomial/[perfect square trinomials]
+- https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/vertex-form-intro[vertex-form]
+- https://www.youtube.com/watch?v=2MKigAgPZMQ[completing the square]
+- https://www.cuemath.com/algebra/polynomials/[cuemath polynomials]
+- https://www.cuemath.com/algebra/standard-form-of-quadratic-equation/[quadradic eqautions]
+- https://www.cuemath.com/algebra/zeros-of-polynomial/[zerso of polynomial]
+- https://www.youtube.com/watch?v=YaU5JTe3cPU[OCT Descartes Rules of Signs]
diff --git a/Algrebra/Polynomials.html b/Algrebra/Polynomials.html
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+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Polynomials</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_standard_form_of_a_polynomial">Standard Form of a Polynomial</a></li>
+<li><a href="#_types_of_polynomials">Types of Polynomials</a>
+<ul class="sectlevel2">
+<li><a href="#_binomial">Binomial</a></li>
+<li><a href="#_quadradic_equation">Quadradic Equation</a></li>
+<li><a href="#_line_slope_intercept_form">Line Slope-Intercept Form</a></li>
+<li><a href="#_line_point_slope_form">Line Point-Slope Form</a></li>
+</ul>
+</li>
+<li><a href="#_zeros_of_polynomial_roots_of_the_equation">Zeros of Polynomial (Roots of the equation)</a>
+<ul class="sectlevel2">
+<li><a href="#_finding_the_number_of_total_real_zeros">Finding the number of total real zeros</a></li>
+<li><a href="#_finding_the_number_of_positive_real_zeros">Finding the number of positive real zeros</a></li>
+<li><a href="#_finding_the_number_of_negative_real_zeros">Finding the number of negative real zeros</a></li>
+<li><a href="#_finding_the_number_of_imaginary_real_zeros">Finding the number of imaginary real zeros</a></li>
+</ul>
+</li>
+<li><a href="#_factoring_trinomials">Factoring Trinomials</a>
+<ul class="sectlevel2">
+<li><a href="#_leading_coeficient_of_one">Leading Coeficient Of One</a></li>
+<li><a href="#_leading_coeficient_other_than_one">Leading Coeficient Other Than One</a></li>
+</ul>
+</li>
+<li><a href="#_quadradic_trinomial_formula">Quadradic Trinomial Formula</a></li>
+<li><a href="#_discriminant">Discriminant</a></li>
+<li><a href="#_important_factoring">Important Factoring</a>
+<ul class="sectlevel2">
+<li><a href="#_difference_of_squares">Difference of Squares</a></li>
+<li><a href="#_factoring_perfact_cubes">Factoring Perfact cubes</a></li>
+</ul>
+</li>
+<li><a href="#_standard_form_to_vertex_form">Standard Form to Vertex Form</a></li>
+<li><a href="#_perfect_square_trinomials">Perfect Square Trinomials</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_standard_form_of_a_polynomial">Standard Form of a Polynomial</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$P(x) = a_{n}X^n + a_{n-1}X^(n-1) + a_{n-2}X^(n-2) + ... + a_{1}X + a_{0}\$
+</div>
+</div>
+<div class="paragraph">
+<p><span class="highlight">\$a_{n},a_{n-1},a_{n-2}\$</span> are coefficients of
+<span class="highlight">\$x^n,x^(n-1),x^(n-2)\$</span>  and constant term respectively and it
+should belong to real number (\$\exists\mathbb{R}\$).</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_types_of_polynomials">Types of Polynomials</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_binomial">Binomial</h3>
+<div class="stemblock">
+<div class="content">
+\$ax^m+bx^n\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_quadradic_equation">Quadradic Equation</h3>
+<div class="stemblock">
+<div class="title">Standard Form</div>
+<div class="content">
+\$ax^2 + bx + c\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Vertex Form</div>
+<div class="content">
+\$y = a(x-h)^2 + k\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Intercept form</div>
+<div class="content">
+\$y = a(x-p)(x - q)\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_line_slope_intercept_form">Line Slope-Intercept Form</h3>
+<div class="stemblock">
+<div class="content">
+\$y = mx + b\$
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">m</dt>
+<dd>
+<p>slope</p>
+</dd>
+<dt class="hdlist1">b</dt>
+<dd>
+<p>y-intercept</p>
+</dd>
+</dl>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_line_point_slope_form">Line <a href="https://www.youtube.com/watch?v=yoHs1h5qtuQ">Point-Slope Form</a></h3>
+<div class="paragraph">
+<p>I you have one point of the line and a slope you can figure out more points on
+the line with this formula</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$(y - y_{1}) = m(x - x_{1})\$
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">m</dt>
+<dd>
+<p>slope</p>
+</dd>
+<dt class="hdlist1">\$x_{1},y_{1}\$</dt>
+<dd>
+<p>any pont on the line</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_zeros_of_polynomial_roots_of_the_equation">Zeros of Polynomial (Roots of the equation)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Values of the variable such that the polynomial equals 0. Often represented as
+\$\alpha\$, \$\beta\$, and y respectively.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/polynomial-roots.png" alt="polynomial roots">
+</div>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+This image is not related to the below equation
+</td>
+</tr>
+</table>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>Positive real root</strong>: is a positive number (like \$1\$, \$5\$, or
+\$\sqrt{2}\$) that, when plugged in for \$x\$, makes \$P(x) = 0\$.</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>Negative real root</strong>: is a negative number (like \$-1\$, \$-3\$, or
+\$-\pi\$) that makes \$P(x) = 0\$.</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>Imaginary roots</strong> (often part of Complex roots, written as \$a \pm bi\$)
+are solutions that involve the imaginary unit \$i\$ (\$\sqrt{-1}\$). You
+cannot plot an imaginary number on a standard Cartesian \$(x,y)\$ grap</p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="title">example equation</div>
+<div class="content">
+\$f(x) = x^3 - 2x^2 - x + 2\$
+</div>
+</div>
+<div class="sect2">
+<h3 id="_finding_the_number_of_total_real_zeros">Finding the number of total real zeros</h3>
+<div class="paragraph">
+<p>The degree of the polynomial represents the total number of real zeros.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_finding_the_number_of_positive_real_zeros">Finding the number of positive real zeros</h3>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Descrates Rule of Signs</dt>
+<dd>
+<p>The number of positive real zeros is either equal to the number of sign
+changes, or less a positive even integer</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>According To <strong>Descartes Rule of signs</strong> The numnber of positive real zeros could
+be 2, because there were to sign changes, or 0 because 2 - 2 is zero and 2 is a
+positive even integer.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_finding_the_number_of_negative_real_zeros">Finding the number of negative real zeros</h3>
+<div class="paragraph">
+<p>First look at the equation as if you added a negative symbol to x.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$f(-x) = (-x)^3 - 2(-x)^2 - (-x) + 2\$
+\$\ = -z^3 - 2x^2 + x + 2\$
+</div>
+</div>
+<div class="paragraph">
+<p>The number of negative real zeros, is either equal to the number of sign
+changes, or less than a positive even integer in the equation.</p>
+</div>
+<div class="paragraph">
+<p>There is one sign change so the answer is one. One is the only answer because
+one minus two would be negative one and you cannot have a negative number of
+negative real zeros.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_finding_the_number_of_imaginary_real_zeros">Finding the number of imaginary real zeros</h3>
+<div class="paragraph">
+<p>In this case if we look at the chart, we know that there has to be three total
+real zeros. If we fall into the situation where there are 0 positive real zeros
+than there would have to be 2 imaginary real zeros.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 14.2857%;">
+<col style="width: 14.2857%;">
+<col style="width: 14.2857%;">
+<col style="width: 57.1429%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Pos</th>
+<th class="tableblock halign-left valign-top">Negative</th>
+<th class="tableblock halign-left valign-top">Imaginary</th>
+<th class="tableblock halign-left valign-top">Total</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">0</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">0</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_factoring_trinomials">Factoring Trinomials</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_leading_coeficient_of_one">Leading Coeficient Of One</h3>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Find two numbers that when multiplied together equal the constant term
+but when added together equal the first degree term.</p>
+</li>
+<li>
+<p>Take the two numbers from step one and put it in the multiplication of two
+binomials in the form of \$(x+a)(x+b)\$, where <code>a</code> is the first number
+number and <code>b</code> is the second term.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Example:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$x^2 + 5x + 6 = 0\$
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">answer</dt>
+<dd>
+<p>\$(x+3)(x+2)\$</p>
+</dd>
+</dl>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_leading_coeficient_other_than_one">Leading Coeficient Other Than One</h3>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Multiply the leading coeficient by the constant.</p>
+</li>
+<li>
+<p>Find two numbers that when multiplied equal the number from step 1, but when
+added equals the first degree term.</p>
+</li>
+<li>
+<p>Replace the first degree term with the two numbers from step 2 added
+together, and make sure they are multiplied by the same variable of the first
+degree term.</p>
+</li>
+<li>
+<p>Factor until you get two binimals mulitiplied together</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Example:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$2x^2 - 3x - 2\$
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>\$2 * -2 = 4\$</p>
+</li>
+<li>
+<p>\$-4 + 1 = -3\ and\ -4 * 1 = -4\$</p>
+</li>
+<li>
+<p>\$2x^2 - 4x + 1x -2\$</p>
+</li>
+<li>
+<p>\$2x(x-2) + 1(x - 2)\$ &#8594; \$(x-2)(2x+1)\$</p>
+</li>
+</ol>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_quadradic_trinomial_formula">Quadradic Trinomial Formula</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\$
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_discriminant">Discriminant</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The part under the square root of the Quadradic Formula.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$D = b^2-4ac\$
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">\$D\gt0\$</dt>
+<dd>
+<p>2 real solutions, factorable if perfect square</p>
+</dd>
+<dt class="hdlist1">\$D=0\$</dt>
+<dd>
+<p>1 real solution</p>
+</dd>
+<dt class="hdlist1">\$D&lt;0\$</dt>
+<dd>
+<p>2 imaginary solutions</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_important_factoring">Important Factoring</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_difference_of_squares">Difference of Squares</h3>
+<div class="stemblock">
+<div class="content">
+\$a^2-b^2 = (a+b)(a-b)\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_factoring_perfact_cubes">Factoring Perfact cubes</h3>
+<div class="stemblock">
+<div class="content">
+\$A^3 + B^3 = (A+B)(A^2-AB+B^2)\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$A^3 - B^3 = (A-B)(A^2+AB+B^2)\$
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_standard_form_to_vertex_form">Standard Form to Vertex Form</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Take the first degree term and take half of it squared added to both sides</p>
+</li>
+<li>
+<p>factor</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>example:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$y = x^2 + 6x - 5\$
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>\$y+3^2=x^2+6x+3^2-5\$</p>
+</li>
+<li>
+<p>\$y + 9 = (x + 3)^2-5\$ &#8594; -9 from both sides &#8594; \$y=(x+3)^2-14\$</p>
+</li>
+</ol>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_perfect_square_trinomials">Perfect Square Trinomials</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$a^2 + 2ab + b^2
+\$
+\$or
+\$
+\$a^2 - 2ab + b^2\$
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Perfect-Square-Trinomial.png" alt="Perfect Square Trinomial">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.desmos.com/calculator">Desmos Graphing</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/algebra/perfect-square-trinomial/">perfect square trinomials</a></p>
+</li>
+<li>
+<p><a href="https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/vertex-form-intro">vertex-form</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=2MKigAgPZMQ">completing the square</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/algebra/polynomials/">cuemath polynomials</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/algebra/standard-form-of-quadratic-equation/">quadradic eqautions</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/algebra/zeros-of-polynomial/">zerso of polynomial</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=YaU5JTe3cPU">OCT Descartes Rules of Signs</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
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--- /dev/null
+++ b/Algrebra/README
@@ -0,0 +1 @@
+https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab[3Blue 1Brown]
diff --git a/Algrebra/Radicals.adoc b/Algrebra/Radicals.adoc
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+= Algrebra | Radicals
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+
+== Core Properties of Radicals
+
+[cols="^.^1a, ^.^1a, <.^2a", frame="none", grid="rows", options="header"]
+|===
+| Rule Name | Math (LaTeX) | Details
+
+| Product Rule
+| 
+[stem]
+++++
+\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}
+++++
+| 
+[%collapsible]
+====
+The root of a product is equal to the product of the roots. This is essential for simplifying radicals.
+====
+
+| Quotient Rule
+| 
+[stem]
+++++
+\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
+++++
+| 
+[%collapsible]
+====
+The root of a fraction is equal to the root of the numerator divided by the root of the denominator. (Note: stem:[b \neq 0]).
+====
+
+| Root of a Power
+| 
+[stem]
+++++
+\sqrt[n]{a\vphantom{[}^m} = (\sqrt[n]{a})^m
+++++
+| 
+[%collapsible]
+====
+You can take the power before or after taking the root; the result remains the same.
+====
+
+| Fractional Exponent
+| 
+[stem]
+++++
+\sqrt[n]{a\vphantom{[}^m} = a^{\frac{m}{n}}
+++++
+| 
+[%collapsible]
+====
+This is the "Bridge" rule. It allows you to convert any radical into a power, making it easier to solve using standard exponent laws.
+====
+
+| Root of a Root
+| 
+[stem]
+++++
+\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}
+++++
+| 
+[%collapsible]
+====
+When nesting radicals, you multiply the indices together. For example:
+
+[stem]
+++++
+\sqrt{\sqrt[3]{x}} = \sqrt[6]{x}
+++++
+
+====
+
+| Inverse Property
+| 
+[stem]
+++++
+(\sqrt[n]{a})^n = a
+++++
+| 
+[%collapsible]
+====
+Raising an stem:[n]-th root to the stem:[n]-th power cancels the radical out.
+====
+|===
+
+== Breaking down roots with Prime factors
+
+[stem]
+++++
+\sqrt{255}
+++++
+
+[stem]
+++++
+\begin{array}{c}
+    225 \\
+    \land \\
+    \bigcirc \kern-0.7em 5 \quad \quad 45 \\
+    \hspace{1.3cm} \land \\
+    \hspace{1.4cm} \bigcirc \kern-0.7em 5 \quad \ 9 \\
+    \hspace{2.2cm} \land \\
+    \hspace{2.3cm} \bigcirc \kern-0.7em 3 \quad \bigcirc \kern-0.7em 3
+\end{array}
+++++
+
+
+[stem]
+++++
+\sqrt{5 * 5 * 3 * 3}
+++++
+
+take a number out of each pair of number
+
+NOTE: If there wasn't numbers under the square root that didn't have pairs, they
+      would be left under the square root.
+
+[stem]
+++++
+5 * 3\sqrt{1}
+++++
+
+[stem]
+++++
+\ = 15
+++++
+
+== Rules
+
+* No negatives under roots.
diff --git a/Algrebra/Radicals.html b/Algrebra/Radicals.html
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+.listingblock.terminal pre .command::before{content:attr(data-prompt);padding-right:.5em;color:inherit;opacity:.5}
+.listingblock.terminal pre .command:not([data-prompt])::before{content:"$"}
+.listingblock pre.highlightjs{padding:0}
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+pre.prettyprint .linenums{line-height:1.45;margin-left:2em}
+pre.prettyprint li{background:none;list-style-type:inherit;padding-left:0}
+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
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+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
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+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
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+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
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+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
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+.imageblock.right{margin:.25em 0 1.25em .625em}
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+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
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+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
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+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
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+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
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+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Radicals</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_core_properties_of_radicals">Core Properties of Radicals</a></li>
+<li><a href="#_breaking_down_roots_with_prime_factors">Breaking down roots with Prime factors</a></li>
+<li><a href="#_rules">Rules</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_core_properties_of_radicals">Core Properties of Radicals</h2>
+<div class="sectionbody">
+<table class="tableblock frame-none grid-rows stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-center valign-middle">Rule Name</th>
+<th class="tableblock halign-center valign-middle">Math (LaTeX)</th>
+<th class="tableblock halign-left valign-middle">Details</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Product Rule</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>The root of a product is equal to the product of the roots. This is essential for simplifying radicals.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Quotient Rule</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>The root of a fraction is equal to the root of the numerator divided by the root of the denominator. (Note: \(b \neq 0\)).</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Root of a Power</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\sqrt[n]{a\vphantom{[}^m} = (\sqrt[n]{a})^m\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>You can take the power before or after taking the root; the result remains the same.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Fractional Exponent</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\sqrt[n]{a\vphantom{[}^m} = a^{\frac{m}{n}}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>This is the "Bridge" rule. It allows you to convert any radical into a power, making it easier to solve using standard exponent laws.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Root of a Root</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>When nesting radicals, you multiply the indices together. For example:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\sqrt{\sqrt[3]{x}} = \sqrt[6]{x}\]
+</div>
+</div>
+</div>
+</details></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="paragraph">
+<p>Inverse Property</p>
+</div></div></td>
+<td class="tableblock halign-center valign-middle"><div class="content"><div class="stemblock">
+<div class="content">
+\[(\sqrt[n]{a})^n = a\]
+</div>
+</div></div></td>
+<td class="tableblock halign-left valign-middle"><div class="content"><details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>Raising an \(n\)-th root to the \(n\)-th power cancels the radical out.</p>
+</div>
+</div>
+</details></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_breaking_down_roots_with_prime_factors">Breaking down roots with Prime factors</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\sqrt{255}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{c}
+    225 \\
+    \land \\
+    \bigcirc \kern-0.7em 5 \quad \quad 45 \\
+    \hspace{1.3cm} \land \\
+    \hspace{1.4cm} \bigcirc \kern-0.7em 5 \quad \ 9 \\
+    \hspace{2.2cm} \land \\
+    \hspace{2.3cm} \bigcirc \kern-0.7em 3 \quad \bigcirc \kern-0.7em 3
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\sqrt{5 * 5 * 3 * 3}\]
+</div>
+</div>
+<div class="paragraph">
+<p>take a number out of each pair of number</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+If there wasn&#8217;t numbers under the square root that didn&#8217;t have pairs, they
+      would be left under the square root.
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[5 * 3\sqrt{1}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\ = 15\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_rules">Rules</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>No negatives under roots.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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+</body>
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\ No newline at end of file
diff --git a/Algrebra/Sequences.adoc b/Algrebra/Sequences.adoc
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+= Algrebra | Sequences
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Arithmetic Sequence
+Has a common difference.
+
+The below has a common difference of 4:
+[stem]
+++++
+3,7,11,15,19,23,27
+++++
+=== Find Specific Index
+
+[stem]
+++++
+a_{n} = a_{1}+(n-1)d
+++++
+
+=== Find Partial Sum
+
+[stem]
+++++
+Sn = (\frac{a_{1}+a_{n}}{2})n
+++++
+
+== Geometric Sequence
+Has a common ratio.
+
+The below has a common ratio of 2:
+[stem]
+++++
+3,6,12,24,48,96,192
+++++
+
+=== Find Specific Index
+[stem]
+++++
+a_{n} = a_{1}(r)^{n-1}
+++++
+
+=== Find Geometric Mean
+
+The goemetric mean of two numbers is another number between the two such that
+there's a common ration between (1) the frist number and the mean and (2) the
+mean and the second number. The goemetric mean of 4 and 9 is 6, because the
+ration stem:[\frac{4}{6}] is the same as the ratio stem:[\frac{6}{9}]. Both
+ratios are equal to stem:[\frac{2}{3}].
+
+image::geometric-mean-and-geometric-sequences.png[]
+
+The goemetric mean will get you the number in between a and b in the geometric
+sequence
+
+==== Formulas
+
+.standard notation
+
+[stem]
+++++
+G = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n}
+++++
+
+.product notation
+
+[stem]
+++++
+G = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}
+++++
+
+
+In many technical applications (like computer science or complex statistics), it
+is easier to calculate the geometric mean by taking the average of the
+logarithms. This prevents "overflow" when multiplying many large numbers
+together:
+
+.Logarithmic Form
+[stem]
+++++
+\ln G = \frac{1}{n} \sum_{i=1}^{n} \ln x_i
+++++
+
+which can be written as:
+
+.Logarithmic Form v2
+[stem]
+++++
+G = \exp \left( \frac{1}{n} \sum_{i=1}^{n} \ln x_i \right)
+++++
+
+.When to use it 
+
+While the arithmetic mean (the standard average) is used for
+additive data, the geometric mean is specifically used for:
+
+
+=== Find Partial Sum
+[stem]
+++++
+Sn = \frac{a_{1}(1-r^n)}{1-r}
+++++
+
+== Links
+
+* https://www.youtube.com/watch?v=xt07TExQNVg[Organic Chemistry tutor | calculating geometric mean]
diff --git a/Algrebra/Sequences.html b/Algrebra/Sequences.html
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+++ b/Algrebra/Sequences.html
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+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
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+p.tableblock:last-child{margin-bottom:0}
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+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
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+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
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+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
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+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
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+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
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+.big{font-size:larger}
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+.black-background{background:#000}
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+.green-background{background:#007d00}
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+.maroon{color:#600000}
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+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
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+.red-background{background:#fa0000}
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+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
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+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
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+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Sequences</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_arithmetic_sequence">Arithmetic Sequence</a>
+<ul class="sectlevel2">
+<li><a href="#_find_specific_index">Find Specific Index</a></li>
+<li><a href="#_find_partial_sum">Find Partial Sum</a></li>
+</ul>
+</li>
+<li><a href="#_geometric_sequence">Geometric Sequence</a>
+<ul class="sectlevel2">
+<li><a href="#_find_specific_index_2">Find Specific Index</a></li>
+<li><a href="#_find_geometric_mean">Find Geometric Mean</a></li>
+<li><a href="#_find_partial_sum_2">Find Partial Sum</a></li>
+</ul>
+</li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_arithmetic_sequence">Arithmetic Sequence</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Has a common difference.</p>
+</div>
+<div class="paragraph">
+<p>The below has a common difference of 4:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[3,7,11,15,19,23,27\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_find_specific_index">Find Specific Index</h3>
+<div class="stemblock">
+<div class="content">
+\[a_{n} = a_{1}+(n-1)d\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_find_partial_sum">Find Partial Sum</h3>
+<div class="stemblock">
+<div class="content">
+\[Sn = (\frac{a_{1}+a_{n}}{2})n\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_geometric_sequence">Geometric Sequence</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Has a common ratio.</p>
+</div>
+<div class="paragraph">
+<p>The below has a common ratio of 2:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[3,6,12,24,48,96,192\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_find_specific_index_2">Find Specific Index</h3>
+<div class="stemblock">
+<div class="content">
+\[a_{n} = a_{1}(r)^{n-1}\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_find_geometric_mean">Find Geometric Mean</h3>
+<div class="paragraph">
+<p>The goemetric mean of two numbers is another number between the two such that
+there&#8217;s a common ration between (1) the frist number and the mean and (2) the
+mean and the second number. The goemetric mean of 4 and 9 is 6, because the
+ration \(\frac{4}{6}\) is the same as the ratio \(\frac{6}{9}\). Both
+ratios are equal to \(\frac{2}{3}\).</p>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/geometric-mean-and-geometric-sequences.png" alt="geometric mean and geometric sequences">
+</div>
+</div>
+<div class="paragraph">
+<p>The goemetric mean will get you the number in between a and b in the geometric
+sequence</p>
+</div>
+<div class="sect3">
+<h4 id="_formulas">Formulas</h4>
+<div class="stemblock">
+<div class="title">standard notation</div>
+<div class="content">
+\[G = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">product notation</div>
+<div class="content">
+\[G = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}\]
+</div>
+</div>
+<div class="paragraph">
+<p>In many technical applications (like computer science or complex statistics), it
+is easier to calculate the geometric mean by taking the average of the
+logarithms. This prevents "overflow" when multiplying many large numbers
+together:</p>
+</div>
+<div class="stemblock">
+<div class="title">Logarithmic Form</div>
+<div class="content">
+\[\ln G = \frac{1}{n} \sum_{i=1}^{n} \ln x_i\]
+</div>
+</div>
+<div class="paragraph">
+<p>which can be written as:</p>
+</div>
+<div class="stemblock">
+<div class="title">Logarithmic Form v2</div>
+<div class="content">
+\[G = \exp \left( \frac{1}{n} \sum_{i=1}^{n} \ln x_i \right)\]
+</div>
+</div>
+<div class="paragraph">
+<div class="title">When to use it</div>
+<p>While the arithmetic mean (the standard average) is used for
+additive data, the geometric mean is specifically used for:</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_find_partial_sum_2">Find Partial Sum</h3>
+<div class="stemblock">
+<div class="content">
+\[Sn = \frac{a_{1}(1-r^n)}{1-r}\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=xt07TExQNVg">Organic Chemistry tutor | calculating geometric mean</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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\ No newline at end of file
diff --git a/Algrebra/Terminology.adoc b/Algrebra/Terminology.adoc
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+= Algrebra | Terminology
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+[%header, cols="1,2a"]
+|===
+| Name | Definition and Example
+| Surd 
+| Irrational numbers that cannot be expressed as exact fractions or
+  terminating decimals, typically represented in square root form. They are used
+  for precise calculations where exact values are necessary. 
+
+[stem]
+++++
+\sqrt{4}, \sqrt[3]{8}, \sqrt{25}
+++++
+
+| Exact Fraction
+| An exact fraction is a fraction that can be expressed as a decimal without
+  repeating or terminating
+
+[stem]
+++++
+\frac{1}{2}, \frac{3}{4}
+++++
+
+NOTE: Fractions with denominators that are powers of 2 or 5 (like 4, 5, 8, 10,
+      etc.) typically have exact decimal representation
+
+| Irrational Numbers
+| A real number that *cannot* be written as a simple fraction
+
+image::rational-irational-numbers-example.png[]
+
+|===
+
+== Links
+* https://www.cuemath.com/numbers/surds/[cuemath | surds]
diff --git a/Algrebra/Terminology.html b/Algrebra/Terminology.html
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+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
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+        scrollPosition: scrollPosition
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Terminology</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Name</th>
+<th class="tableblock halign-left valign-top">Definition and Example</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Surd</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Irrational numbers that cannot be expressed as exact fractions or
+  terminating decimals, typically represented in square root form. They are used
+  for precise calculations where exact values are necessary.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\sqrt{4}, \sqrt[3]{8}, \sqrt{25}\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Exact Fraction</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>An exact fraction is a fraction that can be expressed as a decimal without
+  repeating or terminating</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{1}{2}, \frac{3}{4}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Fractions with denominators that are powers of 2 or 5 (like 4, 5, 8, 10,
+      etc.) typically have exact decimal representation
+</td>
+</tr>
+</table>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Irrational Numbers</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>A real number that <strong>cannot</strong> be written as a simple fraction</p>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/rational-irational-numbers-example.png" alt="rational irational numbers example">
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.cuemath.com/numbers/surds/">cuemath | surds</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
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+MathJax.Hub.Config({
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+</html>
\ No newline at end of file
diff --git a/Algrebra/Triangulation.adoc b/Algrebra/Triangulation.adoc
new file mode 100644 (file)
index 0000000..9374586
--- /dev/null
+++ b/Algrebra/Triangulation.adoc
@@ -0,0 +1,12 @@
+= Algrebra | Triangulation
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Links
+* https://www.youtube.com/watch?v=nIRBApwA-OY&t=14s[YouTube | triangulator with trigonometry]
diff --git a/Algrebra/Triangulation.html b/Algrebra/Triangulation.html
new file mode 100644 (file)
index 0000000..d471210
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+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
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+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
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+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
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+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
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+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
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+<style>
+
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+
+:root {
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+
+
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+body {
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+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
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+
+
+/*
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+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
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+.sect1 h2 {
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+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
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+    text-align: left;
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+    border-bottom: solid;
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+    width: fit-content;
+}
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+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
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+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
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+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
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+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
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+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
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+
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+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
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+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Triangulation</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=nIRBApwA-OY&amp;t=14s">YouTube | triangulator with trigonometry</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
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+MathJax.Hub.Config({
+  messageStyle: "none",
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\ No newline at end of file
diff --git a/Algrebra/complex_numbers.adoc b/Algrebra/complex_numbers.adoc
new file mode 100644 (file)
index 0000000..aa1bf78
--- /dev/null
+++ b/Algrebra/complex_numbers.adoc
@@ -0,0 +1,11 @@
+= Algrebra | Complex numbers
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+
+== Links
+* https://www.youtube.com/watch?v=hPhhxwgk0A0[HackRF Michael Ossmann, Complex Numbers]
diff --git a/Algrebra/complex_numbers.html b/Algrebra/complex_numbers.html
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+.listingblock.terminal pre .command:not([data-prompt])::before{content:"$"}
+.listingblock pre.highlightjs{padding:0}
+.listingblock pre.highlightjs>code{padding:1em;border-radius:4px}
+.listingblock pre.prettyprint{border-width:0}
+.prettyprint{background:#f7f7f8}
+pre.prettyprint .linenums{line-height:1.45;margin-left:2em}
+pre.prettyprint li{background:none;list-style-type:inherit;padding-left:0}
+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
+table.linenotable td.code{padding-left:.75em}
+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
+pre.pygments span.linenos{display:inline-block;margin-right:.75em}
+.quoteblock{margin:0 1em 1.25em 1.5em;display:table}
+.quoteblock:not(.excerpt)>.title{margin-left:-1.5em;margin-bottom:.75em}
+.quoteblock blockquote,.quoteblock p{color:rgba(0,0,0,.85);font-size:1.15rem;line-height:1.75;word-spacing:.1em;letter-spacing:0;font-style:italic;text-align:justify}
+.quoteblock blockquote{margin:0;padding:0;border:0}
+.quoteblock blockquote::before{content:"\201c";float:left;font-size:2.75em;font-weight:bold;line-height:.6em;margin-left:-.6em;color:#7a2518;text-shadow:0 1px 2px rgba(0,0,0,.1)}
+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
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+.verseblock pre strong{font-weight:400}
+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
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+        type: "SCROLL_REQUEST_GET"
+    }, "*");
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+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Complex numbers</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=hPhhxwgk0A0">HackRF Michael Ossmann, Complex Numbers</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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\ No newline at end of file
diff --git a/Algrebra/index.adoc b/Algrebra/index.adoc
new file mode 100644 (file)
index 0000000..f9e59dd
--- /dev/null
+++ b/Algrebra/index.adoc
@@ -0,0 +1,22 @@
+= ./Algrebra
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Polynomials.html;hb=HEAD[Polynomials]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Division.html;hb=HEAD[Division]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Triangulation.html;hb=HEAD[Triangulation]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/linear_inequalities.html;hb=HEAD[linear_inequalities]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Radicals.html;hb=HEAD[Radicals]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Sequences.html;hb=HEAD[Sequences]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Logarithms.html;hb=HEAD[Logarithms]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Angles.html;hb=HEAD[Angles]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Exponents.html;hb=HEAD[Exponents]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/complex_numbers.html;hb=HEAD[complex_numbers]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/rules.html;hb=HEAD[rules]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Terminology.html;hb=HEAD[Terminology]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Fractions.html;hb=HEAD[Fractions]
diff --git a/Algrebra/index.html b/Algrebra/index.html
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index 0000000..04b7495
--- /dev/null
+++ b/Algrebra/index.html
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+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Algrebra</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Polynomials.html;hb=HEAD">Polynomials</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Division.html;hb=HEAD">Division</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Triangulation.html;hb=HEAD">Triangulation</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/linear_inequalities.html;hb=HEAD">linear_inequalities</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Radicals.html;hb=HEAD">Radicals</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Sequences.html;hb=HEAD">Sequences</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Logarithms.html;hb=HEAD">Logarithms</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Angles.html;hb=HEAD">Angles</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Exponents.html;hb=HEAD">Exponents</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/complex_numbers.html;hb=HEAD">complex_numbers</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/rules.html;hb=HEAD">rules</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Terminology.html;hb=HEAD">Terminology</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/Fractions.html;hb=HEAD">Fractions</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Algrebra/linear_inequalities.adoc b/Algrebra/linear_inequalities.adoc
new file mode 100644 (file)
index 0000000..2310a59
--- /dev/null
+++ b/Algrebra/linear_inequalities.adoc
@@ -0,0 +1,16 @@
+= Algrebra | Linear Inequalities
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Multiplying or dividing by negative numbers
+
+image::linear_inequalities.png[]
+
+When you multiply or divide by a negative number when solving for linear
+inequalities make sure to change the direction of the inequality sign.
diff --git a/Algrebra/linear_inequalities.html b/Algrebra/linear_inequalities.html
new file mode 100644 (file)
index 0000000..686835c
--- /dev/null
+++ b/Algrebra/linear_inequalities.html
@@ -0,0 +1,1084 @@
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+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
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+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
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+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+window.addEventListener("message", (e) => {
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+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Linear Inequalities</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_multiplying_or_dividing_by_negative_numbers">Multiplying or dividing by negative numbers</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_multiplying_or_dividing_by_negative_numbers">Multiplying or dividing by negative numbers</h2>
+<div class="sectionbody">
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linear_inequalities.png" alt="linear inequalities">
+</div>
+</div>
+<div class="paragraph">
+<p>When you multiply or divide by a negative number when solving for linear
+inequalities make sure to change the direction of the inequality sign.</p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+</body>
+</html>
\ No newline at end of file
diff --git a/Algrebra/rules.adoc b/Algrebra/rules.adoc
new file mode 100644 (file)
index 0000000..e823dcc
--- /dev/null
+++ b/Algrebra/rules.adoc
@@ -0,0 +1,70 @@
+= Algrebra | Rules
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Properties
+
+.Commutative Property of Addition
+
+[stem]
+++++
+a + b = b + a
+++++
+
+.Commutative Property of Multiplication
+
+[stem]
+++++
+a * b = b * a
+++++
+
+.Associative Property of Addition
+
+[stem]
+++++
+(a + b) + c = a + (b + c)
+++++
+
+.Associative Property of Multiplication
+
+[stem]
+++++
+(a * b) * c = a * (b * c)
+++++
+
+.Distributive Properties of Addition and Multiplication
+
+
+[stem]
+++++
+a * (b + c) = a * b + a * c
+++++
+
+and
+
+[stem]
+++++
+(a + b) * c = a * c + b * c
+++++
+
+
+image::https://o.quizlet.com/3gRs.XX5diEtgknc3VGaNA_b.png[]
+
+=== Links
+* https://www.analyzemath.com/algebra/rules_algebra.html
+
+== Order of Operations (PEMDAS)
+Please excuse my dear aunt sally
+
+- Parentheses
+- Exponents
+- Multiply
+- Divide
+- Add
+- Subtract
diff --git a/Algrebra/rules.html b/Algrebra/rules.html
new file mode 100644 (file)
index 0000000..2d67026
--- /dev/null
+++ b/Algrebra/rules.html
@@ -0,0 +1,1163 @@
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+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
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+        scrollPosition: scrollPosition
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Rules</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_properties">Properties</a>
+<ul class="sectlevel2">
+<li><a href="#_links">Links</a></li>
+</ul>
+</li>
+<li><a href="#_order_of_operations_pemdas">Order of Operations (PEMDAS)</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_properties">Properties</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="title">Commutative Property of Addition</div>
+<div class="content">
+\$a + b = b + a\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Commutative Property of Multiplication</div>
+<div class="content">
+\$a * b = b * a\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Associative Property of Addition</div>
+<div class="content">
+\$(a + b) + c = a + (b + c)\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Associative Property of Multiplication</div>
+<div class="content">
+\$(a * b) * c = a * (b * c)\$
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Distributive Properties of Addition and Multiplication</div>
+<div class="content">
+\$a * (b + c) = a * b + a * c\$
+</div>
+</div>
+<div class="paragraph">
+<p>and</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$(a + b) * c = a * c + b * c\$
+</div>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://o.quizlet.com/3gRs.XX5diEtgknc3VGaNA_b.png" alt="3gRs.XX5diEtgknc3VGaNA b">
+</div>
+</div>
+<div class="sect2">
+<h3 id="_links">Links</h3>
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.analyzemath.com/algebra/rules_algebra.html" class="bare">https://www.analyzemath.com/algebra/rules_algebra.html</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_order_of_operations_pemdas">Order of Operations (PEMDAS)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Please excuse my dear aunt sally</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Parentheses</p>
+</li>
+<li>
+<p>Exponents</p>
+</li>
+<li>
+<p>Multiply</p>
+</li>
+<li>
+<p>Divide</p>
+</li>
+<li>
+<p>Add</p>
+</li>
+<li>
+<p>Subtract</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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\ No newline at end of file
diff --git a/Calculus/Asymptotes.adoc b/Calculus/Asymptotes.adoc
new file mode 100644 (file)
index 0000000..8b3a8a4
--- /dev/null
+++ b/Calculus/Asymptotes.adoc
@@ -0,0 +1,27 @@
+= Calculus | Asymptotes
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Solving for Vertical Asymptotes
+
+[stem]
+++++
+f(x) = frac{x+4}{(x+4)(x-8)}
+++++
+
+* `-4` is removable
+* `8` is non-removable
+
+In this function there would be a hole at `-4`
+
+== Solving for Horizontal and Slant Asymptotes
+
+== Links
+
+* https://youtu.be/kvhpu1TkSjI?si=sZV82P255DY-4mhS[The Orgainic Chemistry Tutor | Horizontal Asymptotes and Slant Asymptotes of Rational Functions]
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new file mode 100644 (file)
index 0000000..bbef716
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+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Asymptotes</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_solving_for_vertical_asymptotes">Solving for Vertical Asymptotes</a></li>
+<li><a href="#_solving_for_horizontal_and_slant_asymptotes">Solving for Horizontal and Slant Asymptotes</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_solving_for_vertical_asymptotes">Solving for Vertical Asymptotes</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$f(x) = frac{x+4}{(x+4)(x-8)}\$
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><code>-4</code> is removable</p>
+</li>
+<li>
+<p><code>8</code> is non-removable</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>In this function there would be a hole at <code>-4</code></p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_solving_for_horizontal_and_slant_asymptotes">Solving for Horizontal and Slant Asymptotes</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://youtu.be/kvhpu1TkSjI?si=sZV82P255DY-4mhS">The Orgainic Chemistry Tutor | Horizontal Asymptotes and Slant Asymptotes of Rational Functions</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
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+</body>
+</html>
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+= Calculus | Derivatives
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Derivative Formula
+
+[stem]
+++++
+\lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = f'(a)
+++++
+
+[.center]
+image::Derivative-Formula.png[]
+
+[.center]
+image::3Blue1Brown-Limit-definition.png[]
+
+Keep in mind that that stem:[\frac{df}{dx}] or stem:[\frac{d}{dx}] is just the
+short had version of this definition.
+
+image::short-hand-derivative.png[]
+
+== Basic Rules
+
+=== Constant Rule
+[stem]
+++++
+\frac{d}{dx} [c] = 0
+++++
+
+=== Power Rule
+[stem]
+++++
+\frac{d}{dx} [x^n] = nx^(n-1)
+++++
+
+=== Product Rule
+[stem]
+++++
+\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
+++++
+
+=== Quotient Rule
+[stem]
+++++
+\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
+++++
+
+=== Chain Rule
+[stem]
+++++
+\frac{d}{dx} [f(g(x))]  = f'(g(x)) * g'(x)
+++++
+
+== Points of Interest
+
+[.center]
+image::points-of-interest-derivatives.png[]
+
+=== Concaving
+
+NOTE: Down looks like a frown. Up looks like a cup.
+
+Whenever a function is concave _down_, it's derivative (and slope) are
+_decreasing_; werever a function is concave up, it's derivative (and slope) are
+_increasing_.
+
+=== Points
+
+[cols="1,1,2a", options="header"]
+|===
+| Concept | Key Points | Mathematical Description
+
+| **Local Maxima**
+| b, g
+| * <- Slope is positive
+* -> Slope is negative
+
+| **Inflection Points**
+| c, e, h
+| * The point at which the slope changes.
+* Steepest or least steep points in their immediate neighborhoods.
+
+| **The Corner**
+| j
+| * No tangent line means no slope.
+* No slope means no derivative.
+* The derivative is undefined at this point.
+
+| **Critical Points & Numbers**
+| b, d, g, i, j
+| * **Critical Points:** Where the derivative is either zero or undefined.
+* **Critical Numbers:** The x-values of these critical points.
+|===
+
+Local Extrma::
+    local maximums and minimums
+
+:stem: latexmath
+
+== Finding Critical numbers
+
+Find the critical numbers of
+[stem]
+++++
+f(x) = 3x^5 - 20x^3
+++++
+
+1. Find the first derivative of f using the power rule.
+
+[stem]
+++++
+f(x) = 3x^5 - 20x^3
+\
+f'(x) = 15x^4 - 60x^2
+++++
+
+[start=2]
+2. Set the derivative equal to zero and solve for x.
+
+[stem]
+++++
+\begin{aligned}
+    & 15x^4 - 60x^2 = 0 \\
+    & 15x^2(x^2 - 4) = 0 \\
+    & 15x^2(x+2)(x-2) = 0 \\
+    & 15x^2 = 0\ \ or\ \ x + 2 = 0\ \ or\ \ x - 2 = 0 \\
+    & x = 0\ \ or\ \ x = -2\ \ or\ \ x = 2
+\end{aligned}
+++++
+
+These three X-Values are crtical number of f.
+
+WARNING: Additional critical numbers could exists if the first derivate were
+         undefined at some x-values, but that is not the case here.
+
+[.center]
+image::Finding-Critical-Numbers.png[]
+
+=== The first derivative test
+
+Take a number line and put down the critical numbers you found above 0, -2, and
+2.
+
+[.center]
+image::critical-numbers-number-line.png[]
+
+Pick a value from each region, plug it into the first dierivate and note whether
+your result is positive or negative
+
+[stem]
+++++
+\begin{aligned}
+    & f'(x) = 15x^4 - 60x^2 \\[10pt]
+    & f'(-3) = 15(-3)^4 - 60(-3)^2 = 15 \cdot 81 - 60 \cdot 9 = 675 \\
+    & f'(-1) = 15(-1)^4 - 60(-1)^2 = 15 - 60 = -45 \\
+    & f'(1) = 15(1)^4 - 60(1)^2 = 15 - 60 = -45 \\
+    & f'(3) = 15(3)^4 - 60(3)^2 = 15 \cdot 81 - 60 \cdot 9 = 675
+\end{aligned}
+++++
+
+Mark were the positives and negatives are.
+
+[.center]
+image::critical-numbers-number-line-2.png[]
+
+* *local maximum*: -2
+* *local minimum*: 2
+
+Lastly, plug in the x values
+
+[stem]
+++++
+\begin{aligned}
+    & f(x) = 3x^5 - 20x^3 \\[15pt]
+    & f(-2) = 3(-2)^5 - 20(-2)^3 = 64 \\
+    & f(2) = 3(2)^5 - 20(2)^3 = -64
+\end{aligned}
+++++
+
+The local max is located at stem:[(-2,64)] and the local min is at
+stem:[(2,-64)].
+
+=== The second derivative test
+
+* *positive value*: concave up
+
+* *negative value*: concave down
+
+* *zero value*: _inconclusive_. Could be concave up, concave down, or there
+  could be an inflection point there.
+
+Find the second derivative and then plug in the critical numbers you found
+
+* *-2*: concave down
+* *2*: concave up
+* *0*: inconclusive
+
+[stem]
+++++
+\begin{aligned}
+    f(x) &= 3x^5 - 20x^3 \\[5pt]
+    f'(x) &= 15x^4 - 60x^2 && \text{(power rule)} \\
+    f''(x) &= 60x^3 - 120x && \text{(power rule)} \\[15pt]
+    f''(-2) &= 60(-2)^3 - 120(-2) = -240 \\
+    f''(0) &= 60(0)^3 - 120(0) = 0 \\
+    f''(2) &= 60(2)^3 - 120(2) = 240
+\end{aligned}
+++++
+
+* *-2*: concave down
+* *2*: concave up
+
+WARNING: The second derivative test is no help where the first derivative is
+         undefined.
+
+=== Examples
+
+[%collapsible]
+====
+Find the local exteme of stem:[g(x) = 2x - 3x^{2/3} + 4]
+
+[.center]
+image::derivative-figure-11-5.png[]
+
+1. *Find the first derivative of g.*
+
++
+[stem]
+++++
+\begin{aligned}
+    g(x)  &= 2x - 3x^{2/3} + 4 \\[5pt]
+    g'(x) &= 2 - 2x^{1/3} && \text{(power rule)} \\
+\end{aligned}
+++++
+
+2. *Set the derivative equal to zero and solve*
+
++
+[stem]
+++++
+\begin{aligned}
+    2 - 2x^{-1/3} &= 0 \\
+    -2x^{-1/3} &= -2 \\
+    x^{-1/3} &= 1 \\
+    (x^{-1/3})^{-3} &= 1^{-3} \\
+    x &= 1
+\end{aligned}
+++++
+
++
+Thus 1 is a critical number
+
+
+3. *Determine whether the first derivative is undefined for any x-values*
+
++
+stem:[2x^{-1/3}] equals latexmath:[\frac{2}{\sqrt[3\]{x}}], you'd have 
+latexmath:[\frac{2}{0}] which is undefined. +
+
++
+stem:[0] is another critical number.
+
+4. *Plot the critical numbers on a number line, and then use the first*
+   *derivative test to figure out the sign of each region.*
+
++
+You can use -1, 0.5, and 2 as test numbers:
+
++
+[stem]
+++++
+\begin{aligned}
+    g'(x) &= 2 - 2x^{-1/3} \\[15pt]
+    g'(-1) &= 4 && \text{(pos.)} \\
+    g'(0.5) &\approx -0.52 && \text{(neg.)} \\
+    g'(2) &\approx 0.41 && \text{(pos.)}
+\end{aligned}
+++++
+
++
+[.center]
+image::figure-11-6-critical-points-graph.png[]
+
++
+Because the derivative of _g_ switches from positive to negative at 0 there's a
+local max there. And becaause the first derivative switches from negative to
+positive atr 1, there's a local min at stem:[x = 1].
+
+5. *Plug the critical numbers into g to obtain the function values (the*
+   *heights) of these two local extrema.*
+
++
+[stem]
+++++
+\begin{aligned}
+    g(x) = 2x - 3x^{2/3} + 4 \quad & g(0) = 4 \\
+    & g(1) = 3
+\end{aligned}
+++++
+
+So, there's a #local max at (0,4)# and a #local min at (1,3)#. You're done.
+
+You could have used the second derivative test instead of the first derivative
+test in Step 4. First, you need the second derivative of latexmath:[g], which
+is, as you know, the derivative of its first derivative:
+
+[latexmath]
+++++
+\begin{aligned}
+    g'(x) &= 2 - 2x^{-1/3} \\
+    g''(x) &= \frac{2}{3}x^{-4/3}
+\end{aligned}
+++++
+
+Evaluate the second derivative at latexmath:[x = 1] (the critical number from
+Step 2):
+
+[latexmath]
+++++
+g''(1) = \frac{2}{3}
+++++
+
+:stem: latexmath
+
+Because latexmath:[g''(1)] is positive, you know that latexmath:[g] is concave
+up at latexmath:[x = 1] and, therefore, that there's a local min there.
+
+At the other critical number, latexmath:[x = 0] (from Step 3), the first
+derivative is undefined. The second derivative test is no help where the first
+derivative is undefined, so you've got to use the first derivative test for that
+critical number.
+
+====
+
+:stem:
+
+== Derivatives to Memorize
+
+=== Logrithmic Derivative
+
+[.inline]
+- stem:[\frac{d}{dx}\[\ln(u)\] = \frac{u'}{u}]
+
+=== Exponential Derivative
+
+[.inline]
+- stem:[\frac{d}{dx}\[e^u\] = e^u * u']
+
+
+=== Trigonometric Derivatives
+
+[.inline]
+- stem:[\frac{d}{dx}\[\sinx\] = \cosx]
+
+[.inline]
+- stem:[\frac{d}{dx}\[\cosx\] = -\sinx]
+
+[.inline]
+- stem:[\frac{d}{dx}\[\tanx\] = \sec^2x]
+
+[.inline]
+- stem:[\frac{d}{dx}\[\secx\] = \secx\tanx]
+
+[.inline]
+- stem:[\frac{d}{dx}\[\cotx\] = -\csc^2x]
+
+[.inline]
+- stem:[\frac{d}{dx}\[\cscx\] = -\cscx\cotx]
+
+=== Inverse Trigonometric Derivatives
+
+[.inline]
+- stem:[\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}\ \text{(for |x| < 1)}]
+
+[.inline]
+- stem:[\arccos(x) = \frac{-1}{\sqrt{1 - x^2}}\ \text{(for |x| < 1)}]
+
+[.inline]
+- stem:[\arctan(x) = \frac{1}{1 - x^2}\ \text{(for all x)}]
+
+[.inline]
+- stem:[\arccot(x) = \frac{-1}{1 + x^2}\ \text{(for all x)}]
+
+[.inline]
+- stem:[\arcsec(x) = \frac{1}{|x| \sqrt{x^2 - 1}}\ \text{(for |x| > 1)}]
+
+[.inline]
+- stem:[\arccsc(x) = \frac{-1}{|x| \sqrt{x^2 - 1}}\ \text{(for |x| > 1)}]
+
+:stem: latexmath
+
+== Examples
+
+.Problem 1
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Find the equation of the tangent line to the curve } x^3 + 4xy^2 + y^3 = 107 \text{ at the point } (2, 3) \\
+    & \text{using implicit differentiation.} \\[10pt]
+    & \frac{d}{dx} \left[ x^3 + (4x)(y^2) + y^3 = 107 \right] \\[10pt]
+    & 3x^2 + 4(y^2) + 4x(2y) \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0 \\[10pt]
+    & 3(2)^2 + 4(3)^2 + 4(2)(2 \cdot 3) \frac{dy}{dx} + 3(3)^2 \frac{dy}{dx} = 0 \\[10pt]
+    & 12 + 36 + 8(6) \frac{dy}{dx} + 27 \frac{dy}{dx} = 0 \\[10pt]
+    & 48 \frac{dy}{dx} + 27 \frac{dy}{dx} = -48
+\end{aligned}
+++++
+
+NOTE: Whenever you differentiate a *stem:[y]* variable with respect to
+      *stem:[x]* you need to add a stem:[\frac{dy}{dx}] in front of it
+
+[stem]
+++++
+\begin{aligned}
+    & m_{\text{tan}} = \frac{dy}{dx} = \frac{-48 \div 3}{75 \div 3} = -\frac{16}{25} \\[15pt]
+    & y - y_1 = m(x - x_1) \\[10pt]
+    & \boxed{y - 3 = -\frac{16}{25}(x - 2)}
+\end{aligned}
+++++
+
+====
+
+.Problem 2
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \qquad f(x) = \sin x \\[15pt]
+    & f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\[15pt]
+    & f'(x) = ? \qquad f'(x) = \boxed{\cos x}
+\end{aligned}
+++++
+
+NOTE: Notice how the limit is similar to the formal definition of a derivative
+====
+
+.Critical numbers
+[%collapsible]
+====
+[stem]
+++++
+\text{10. Identify all intervals where } f(x) \text{ is increasing given } f(x) = x^3 + \frac{3}{2}x^2 - 36x - 9.
+++++
+
+[stem]
+++++
+\begin{aligned}
+    & f'(c) = 0 \quad f'(x) = 3x^2 + \frac{3}{2}(2x) - 36(1) + 0 \\
+    & f'(x) = 3x^2 + 3x - 36 = 0 \\
+    & 0 = 3(x^2 + 1x - 12) \qquad \begin{aligned} c &= -4 \\ c &= 3 \end{aligned} \\
+    & 0 = 3(x + 4)(x - 3)
+\end{aligned}
+++++
+
+Now plug in a number for each section of this number line, that will give you
+whether numbers are increasing or decreasing in that section.
+
+[.center]
+image::critical-numbers-example-1.png[]
+
+[stem]
+++++
+\text{Inc } \boxed{(-\infty, -4) \cup (3, \infty)}
+++++
+
+You can identify your local maximum and minimums like this
+
+[.center]
+image::local-minimum-and-maximum.png[]
+
+====
+
+.Related Rates 1
+[%collapsible]
+====
+[.center]
+image::water-cylinder.png[]
+
+[stem]
+++++
+\begin{aligned}
+    & \text{9. Water is flowing into a cylinder with a diameter of } 6\text{ft and a height of } 10\text{ft.} \\
+    & \text{If the height of the water in the cylinder is increasing at } 3\text{ft/min, at what} \\
+    & \text{rate is the volume of the water in the cylinder changing?} \\[10pt]
+    & V = Ah \qquad \frac{dh}{dt} = +3 \text{ ft/min} \\
+    & [V = (\pi R^2)h] \frac{d}{dt} \\[10pt]
+    & \frac{dV}{dt} = \pi R^2 (1) \frac{dh}{dt} \\[10pt]
+    & \frac{dV}{dt} = \pi(3)^2(3) \\[10pt]
+    & \frac{dV}{dt} = 27\pi \text{ ft}^3\text{/min}
+\end{aligned}
+++++
+
+
+NOTE: Constants don't get derived but variables do. A constant is something that
+      doesn't change while a variable is changing.
+====
+
+
+== Links
+- https://www.derivative-calculator.net/[Derivative Calculator]
+- https://www.youtube.com/playlist?list=PLTEBkBE3HYK9XpUFla9Th86wPfR1XxFlQ[The Organic Chemistry Tutor | Related Rates]
+- https://www.youtube.com/watch?v=WmBzmHru78w[The Organic Chemistry Tutor | Calculus 1 Final Exam]
diff --git a/Calculus/Derivatives.html b/Calculus/Derivatives.html
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+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Derivatives</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_derivative_formula">Derivative Formula</a></li>
+<li><a href="#_basic_rules">Basic Rules</a>
+<ul class="sectlevel2">
+<li><a href="#_constant_rule">Constant Rule</a></li>
+<li><a href="#_power_rule">Power Rule</a></li>
+<li><a href="#_product_rule">Product Rule</a></li>
+<li><a href="#_quotient_rule">Quotient Rule</a></li>
+<li><a href="#_chain_rule">Chain Rule</a></li>
+</ul>
+</li>
+<li><a href="#_points_of_interest">Points of Interest</a>
+<ul class="sectlevel2">
+<li><a href="#_concaving">Concaving</a></li>
+<li><a href="#_points">Points</a></li>
+</ul>
+</li>
+<li><a href="#_finding_critical_numbers">Finding Critical numbers</a>
+<ul class="sectlevel2">
+<li><a href="#_the_first_derivative_test">The first derivative test</a></li>
+<li><a href="#_the_second_derivative_test">The second derivative test</a></li>
+<li><a href="#_examples">Examples</a></li>
+</ul>
+</li>
+<li><a href="#_derivatives_to_memorize">Derivatives to Memorize</a>
+<ul class="sectlevel2">
+<li><a href="#_logrithmic_derivative">Logrithmic Derivative</a></li>
+<li><a href="#_exponential_derivative">Exponential Derivative</a></li>
+<li><a href="#_trigonometric_derivatives">Trigonometric Derivatives</a></li>
+<li><a href="#_inverse_trigonometric_derivatives">Inverse Trigonometric Derivatives</a></li>
+</ul>
+</li>
+<li><a href="#_examples_2">Examples</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_derivative_formula">Derivative Formula</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = f'(a)\$
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Derivative-Formula.png" alt="Derivative Formula">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3Blue1Brown-Limit-definition.png" alt="3Blue1Brown Limit definition">
+</div>
+</div>
+<div class="paragraph">
+<p>Keep in mind that that \$\frac{df}{dx}\$ or \$\frac{d}{dx}\$ is just the
+short had version of this definition.</p>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/short-hand-derivative.png" alt="short hand derivative">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_basic_rules">Basic Rules</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_constant_rule">Constant Rule</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{d}{dx} [c] = 0\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_power_rule">Power Rule</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{d}{dx} [x^n] = nx^(n-1)\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_product_rule">Product Rule</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_quotient_rule">Quotient Rule</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\$
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_chain_rule">Chain Rule</h3>
+<div class="stemblock">
+<div class="content">
+\$\frac{d}{dx} [f(g(x))]  = f'(g(x)) * g'(x)\$
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_points_of_interest">Points of Interest</h2>
+<div class="sectionbody">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/points-of-interest-derivatives.png" alt="points of interest derivatives">
+</div>
+</div>
+<div class="sect2">
+<h3 id="_concaving">Concaving</h3>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Down looks like a frown. Up looks like a cup.
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>Whenever a function is concave <em>down</em>, it&#8217;s derivative (and slope) are
+<em>decreasing</em>; werever a function is concave up, it&#8217;s derivative (and slope) are
+<em>increasing</em>.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_points">Points</h3>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Concept</th>
+<th class="tableblock halign-left valign-top">Key Points</th>
+<th class="tableblock halign-left valign-top">Mathematical Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Local Maxima</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">b, g</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="ulist">
+<ul>
+<li>
+<p>&#8592; Slope is positive</p>
+</li>
+<li>
+<p>&#8594; Slope is negative</p>
+</li>
+</ul>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Inflection Points</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">c, e, h</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="ulist">
+<ul>
+<li>
+<p>The point at which the slope changes.</p>
+</li>
+<li>
+<p>Steepest or least steep points in their immediate neighborhoods.</p>
+</li>
+</ul>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>The Corner</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">j</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="ulist">
+<ul>
+<li>
+<p>No tangent line means no slope.</p>
+</li>
+<li>
+<p>No slope means no derivative.</p>
+</li>
+<li>
+<p>The derivative is undefined at this point.</p>
+</li>
+</ul>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Critical Points &amp; Numbers</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">b, d, g, i, j</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="ulist">
+<ul>
+<li>
+<p><strong>Critical Points:</strong> Where the derivative is either zero or undefined.</p>
+</li>
+<li>
+<p><strong>Critical Numbers:</strong> The x-values of these critical points.</p>
+</li>
+</ul>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Local Extrma</dt>
+<dd>
+<p>local maximums and minimums</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_finding_critical_numbers">Finding Critical numbers</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Find the critical numbers of</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[f(x) = 3x^5 - 20x^3\]
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Find the first derivative of f using the power rule.</p>
+</li>
+</ol>
+</div>
+<div class="stemblock">
+<div class="content">
+\[f(x) = 3x^5 - 20x^3
+\
+f'(x) = 15x^4 - 60x^2\]
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic" start="2">
+<li>
+<p>Set the derivative equal to zero and solve for x.</p>
+</li>
+</ol>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; 15x^4 - 60x^2 = 0 \\
+    &amp; 15x^2(x^2 - 4) = 0 \\
+    &amp; 15x^2(x+2)(x-2) = 0 \\
+    &amp; 15x^2 = 0\ \ or\ \ x + 2 = 0\ \ or\ \ x - 2 = 0 \\
+    &amp; x = 0\ \ or\ \ x = -2\ \ or\ \ x = 2
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>These three X-Values are crtical number of f.</p>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+Additional critical numbers could exists if the first derivate were
+         undefined at some x-values, but that is not the case here.
+</td>
+</tr>
+</table>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Finding-Critical-Numbers.png" alt="Finding Critical Numbers">
+</div>
+</div>
+<div class="sect2">
+<h3 id="_the_first_derivative_test">The first derivative test</h3>
+<div class="paragraph">
+<p>Take a number line and put down the critical numbers you found above 0, -2, and
+2.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/critical-numbers-number-line.png" alt="critical numbers number line">
+</div>
+</div>
+<div class="paragraph">
+<p>Pick a value from each region, plug it into the first dierivate and note whether
+your result is positive or negative</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; f'(x) = 15x^4 - 60x^2 \\[10pt]
+    &amp; f'(-3) = 15(-3)^4 - 60(-3)^2 = 15 \cdot 81 - 60 \cdot 9 = 675 \\
+    &amp; f'(-1) = 15(-1)^4 - 60(-1)^2 = 15 - 60 = -45 \\
+    &amp; f'(1) = 15(1)^4 - 60(1)^2 = 15 - 60 = -45 \\
+    &amp; f'(3) = 15(3)^4 - 60(3)^2 = 15 \cdot 81 - 60 \cdot 9 = 675
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Mark were the positives and negatives are.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/critical-numbers-number-line-2.png" alt="critical numbers number line 2">
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>local maximum</strong>: -2</p>
+</li>
+<li>
+<p><strong>local minimum</strong>: 2</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Lastly, plug in the x values</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; f(x) = 3x^5 - 20x^3 \\[15pt]
+    &amp; f(-2) = 3(-2)^5 - 20(-2)^3 = 64 \\
+    &amp; f(2) = 3(2)^5 - 20(2)^3 = -64
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>The local max is located at \((-2,64)\) and the local min is at
+\((2,-64)\).</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_the_second_derivative_test">The second derivative test</h3>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>positive value</strong>: concave up</p>
+</li>
+<li>
+<p><strong>negative value</strong>: concave down</p>
+</li>
+<li>
+<p><strong>zero value</strong>: <em>inconclusive</em>. Could be concave up, concave down, or there
+could be an inflection point there.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Find the second derivative and then plug in the critical numbers you found</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>-2</strong>: concave down</p>
+</li>
+<li>
+<p><strong>2</strong>: concave up</p>
+</li>
+<li>
+<p><strong>0</strong>: inconclusive</p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    f(x) &amp;= 3x^5 - 20x^3 \\[5pt]
+    f'(x) &amp;= 15x^4 - 60x^2 &amp;&amp; \text{(power rule)} \\
+    f''(x) &amp;= 60x^3 - 120x &amp;&amp; \text{(power rule)} \\[15pt]
+    f''(-2) &amp;= 60(-2)^3 - 120(-2) = -240 \\
+    f''(0) &amp;= 60(0)^3 - 120(0) = 0 \\
+    f''(2) &amp;= 60(2)^3 - 120(2) = 240
+\end{aligned}\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>-2</strong>: concave down</p>
+</li>
+<li>
+<p><strong>2</strong>: concave up</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+The second derivative test is no help where the first derivative is
+         undefined.
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_examples">Examples</h3>
+<details>
+<summary class="title">Details</summary>
+<div class="content">
+<div class="paragraph">
+<p>Find the local exteme of \(g(x) = 2x - 3x^{2/3} + 4\)</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/derivative-figure-11-5.png" alt="derivative figure 11 5">
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p><strong>Find the first derivative of g.</strong></p>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    g(x)  &amp;= 2x - 3x^{2/3} + 4 \\[5pt]
+    g'(x) &amp;= 2 - 2x^{1/3} &amp;&amp; \text{(power rule)} \\
+\end{aligned}\]
+</div>
+</div>
+</li>
+<li>
+<p><strong>Set the derivative equal to zero and solve</strong></p>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    2 - 2x^{-1/3} &amp;= 0 \\
+    -2x^{-1/3} &amp;= -2 \\
+    x^{-1/3} &amp;= 1 \\
+    (x^{-1/3})^{-3} &amp;= 1^{-3} \\
+    x &amp;= 1
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Thus 1 is a critical number</p>
+</div>
+</li>
+<li>
+<p><strong>Determine whether the first derivative is undefined for any x-values</strong></p>
+<div class="paragraph">
+<p>\(2x^{-1/3}\) equals \(\frac{2}{\sqrt[3]{x}}\), you&#8217;d have
+\(\frac{2}{0}\) which is undefined.<br></p>
+</div>
+<div class="paragraph">
+<p>\(0\) is another critical number.</p>
+</div>
+</li>
+<li>
+<p><strong>Plot the critical numbers on a number line, and then use the first</strong>
+<strong>derivative test to figure out the sign of each region.</strong></p>
+<div class="paragraph">
+<p>You can use -1, 0.5, and 2 as test numbers:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    g'(x) &amp;= 2 - 2x^{-1/3} \\[15pt]
+    g'(-1) &amp;= 4 &amp;&amp; \text{(pos.)} \\
+    g'(0.5) &amp;\approx -0.52 &amp;&amp; \text{(neg.)} \\
+    g'(2) &amp;\approx 0.41 &amp;&amp; \text{(pos.)}
+\end{aligned}\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/figure-11-6-critical-points-graph.png" alt="figure 11 6 critical points graph">
+</div>
+</div>
+<div class="paragraph">
+<p>Because the derivative of <em>g</em> switches from positive to negative at 0 there&#8217;s a
+local max there. And becaause the first derivative switches from negative to
+positive atr 1, there&#8217;s a local min at \(x = 1\).</p>
+</div>
+</li>
+<li>
+<p><strong>Plug the critical numbers into g to obtain the function values (the</strong>
+<strong>heights) of these two local extrema.</strong></p>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    g(x) = 2x - 3x^{2/3} + 4 \quad &amp; g(0) = 4 \\
+    &amp; g(1) = 3
+\end{aligned}\]
+</div>
+</div>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>So, there&#8217;s a <mark>local max at (0,4)</mark> and a <mark>local min at (1,3)</mark>. You&#8217;re done.</p>
+</div>
+<div class="paragraph">
+<p>You could have used the second derivative test instead of the first derivative
+test in Step 4. First, you need the second derivative of \(g\), which
+is, as you know, the derivative of its first derivative:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    g'(x) &amp;= 2 - 2x^{-1/3} \\
+    g''(x) &amp;= \frac{2}{3}x^{-4/3}
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Evaluate the second derivative at \(x = 1\) (the critical number from
+Step 2):</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[g''(1) = \frac{2}{3}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Because \(g''(1)\) is positive, you know that \(g\) is concave
+up at \(x = 1\) and, therefore, that there&#8217;s a local min there.</p>
+</div>
+<div class="paragraph">
+<p>At the other critical number, \(x = 0\) (from Step 3), the first
+derivative is undefined. The second derivative test is no help where the first
+derivative is undefined, so you&#8217;ve got to use the first derivative test for that
+critical number.</p>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_derivatives_to_memorize">Derivatives to Memorize</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_logrithmic_derivative">Logrithmic Derivative</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\ln(u)] = \frac{u'}{u}\$</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_exponential_derivative">Exponential Derivative</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[e^u] = e^u * u'\$</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_trigonometric_derivatives">Trigonometric Derivatives</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\sinx] = \cosx\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\cosx] = -\sinx\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\tanx] = \sec^2x\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\secx] = \secx\tanx\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\cotx] = -\csc^2x\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{d}{dx}\[\cscx] = -\cscx\cotx\$</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_inverse_trigonometric_derivatives">Inverse Trigonometric Derivatives</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}\ \text{(for |x| &lt; 1)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arccos(x) = \frac{-1}{\sqrt{1 - x^2}}\ \text{(for |x| &lt; 1)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arctan(x) = \frac{1}{1 - x^2}\ \text{(for all x)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arccot(x) = \frac{-1}{1 + x^2}\ \text{(for all x)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arcsec(x) = \frac{1}{|x| \sqrt{x^2 - 1}}\ \text{(for |x| &gt; 1)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\arccsc(x) = \frac{-1}{|x| \sqrt{x^2 - 1}}\ \text{(for |x| &gt; 1)}\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_examples_2">Examples</h2>
+<div class="sectionbody">
+<details>
+<summary class="title">Problem 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Find the equation of the tangent line to the curve } x^3 + 4xy^2 + y^3 = 107 \text{ at the point } (2, 3) \\
+    &amp; \text{using implicit differentiation.} \\[10pt]
+    &amp; \frac{d}{dx} \left[ x^3 + (4x)(y^2) + y^3 = 107 \right] \\[10pt]
+    &amp; 3x^2 + 4(y^2) + 4x(2y) \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0 \\[10pt]
+    &amp; 3(2)^2 + 4(3)^2 + 4(2)(2 \cdot 3) \frac{dy}{dx} + 3(3)^2 \frac{dy}{dx} = 0 \\[10pt]
+    &amp; 12 + 36 + 8(6) \frac{dy}{dx} + 27 \frac{dy}{dx} = 0 \\[10pt]
+    &amp; 48 \frac{dy}{dx} + 27 \frac{dy}{dx} = -48
+\end{aligned}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Whenever you differentiate a <strong>\(y\)</strong> variable with respect to
+      <strong>\(x\)</strong> you need to add a \(\frac{dy}{dx}\) in front of it
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; m_{\text{tan}} = \frac{dy}{dx} = \frac{-48 \div 3}{75 \div 3} = -\frac{16}{25} \\[15pt]
+    &amp; y - y_1 = m(x - x_1) \\[10pt]
+    &amp; \boxed{y - 3 = -\frac{16}{25}(x - 2)}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Problem 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \qquad f(x) = \sin x \\[15pt]
+    &amp; f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\[15pt]
+    &amp; f'(x) = ? \qquad f'(x) = \boxed{\cos x}
+\end{aligned}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Notice how the limit is similar to the formal definition of a derivative
+</td>
+</tr>
+</table>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Critical numbers</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\text{10. Identify all intervals where } f(x) \text{ is increasing given } f(x) = x^3 + \frac{3}{2}x^2 - 36x - 9.\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; f'(c) = 0 \quad f'(x) = 3x^2 + \frac{3}{2}(2x) - 36(1) + 0 \\
+    &amp; f'(x) = 3x^2 + 3x - 36 = 0 \\
+    &amp; 0 = 3(x^2 + 1x - 12) \qquad \begin{aligned} c &amp;= -4 \\ c &amp;= 3 \end{aligned} \\
+    &amp; 0 = 3(x + 4)(x - 3)
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Now plug in a number for each section of this number line, that will give you
+whether numbers are increasing or decreasing in that section.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/critical-numbers-example-1.png" alt="critical numbers example 1">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\text{Inc } \boxed{(-\infty, -4) \cup (3, \infty)}\]
+</div>
+</div>
+<div class="paragraph">
+<p>You can identify your local maximum and minimums like this</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/local-minimum-and-maximum.png" alt="local minimum and maximum">
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Related Rates 1</summary>
+<div class="content">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/water-cylinder.png" alt="water cylinder">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{9. Water is flowing into a cylinder with a diameter of } 6\text{ft and a height of } 10\text{ft.} \\
+    &amp; \text{If the height of the water in the cylinder is increasing at } 3\text{ft/min, at what} \\
+    &amp; \text{rate is the volume of the water in the cylinder changing?} \\[10pt]
+    &amp; V = Ah \qquad \frac{dh}{dt} = +3 \text{ ft/min} \\
+    &amp; [V = (\pi R^2)h] \frac{d}{dt} \\[10pt]
+    &amp; \frac{dV}{dt} = \pi R^2 (1) \frac{dh}{dt} \\[10pt]
+    &amp; \frac{dV}{dt} = \pi(3)^2(3) \\[10pt]
+    &amp; \frac{dV}{dt} = 27\pi \text{ ft}^3\text{/min}
+\end{aligned}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Constants don&#8217;t get derived but variables do. A constant is something that
+      doesn&#8217;t change while a variable is changing.
+</td>
+</tr>
+</table>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.derivative-calculator.net/">Derivative Calculator</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/playlist?list=PLTEBkBE3HYK9XpUFla9Th86wPfR1XxFlQ">The Organic Chemistry Tutor | Related Rates</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=WmBzmHru78w">The Organic Chemistry Tutor | Calculus 1 Final Exam</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 22:00:50 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+  },
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+    delimiters: [["\\$", "\\$"]],
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+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
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+= Calculus | Integrals
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Funamental Theorem of Calculus
+
+image::fundemental-theorem-of-calculus-v1.png[width=80%]
+
+.version 1
+[stem]
+++++
+A_{f}(x) = \int_{s}^{x} f(t)\ dt
+++++
+
+stem:[A_{f}(x)]::
+    The area under the curve at point `x` for funtion `f()`
+
+`integral s`::
+    The starting point
+
+`integral x`::
+    The end point
+
+`dt`::
+    An infinitely small width of each section
+
+// image::fundemental-theorem-of-calculus-v2-minus.png[width=75%]
+
+.version 2
+[stem]
+++++
+\int_{a}^{b} f(x) \, dx = F(b) - F(a)
+++++
+
+`F()`::
+    The antiderivative of `f()`
+`x`::
+    Represents all the values in the interval stem:[[a, b]]
+
+
+.example
+[%collapsible]
+====
+
+Find the area under the graph between 2 and 3:
+
+[.center]
+image::fundemental-theorem-of-calculus-v2-example.png[]
+
+[stem]
+++++
+\int_{2}^{3} (x^2+1)dx = F(3) - F(2)
+++++
+
+the antiderivative of stem:[x^2 + 1] is stem:[\frac{1}{3}x^3 + x]
+
+stem:[F(3) - F(2)] can be written as stem:[ \left[ \frac{1}{3}x^3 + x
+\right\]_2^3 ], and thus,
+
+[stem]
+++++
+\begin{aligned}
+\int_{2}^{3} (x^2+1)dx &= \left[ \frac{1}{3}x^3 + x \right]_2^3 \\
+&= \left( \frac{1}{3} \cdot 3^3 + 3 \right) - \left( \frac{1}{3} \cdot 2^3 + 2 \right) \\
+&= 12 - 4\frac{2}{3} \\
+&= 7\frac{1}{3}
+\end{aligned}
+++++
+====
+
+== Power Rule
+
+1. Bring the power in front where it will multiply the rest of the derivative
+
++
+[stem]
+++++
+5x^4 \rightarrow 4 * 5x^4
+++++
+
+2. Reduce the power by one and simplify
+
++
+[stem]
+++++
+4 * 5x^4 \rightarrow 4 * 5x^3 = 20x^3
+++++
+
++
+thus, stem:[y'] = stem:[20x^3]
+
+== Reverse Power Rule
+
+1. Increase the power by one.
+
++
+The 3 becomes a 4.
+
++
+[stem]
+++++
+20x^3 \rightarrow 20x^4
+++++
+
+2. Divide by th enew power and simplify
+
++
+[stem]
+++++
+20x^4 \rightarrow \frac{20}{4}x^4 = 5x^4
+++++
+
++
+WARNING: The reverse power rule does not work for a power of negative one.
+
+
+== "u" substitution
+
+find
+
+[stem]
+++++
+\lim 2x \cos(x^2) dx
+++++
+
+. *Set _u_ equal to the argument of the main function.*
+
++
+The argument of stem:[\cos(x^2)] is stem:[x^2], so you set stem:[u] equal to
+stem:[x^2].
+
+. *Take the derivative of _u_ with respect to _x_.*
+
++
+[stem]
+++++
+u = x^2\ \text{so}\ \frac{du}{dx} = 2x
+++++
+
+. *Solve for _dx_.*
+
++
+[stem]
+++++
+\begin{aligned}
+ & \frac{du}{dx} = \frac{2x}{1} \\[10pt]
+ & du = 2xdx\ \text{(cross multiplication)} \\
+ & \frac{du}{2x}\ = dx\ \text{(dviding both sides by 2x)}
+\end{aligned}
+++++
+
+. *Make substitutions*
+
++
+In stem:[\int 2x \cos(x^2)dx], u takes the place of stem:[x^2] and
+stem:[\frac{du}{2x}] takes the place of stem:[dx]. So now you've got
+stem:[\int 2x\cos u \frac{du}{2x}]. The two stem:[2x]s cancel, giving you
+stem:[\int \cos\ u\ du].
+
+. *Anitdifferentiate using the simple reverse rule.*
+
++
+[stem]
+++++
+\lim \cos\ u\ du = \sin\ u + C
+++++
+
+. *Substitute stem:[x^2] back in for _u_, coming full circle.*
+
++
+_u_ equals stem:[x^2], so stem:[x^2] goes in for the _u_:
+
++
+[stem]
+++++
+\lim \cos\ u\ du - \sin(x^2) + C
+++++
+
++
+That's it. So stem:[\lim\ 2x\ cos(x^2)dx = \sin(x^2) + C]
+
+=== Examples
+
+.Example 1
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{8. Evaluate the integral shown below:} \\[10pt]
+    & \int 2x \sqrt{3x^2 + 5} \, dx \qquad \begin{aligned} u &= 3x^2 + 5 \\ du &= 6x \, dx \\ \frac{du}{6x} &= dx \end{aligned} \\[15pt]
+    & \int 2x \sqrt{u} \, \frac{du}{6x} \\[15pt]
+    & \frac{1}{3} \int u^{1/2} \, du = \frac{1}{3} \cdot \frac{u^{3/2}}{3/2} + C \\[15pt]
+    & = \frac{2}{9} u^{3/2} + C = \boxed{\frac{2}{9} (3x^2 + 5)^{3/2} + C}
+\end{aligned}
+++++
+====
+
+
+== Integration by Parts
+
+[stem]
+++++
+\int u \, dv = uv - \int v \, du
+++++
+
+NOTE: You can check "calculus for dumies" chapter 16
+
+////
+[stem]
+++++
+\begin{array}{|c|c|}
+\hline
+\rule[-1.2em]{0pt}{3em} \quad u \quad & \quad v \quad \\ \hline
+\rule[-1.2em]{0pt}{3em} \quad du \quad & \quad dv \quad \\ \hline
+\end{array}
+++++
+////
+
+[stem]
+++++
+\begin{array}{ccc}
+\begin{array}{c} \text{diff.} \\ \downarrow \end{array} & 
+\begin{array}{|c|c|}
+\hline
+\rule[-1.2em]{0pt}{3em} \quad u \quad & \quad v \quad \\ \hline
+\rule[-1.2em]{0pt}{3em} \quad du \quad & \quad dv \quad \\ \hline
+\end{array} & 
+\begin{array}{c} \text{int.} \\ \uparrow \end{array}
+\end{array}
+++++
+
+[stem]
+++++
+u \cdot v - \int v \cdot du
+++++
+
+[.center]
+image::integration-by-parts-7.png[]
+
+=== Picking your "u" and "dv"
+
+
+You can use thsi mnemonic to pick you're "u", anything that isn't you're "u" is
+you're "dv".
+
+.LIATE mnemonic
+[cols="1,2,2a", options="header"]
+|===
+| Letter | Function Type | Example
+
+| **L**
+| Logarithmic
+|
+[stem]
+++++
+\ln(x), \log(x)
+++++
+
+| **I**
+| Inverse Trigonometric
+|
+[stem]
+++++
+\arctan(x), \arcsin(x)
+++++
+
+| **A**
+| Algebraic
+|
+[stem]
+++++
+x^2, 5x^2 + 3
+++++
+
+| **T**
+| Trigonometric
+|
+[stem]
+++++
+\cos(x), \sin(x)
+++++
+
+| **E**
+| Exponential
+|
+[stem]
+++++
+e^x, 10^x
+++++
+|===
+
+.example
+[%collapsible]
+====
+
+Integrate this 
+
+[stem]
+++++
+\int \arctan(x)dx
+++++
+
+You see that there are no logarithmic functions in arctan(x)dx, but
+there is an inverse trigonometric function, arctan(x). So that’s your u.
+Everything else is your dv, namely, plain old dx.
+
+[stem]
+++++
+u = archtan(x) \\
+dv = dx
+++++
+
+[stem]
+++++
+\begin{aligned}
+\text{diff.} \downarrow \quad
+\begin{array}{|wc{7em}|wc{7em}|}
+\hline
+\rule[-1.5em]{0pt}{4em} \arctan(x) & x \\ \hline
+\rule[-1.5em]{0pt}{4em} \frac{1}{1+x^2} \, dx & dx \\ \hline
+\end{array}
+\hspace{1em} \uparrow \text{int.}
+\end{aligned}
+++++
+
+showing integration from dx to x
+
+[stem]
+++++
+\int 1 dx \\
+\int x^0 dx =
+x^1 = x
+++++
+
+Do the seven thing.
+
+[.center]
+image::integration-by-parts-7.png[]
+
+[stem]
+++++
+\int udv = uv - \int vdu \\
+\int \arctan(x)dx = x \arctan(x) - \int x \cdot \frac{1}{1 + x^2}dx
+++++
+
+Now you can finish this problem by integrating 
+stem:[\int x \cdot \frac{1}{1 + x^2}dx]with the substitution method, setting
+u=1+x2.
+
+NOTE: You can check "calculus for dumies" chapter 15
+
+WARNING:  _u_ in u=1+x2 has nothing to do with the integration-by-parts _u_.
+
+solution
+
+[stem]
+++++
+\int \arctan(x)dx = x\arctan(x) - \frac{1}{2}\ln(1 + xx^2) + C.
+++++
+====
+
+== Basic Antiderivative Formulas
+
+[.inline]
+- stem:[\int x^n\ dx = \frac{x^n+1}{n+1} + C\ (n \neq -1)]
+
+[.inline]
+- stem:[\int \sin x\ dx = -\cos x + C]
+
+[.inline]
+- stem:[\int \frac{dx}{x} = \ln|x| + C]
+
+[.inline]
+- stem:[\int e^x\ dx = -\cos x + C]
+
+[.inline]
+- stem:[\int a^x\ dx = \frac{1}{\ln a} a^x + C]
+
+[.inline]
+- stem:[\int \sin x\ dx = -\cos x + C]
+
+[.inline]
+- stem:[\int \cos x\ dx = \sin x + C]
+
+[.inline]
+- stem:[\int \sec^2 x\ dx = \tan x + C]
+
+[.inline]
+- stem:[\int \csc^2 x\ dx = - \cot x + C]
+
+[.inline]
+- stem:[\int \sec x \tan x\ dx = \sec x + C]
+
+[.inline]
+- stem:[\int \csc x \cot x\ dx = -\csc x + C]
+
+[.inline]
+- stem:[\int \frac{dx}{x\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C]
+
+[.inline]
+- stem:[\int \frac{dx}{a^2 - x^2} = \frac{1}{a}\arctan \frac{x}{a} + C]
+
+[.inline]
+- stem:[\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a} \text{arcsec} \frac{|x|}{a} + C]
+
+NOTE: remember stem:[\frac{d}{dx}] means derivative which is similar to the the
+      change in y divided by the change in x (stem:[\frac{\Delta Y}{\Delta X}]
+      rise over run), except the scale of stem:[\frac{d}{dx}] is instantaneous,
+      where the scale of stem:[\frac{\Delta Y}{\Delta X}] is between two
+      distinct points on a line.
+
+== Examples
+
+.Example 1
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \int \frac{4x^5 + x^4 - 3x^2}{x^2} \, dx = \int (4x^3 + x^2 - 3) \, dx \\[15pt]
+    & = \frac{4x^4}{4} + \frac{x^3}{3} - 3x + C \\[15pt]
+    & = \boxed{x^4 + \frac{1}{3}x^3 - 3x + C}
+\end{aligned}
+++++
+====
diff --git a/Calculus/Integrals.html b/Calculus/Integrals.html
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Integrals</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_funamental_theorem_of_calculus">Funamental Theorem of Calculus</a></li>
+<li><a href="#_power_rule">Power Rule</a></li>
+<li><a href="#_reverse_power_rule">Reverse Power Rule</a></li>
+<li><a href="#_u_substitution">"u" substitution</a>
+<ul class="sectlevel2">
+<li><a href="#_examples">Examples</a></li>
+</ul>
+</li>
+<li><a href="#_integration_by_parts">Integration by Parts</a>
+<ul class="sectlevel2">
+<li><a href="#_picking_your_u_and_dv">Picking your "u" and "dv"</a></li>
+</ul>
+</li>
+<li><a href="#_basic_antiderivative_formulas">Basic Antiderivative Formulas</a></li>
+<li><a href="#_examples_2">Examples</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_funamental_theorem_of_calculus">Funamental Theorem of Calculus</h2>
+<div class="sectionbody">
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/fundemental-theorem-of-calculus-v1.png" alt="fundemental theorem of calculus v1" width="80%">
+</div>
+</div>
+<div class="stemblock">
+<div class="title">version 1</div>
+<div class="content">
+\[A_{f}(x) = \int_{s}^{x} f(t)\ dt\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">\(A_{f}(x)\)</dt>
+<dd>
+<p>The area under the curve at point <code>x</code> for funtion <code>f()</code></p>
+</dd>
+<dt class="hdlist1"><code>integral s</code></dt>
+<dd>
+<p>The starting point</p>
+</dd>
+<dt class="hdlist1"><code>integral x</code></dt>
+<dd>
+<p>The end point</p>
+</dd>
+<dt class="hdlist1"><code>dt</code></dt>
+<dd>
+<p>An infinitely small width of each section</p>
+</dd>
+</dl>
+</div>
+<div class="stemblock">
+<div class="title">version 2</div>
+<div class="content">
+\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><code>F()</code></dt>
+<dd>
+<p>The antiderivative of <code>f()</code></p>
+</dd>
+<dt class="hdlist1"><code>x</code></dt>
+<dd>
+<p>Represents all the values in the interval \([a, b\)]</p>
+</dd>
+</dl>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="paragraph">
+<p>Find the area under the graph between 2 and 3:</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/fundemental-theorem-of-calculus-v2-example.png" alt="fundemental theorem of calculus v2 example">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\int_{2}^{3} (x^2+1)dx = F(3) - F(2)\]
+</div>
+</div>
+<div class="paragraph">
+<p>the antiderivative of \(x^2 + 1\) is \(\frac{1}{3}x^3 + x\)</p>
+</div>
+<div class="paragraph">
+<p>\(F(3) - F(2)\) can be written as \( \left[ \frac{1}{3}x^3 + x
+\right]_2^3 \), and thus,</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+\int_{2}^{3} (x^2+1)dx &amp;= \left[ \frac{1}{3}x^3 + x \right]_2^3 \\
+&amp;= \left( \frac{1}{3} \cdot 3^3 + 3 \right) - \left( \frac{1}{3} \cdot 2^3 + 2 \right) \\
+&amp;= 12 - 4\frac{2}{3} \\
+&amp;= 7\frac{1}{3}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_power_rule">Power Rule</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Bring the power in front where it will multiply the rest of the derivative</p>
+<div class="stemblock">
+<div class="content">
+\[5x^4 \rightarrow 4 * 5x^4\]
+</div>
+</div>
+</li>
+<li>
+<p>Reduce the power by one and simplify</p>
+<div class="stemblock">
+<div class="content">
+\[4 * 5x^4 \rightarrow 4 * 5x^3 = 20x^3\]
+</div>
+</div>
+<div class="paragraph">
+<p>thus, \(y'\) = \(20x^3\)</p>
+</div>
+</li>
+</ol>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_reverse_power_rule">Reverse Power Rule</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Increase the power by one.</p>
+<div class="paragraph">
+<p>The 3 becomes a 4.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[20x^3 \rightarrow 20x^4\]
+</div>
+</div>
+</li>
+<li>
+<p>Divide by th enew power and simplify</p>
+<div class="stemblock">
+<div class="content">
+\[20x^4 \rightarrow \frac{20}{4}x^4 = 5x^4\]
+</div>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+The reverse power rule does not work for a power of negative one.
+</td>
+</tr>
+</table>
+</div>
+</li>
+</ol>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_u_substitution">"u" substitution</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>find</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim 2x \cos(x^2) dx\]
+</div>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p><strong>Set <em>u</em> equal to the argument of the main function.</strong></p>
+<div class="paragraph">
+<p>The argument of \(\cos(x^2)\) is \(x^2\), so you set \(u\) equal to
+\(x^2\).</p>
+</div>
+</li>
+<li>
+<p><strong>Take the derivative of <em>u</em> with respect to <em>x</em>.</strong></p>
+<div class="stemblock">
+<div class="content">
+\[u = x^2\ \text{so}\ \frac{du}{dx} = 2x\]
+</div>
+</div>
+</li>
+<li>
+<p><strong>Solve for <em>dx</em>.</strong></p>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+ &amp; \frac{du}{dx} = \frac{2x}{1} \\[10pt]
+ &amp; du = 2xdx\ \text{(cross multiplication)} \\
+ &amp; \frac{du}{2x}\ = dx\ \text{(dviding both sides by 2x)}
+\end{aligned}\]
+</div>
+</div>
+</li>
+<li>
+<p><strong>Make substitutions</strong></p>
+<div class="paragraph">
+<p>In \(\int 2x \cos(x^2)dx\), u takes the place of \(x^2\) and
+\(\frac{du}{2x}\) takes the place of \(dx\). So now you&#8217;ve got
+\(\int 2x\cos u \frac{du}{2x}\). The two \(2x\)s cancel, giving you
+\(\int \cos\ u\ du\).</p>
+</div>
+</li>
+<li>
+<p><strong>Anitdifferentiate using the simple reverse rule.</strong></p>
+<div class="stemblock">
+<div class="content">
+\[\lim \cos\ u\ du = \sin\ u + C\]
+</div>
+</div>
+</li>
+<li>
+<p><strong>Substitute \(x^2\) back in for <em>u</em>, coming full circle.</strong></p>
+<div class="paragraph">
+<p><em>u</em> equals \(x^2\), so \(x^2\) goes in for the <em>u</em>:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim \cos\ u\ du - \sin(x^2) + C\]
+</div>
+</div>
+<div class="paragraph">
+<p>That&#8217;s it. So \(\lim\ 2x\ cos(x^2)dx = \sin(x^2) + C\)</p>
+</div>
+</li>
+</ol>
+</div>
+<div class="sect2">
+<h3 id="_examples">Examples</h3>
+<details>
+<summary class="title">Example 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{8. Evaluate the integral shown below:} \\[10pt]
+    &amp; \int 2x \sqrt{3x^2 + 5} \, dx \qquad \begin{aligned} u &amp;= 3x^2 + 5 \\ du &amp;= 6x \, dx \\ \frac{du}{6x} &amp;= dx \end{aligned} \\[15pt]
+    &amp; \int 2x \sqrt{u} \, \frac{du}{6x} \\[15pt]
+    &amp; \frac{1}{3} \int u^{1/2} \, du = \frac{1}{3} \cdot \frac{u^{3/2}}{3/2} + C \\[15pt]
+    &amp; = \frac{2}{9} u^{3/2} + C = \boxed{\frac{2}{9} (3x^2 + 5)^{3/2} + C}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_integration_by_parts">Integration by Parts</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\int u \, dv = uv - \int v \, du\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+You can check "calculus for dumies" chapter 16
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{ccc}
+\begin{array}{c} \text{diff.} \\ \downarrow \end{array} &amp;
+\begin{array}{|c|c|}
+\hline
+\rule[-1.2em]{0pt}{3em} \quad u \quad &amp; \quad v \quad \\ \hline
+\rule[-1.2em]{0pt}{3em} \quad du \quad &amp; \quad dv \quad \\ \hline
+\end{array} &amp;
+\begin{array}{c} \text{int.} \\ \uparrow \end{array}
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[u \cdot v - \int v \cdot du\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/integration-by-parts-7.png" alt="integration by parts 7">
+</div>
+</div>
+<div class="sect2">
+<h3 id="_picking_your_u_and_dv">Picking your "u" and "dv"</h3>
+<div class="paragraph">
+<p>You can use thsi mnemonic to pick you&#8217;re "u", anything that isn&#8217;t you&#8217;re "u" is
+you&#8217;re "dv".</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<caption class="title">Table 1. LIATE mnemonic</caption>
+<colgroup>
+<col style="width: 20%;">
+<col style="width: 40%;">
+<col style="width: 40%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Letter</th>
+<th class="tableblock halign-left valign-top">Function Type</th>
+<th class="tableblock halign-left valign-top">Example</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>L</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Logarithmic</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\ln(x), \log(x)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>I</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inverse Trigonometric</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\arctan(x), \arcsin(x)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>A</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Algebraic</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[x^2, 5x^2 + 3\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>T</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Trigonometric</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[\cos(x), \sin(x)\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>E</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Exponential</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="stemblock">
+<div class="content">
+\[e^x, 10^x\]
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="paragraph">
+<p>Integrate this</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\int \arctan(x)dx\]
+</div>
+</div>
+<div class="paragraph">
+<p>You see that there are no logarithmic functions in arctan(x)dx, but
+there is an inverse trigonometric function, arctan(x). So that’s your u.
+Everything else is your dv, namely, plain old dx.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[u = archtan(x) \\
+dv = dx\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+\text{diff.} \downarrow \quad
+\begin{array}{|wc{7em}|wc{7em}|}
+\hline
+\rule[-1.5em]{0pt}{4em} \arctan(x) &amp; x \\ \hline
+\rule[-1.5em]{0pt}{4em} \frac{1}{1+x^2} \, dx &amp; dx \\ \hline
+\end{array}
+\hspace{1em} \uparrow \text{int.}
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>showing integration from dx to x</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\int 1 dx \\
+\int x^0 dx =
+x^1 = x\]
+</div>
+</div>
+<div class="paragraph">
+<p>Do the seven thing.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/integration-by-parts-7.png" alt="integration by parts 7">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\int udv = uv - \int vdu \\
+\int \arctan(x)dx = x \arctan(x) - \int x \cdot \frac{1}{1 + x^2}dx\]
+</div>
+</div>
+<div class="paragraph">
+<p>Now you can finish this problem by integrating
+\(\int x \cdot \frac{1}{1 + x^2}dx\)with the substitution method, setting
+u=1+x2.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+You can check "calculus for dumies" chapter 15
+</td>
+</tr>
+</table>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+<em>u</em> in u=1+x2 has nothing to do with the integration-by-parts <em>u</em>.
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>solution</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\int \arctan(x)dx = x\arctan(x) - \frac{1}{2}\ln(1 + xx^2) + C.\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_basic_antiderivative_formulas">Basic Antiderivative Formulas</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int x^n\ dx = \frac{x^n+1}{n+1} + C\ (n \neq -1)\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \sin x\ dx = -\cos x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \frac{dx}{x} = \ln|x| + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int e^x\ dx = -\cos x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int a^x\ dx = \frac{1}{\ln a} a^x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \sin x\ dx = -\cos x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \cos x\ dx = \sin x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \sec^2 x\ dx = \tan x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \csc^2 x\ dx = - \cot x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \sec x \tan x\ dx = \sec x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \csc x \cot x\ dx = -\csc x + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \frac{dx}{x\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \frac{dx}{a^2 - x^2} = \frac{1}{a}\arctan \frac{x}{a} + C\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a} \text{arcsec} \frac{|x|}{a} + C\)</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+remember \(\frac{d}{dx}\) means derivative which is similar to the the
+      change in y divided by the change in x (\(\frac{\Delta Y}{\Delta X}\)
+      rise over run), except the scale of \(\frac{d}{dx}\) is instantaneous,
+      where the scale of \(\frac{\Delta Y}{\Delta X}\) is between two
+      distinct points on a line.
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_examples_2">Examples</h2>
+<div class="sectionbody">
+<details>
+<summary class="title">Example 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \int \frac{4x^5 + x^4 - 3x^2}{x^2} \, dx = \int (4x^3 + x^2 - 3) \, dx \\[15pt]
+    &amp; = \frac{4x^4}{4} + \frac{x^3}{3} - 3x + C \\[15pt]
+    &amp; = \boxed{x^4 + \frac{1}{3}x^3 - 3x + C}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
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diff --git a/Calculus/Intermediate-Value-Theorem.adoc b/Calculus/Intermediate-Value-Theorem.adoc
new file mode 100644 (file)
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+= Calculus | Intermediate Value Theorem
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Intermediate Value Theorem
+
+Suppose stem:[f] is continuous on stem:[[a,b]], and stem:[N], a number b/t
+stem:[f(a)] and stem:[f(b)], where stem:[f(a)\ \ne\ f(b)], then there exists a
+number stem:[c] in stem:[(a,b)] s.t stem:[f(c) = N].
+
+[.center]
+image::Intermediate-Value-Theorem.png[]
+
+== Links
+* https://www.youtube.com/watch?v=9wEHwFrUyOU[TOCT | Intermediate Value Theorem]
diff --git a/Calculus/Intermediate-Value-Theorem.html b/Calculus/Intermediate-Value-Theorem.html
new file mode 100644 (file)
index 0000000..fb5f5e8
--- /dev/null
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+.listingblock.terminal pre .command::before{content:attr(data-prompt);padding-right:.5em;color:inherit;opacity:.5}
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+pre.prettyprint .linenums{line-height:1.45;margin-left:2em}
+pre.prettyprint li{background:none;list-style-type:inherit;padding-left:0}
+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
+table.linenotable td.code{padding-left:.75em}
+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
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+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
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+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
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+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
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+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
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+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Intermediate Value Theorem</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_intermediate_value_theorem">Intermediate Value Theorem</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_intermediate_value_theorem">Intermediate Value Theorem</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Suppose \(f\) is continuous on \([a,b\)], and \(N\), a number b/t
+\(f(a)\) and \(f(b)\), where \(f(a)\ \ne\ f(b)\), then there exists a
+number \(c\) in \((a,b)\) s.t \(f(c) = N\).</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Intermediate-Value-Theorem.png" alt="Intermediate Value Theorem">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=9wEHwFrUyOU">TOCT | Intermediate Value Theorem</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
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diff --git a/Calculus/Limits.adoc b/Calculus/Limits.adoc
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+= Calculus | Limits
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Delta, Epsilon Definition of a Limit
+
+[stem]
+++++
+\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. } \forall x \in \mathbb{R} 
+\text{ with } 0 < |x - a| < \delta, \text{ we have } |f(x) - L| < \epsilon
+++++
+
+[.center]
+image::Epsilon-Delta-Definition-of-a-Limit.png[]
+
+
+
+== Limits to Memorize
+
+[.inline]
+- stem:[\lim_{x \to 0^+} \frac{1}{x} = \infty]
+
+[.inline]
+- stem:[\lim_{x \to 0^-} \frac{1}{x} = -\infty]
+
+[.inline]
+- stem:[\lim_{x \to \infty} \frac{1}{x} = 0]
+
+[.inline]
+- stem:[\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{x}{\sin x} = 1]
+
+[.inline]
+- stem:[\lim_{x \to 0} \frac{\cos x - 1}{x} = 0]
+
+[.inline]
+- stem:[\lim_{x \to \infty} (\frac{1 + 1}{x})^x = e \approx 2.718]
+
+NOTE: 1 is the placeholder for any constant (any number)
+
+== When Limits exist
+
+[.center]
+image::when-limits-exist.png[]
+
+.Left Side Limit
+[%collapsible]
+====
+When approaching the limit from the left 
+
+[stem]
+++++
+\lim_{x \to 3^{-}}
+++++
+
+you get stem:[1], even though there is a hole their. That is because it is only
+"as it is approaching 3".
+====
+
+.Right side limit
+[%collapsible]
+====
+When approaching the limit from the Right
+
+[stem]
+++++
+\lim_{x \to 3^{+}}
+++++
+
+you get stem:[1],
+====
+
+.Regular Limit
+[%collapsible]
+====
+When approaching the limit from both sides 
+
+[stem]
+++++
+\lim_{x \to 3}
+++++
+
+the answer is
+DNE (Does not exist). That is because you cannot have two answers for a limit
+====
+
+== Bounded Function and Limit Proof with Squeeze Theorom
+
+[stem]
+++++
+\begin{aligned}
+    & \text{Prove that if } g \text{ is bounded and } \lim_{x \to c} f(x) = 0, \text{ then } \\
+    & \lim_{x \to c} (f(x)g(x)) = 0. \\[10pt]
+    & \underline{\text{Proof}} \text{ Spse. } g \text{ is bounded by say } M \text{ and spse. } \\
+    & \lim_{x \to c} f(x) = 0. \qquad (|g(x)| \leq M) \\[10pt]
+    & \text{Then } 0 \leq |f(x)g(x)| = |f(x)| \cdot |g(x)| \leq |f(x)| \cdot M \\[10pt]
+    & \text{Then } \lim_{x \to c} 0 = 0, \quad \lim_{x \to c} |f(x)| \cdot M = |0| \cdot M = 0 \\[10pt]
+    & \text{Thus by squeeze, } \lim_{x \to c} |f(x)g(x)| = 0 \\[10pt]
+    & \text{So } \lim_{x \to c} f(x)g(x) = 0. \quad \square
+\end{aligned}
+++++
+
+NOTE: Whenever you're trying to show something is equal to zero if you take the
+      absolute value and show that's zero, then whatever is inside is also zero
+
+== Solving Limits
+
+=== Plug in the number
+
+Try to plug in the number into the limit. If you can solve it than that's
+you're answer. Always try this first.
+
+.example
+[%collapsible]
+====
+we plug 1 into the limit
+
+[stem]
+++++
+\lim_{x \to 1} (3x + 2) = 3 \cdot 1 + 2 = 5
+++++
+
+In this case we can solve so the solution is stem:[5]
+====
+
+If this doen't work then move on to the other steps
+
+=== Other Methods
+
+.Difference of squares
+[%collapsible]
+====
+
+Using the difference of squares
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \\
+    & \lim_{x \to 3} \frac{\cancel{(x - 3)}(x + 3)}{\cancel{x - 3}} \\
+    & \lim_{x \to 3} (x + 3) = 3 + 3 = 6
+\end{aligned}
+++++
+
+NOTE: You only get rid of the limit sign when you take the limit (plug in the
+      number).
+====
+
+.Difference of cubes
+[%collapsible]
+====
+
+[stem]
+++++
+\lim_{x \to 2} \frac{x^3 - 8}{x-2}
+++++
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & a^3 - b^3 = (a - b)(a^2 + ab + b^2) \\[10pt]
+    & \lim_{x \to 2} \frac{\cancel{(x - 2)}(x^2 + 2x + 4)}{\cancel{x - 2}} \\[10pt]
+    & \lim_{x \to 2} (x^2 + 2x + 4) = 2^2 + 2(2) + 4 \\
+    & \phantom{\lim_{x \to 2} (x^2 + 2x + 4)} = 4 + 4 + 4 \\
+    & \phantom{\lim_{x \to 2} (x^2 + 2x + 4)} = 12
+\end{aligned}
+++++
+====
+
+.Absolute Value 1
+[%collapsible]
+====
+
+This is actually a piecewise function
+
+NOTE: This piecewise function works is the absolute value is in the numerator or
+      denominator
+
+[stem]
+++++
+\begin{aligned}
+    & \text{Find } \lim_{x \to 3^-} \frac{|x - 3|}{x - 3} \\[10pt]
+    & \text{Sol. } |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}
+\end{aligned}
+++++
+
+Because stem:[x] is approaching from the left, stem:[x] must be less then three,
+so stem:[x-3 \lt 0]. Looking at the piecewise function we see that we can
+substute the top of the fraction with stem:[-(x-3)].
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    \lim_{x \to 3^-} \frac{-(x - 3)}{x - 3} & = \lim_{x \to 3^-} \frac{-\cancel{(x - 3)}}{\cancel{x - 3}} \\
+    & = \lim_{x \to 3^-} (-1) \\
+    & = -1
+\end{aligned}
+++++
+
+====
+
+.Absolute Value 2
+[%collapsible]
+====
+
+[stem]
+++++
+\lim_{x \to -6} \frac{2x + 12}{|x + 6|} \\[10pt]
+++++
+
+We can use the distributive property
+
+[stem]
+++++
+\lim_{x \to -6} \frac{2(x + 6)}{|x + 6|}
+++++
+
+The piecewise function
+
+[stem]
+++++
+\begin{aligned}
+    |x| &= \begin{cases} 
+        x, & \text{if } x \geq 0 \\ 
+        -x, & \text{if } x < 0 
+    \end{cases} \\[15pt]
+    |x + 6| &= \begin{cases} 
+        x + 6, & \text{if } x + 6 \geq 0 \implies x \geq -6 \\ 
+        -(x + 6), & \text{if } x + 6 < 0 \implies x < -6 
+    \end{cases}
+\end{aligned}
+++++
+
+To solve this we are going to approach from the left and right. First we are
+going to to approach from the left, so we'll use stem:[-(x + 6)]. 
+
+NOTE: Whenever you have an absoluate value and you're solving for a normal limit
+      you want to use one sided limits, you take the limit from the left first
+      and then the limit from the right. If the limits are the same then that'll
+      be the limit and the limit exists other with it does not exist (DNE).
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    \lim_{x \to -6^-} \frac{2(x + 6)}{-(x + 6)} &= \lim_{x \to -6^-} \frac{2\cancel{(x + 6)}}{-\cancel{(x + 6)}} \\[10pt]
+    &= \lim_{x \to -6^-} (-2) \\[10pt]
+    &= -2
+\end{aligned}
+++++
+
+Now we'll solve the right side. Looking at are peicewise function we can use
+stem:[x + 6] for our denominator because we are approaching form the right.
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    \lim_{x \to -6^+} \frac{2(x + 6)}{x + 6} &= \lim_{x \to -6^+} \frac{2\cancel{(x + 6)}}{\cancel{x + 6}} \\[10pt]
+    &= \lim_{x \to -6^+} (2) \\[10pt]
+    &= 2
+\end{aligned}
+++++
+
+The limit from the left is stem:[-2] and the limimt from the right is stem:[2]
+so the limit doesn not exist.
+
+[stem]
+++++
+\lim_{x \to -6} \frac{2x + 12}{|x + 6|} \\[10pt] = DNE
+++++
+
+====
+
+.Limit of rational function example 1
+[%collapsible]
+====
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \lim_{x \to 0} \frac{x^2 + 3x}{x} \\
+    & \lim_{x \to 0} \frac{\cancel{x}(x + 3)}{\cancel{x}} \\
+    & \lim_{x \to 0} (x + 3) = 0 + 3 \\
+    & \phantom{\lim_{x \to 0} (x + 3)} = 3
+\end{aligned}
+++++
+====
+
+.Limit of rational function example 2
+[%collapsible]
+====
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \lim_{x \to 2} \frac{2 - x}{x^2 - 4} \\
+    & \lim_{x \to 2} \frac{-1(x - 2)}{(x - 2)(x + 2)} \\
+    & \lim_{x \to 2} \frac{-1\cancel{(x - 2)}}{\cancel{(x - 2)}(x + 2)} \\
+    & \lim_{x \to 2} \frac{-1}{x + 2} = \frac{-1}{2 + 2} \\
+    & \phantom{\lim_{x \to 2} \frac{-1}{x + 2}} = -\frac{1}{4}
+\end{aligned}
+++++
+====
+
+.Limit by rationalizing
+[%collapsible]
+====
+
+We Multiple by the conjugate to get the radical on the bottom. 
+stem:[\sqrt{x + 3} - \sqrt{3}] is the difference of squares.
+
+* stem:[x + 3] is a
+* stem:[3] is b
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \text{Compute } \lim_{x \to 0} \frac{\sqrt{x+3} - \sqrt{3}}{x} \\[10pt]
+    & \text{Sol. } \lim_{x \to 0} \frac{\sqrt{x+3} - \sqrt{3}}{x} \cdot \left( \frac{\sqrt{x+3} + \sqrt{3}}{\sqrt{x+3} + \sqrt{3}} \right) \\[10pt]
+    & = \lim_{x \to 0} \frac{(x + 3) - 3}{x(\sqrt{x+3} + \sqrt{3})} \\
+    & = \lim_{x \to 0} \frac{\cancel{x}}{\cancel{x}(\sqrt{x+3} + \sqrt{3})} \\
+    & = \lim_{x \to 0} \frac{1}{\sqrt{x+3} + \sqrt{3}} = \frac{1}{\sqrt{0+3} + \sqrt{3}} \\
+    & = \frac{1}{2\sqrt{3}}
+\end{aligned}
+++++
+====
+
+.Limit by rationalizing 2
+[%collapsible]
+====
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \lim_{u \to 2} \frac{\sqrt{4u+1} - 3}{u - 2} \\[10pt]
+    & \text{Sol. } \lim_{u \to 2} \frac{\sqrt{4u+1} - 3}{u - 2} \cdot \left( \frac{\sqrt{4u+1} + 3}{\sqrt{4u+1} + 3} \right) \\[10pt]
+    & = \lim_{u \to 2} \frac{(4u + 1) - 9}{(u - 2)(\sqrt{4u+1} + 3)} \\
+    & = \lim_{u \to 2} \frac{4u - 8}{(u - 2)(\sqrt{4u+1} + 3)} \\
+    & = \lim_{u \to 2} \frac{4\cancel{(u - 2)}}{\cancel{(u - 2)}(\sqrt{4u+1} + 3)} \\
+    & = \lim_{u \to 2} \frac{4}{\sqrt{4u+1} + 3} = \frac{4}{\sqrt{4(2)+1} + 3} \\
+    & = \frac{4}{3 + 3} = \frac{4}{6} = \frac{2}{3}
+\end{aligned}
+++++
+====
+
+.Limit of a piecewise function
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & f(x) = \begin{cases} x^2 - 4x + 6, & \text{if } x < 2 \\ -x^2 + 4x - 2, & \text{if } x \geq 2 \end{cases} \\[15pt]
+    & \text{Find } \lim_{x \to 2} f(x)
+\end{aligned}
+++++
+
+We solve for the left and right.
+
+[stem]
+++++
+\begin{aligned}
+    & \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 - 4x + 6) = 2^2 - 4(2) + 6 = 2 \\
+    & \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (-x^2 + 4x - 2) = -(2^2) + 4(2) - 2 = 2 \\
+    & \therefore \lim_{x \to 2} f(x) = 2
+\end{aligned}
+++++
+
+The left and right are equal so we get a solution.
+
+====
+
+.Limit of a Difference Quotient with Delta x
+[%collapsible]
+====
+[stem]
+++++
+\lim_{\Delta x \to 0^+} \frac{\frac{2}{x + \Delta x} - \frac{2}{x}}{\Delta x}
+++++
+
+Get the LCD 
+
+[stem]
+++++
+\begin{aligned}
+    & \frac{x \cdot 2}{x \cdot (x + \Delta x)} - \frac{2 \cdot (x + \Delta x)}{x \cdot (x + \Delta x)} \\[15pt]
+    & = \frac{2x - 2(x + \Delta x)}{x(x + \Delta x)}
+\end{aligned}
+++++
+
+Distribute.
+
+NOTE: Dividing fractions is the same as multiplying by the reciprical
+
+[stem]
+++++
+\require{cancel}
+\begin{aligned}
+    & \lim_{\Delta x \to 0^+} \frac{\frac{2x - 2(x + \Delta x)}{x(x + \Delta x)}}{\Delta x} \\
+    & = \lim_{\Delta x \to 0^+} \frac{\frac{\cancel{2x - 2x} - 2\Delta x}{x(x + \Delta x)}}{\Delta x} \\
+    & = \lim_{\Delta x \to 0^+} \frac{\frac{-2\Delta x}{x(x + \Delta x)}}{\Delta x} \\
+    & = \lim_{\Delta x \to 0^+} \frac{-2 \cancel{\Delta x}}{(x + \Delta x)x} \cdot \frac{1}{\cancel{\Delta x}} \\
+    & = \lim_{\Delta x \to 0^+} \frac{-2}{(x + \Delta x)x} \\
+    & = \frac{-2}{(x + 0)x} = \frac{-2}{x^2}
+\end{aligned}
+++++
+
+====
+
+.Squeeze Theorem Example 1
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Find } \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) \\[10pt]
+    & -1 \leq \cos\left(\frac{1}{x}\right) \leq 1 \\
+    & -x^2 \leq x^2 \cos\left(\frac{1}{x}\right) \leq x^2 \\[10pt]
+    & \lim_{x \to 0} (-x^2) = 0, \quad \lim_{x \to 0} x^2 = 0. \text{ So by the} \\
+    & \text{squeeze thm, } \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) = 0.
+\end{aligned}
+++++
+====
+
+.Squeeze Theorem Example 2
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Prove } \lim_{x \to 0} x^4 \cos\left(\frac{17}{x}\right) = 0. \\[10pt]
+    & \underline{\text{Proof}} \qquad (-1 \leq \cos x \leq 1) \\[10pt]
+    & -x^4 \leq x^4 \cos\left(\frac{17}{x}\right) \leq x^4 \\[10pt]
+    & \lim_{x \to 0} (-x^4) = 0, \quad \lim_{x \to 0} x^4 = 0 \\[10pt]
+    & \text{Thus by the squeeze thm,} \\
+    & \lim_{x \to 0} x^4 \cos\left(\frac{17}{x}\right) = 0
+\end{aligned}
+++++
+====
+
+.Squeeze Theorem Example 3
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Prove } \lim_{x \to 1} (x-1)^4 \sin\left(\frac{1}{x-1}\right) = 0 \\[10pt]
+    & \underline{\text{Proof}} \\[10pt]
+    & -1 \leq \sin\left(\frac{1}{x-1}\right) \leq 1 \\
+    & -(x-1)^4 \leq (x-1)^4 \sin\left(\frac{1}{x-1}\right) \leq (x-1)^4 \\[10pt]
+    & \lim_{x \to 1} -(x-1)^4 = -(1-1)^4 = -0^4 = 0 \\
+    & \lim_{x \to 1} (x-1)^4 = (1-1)^4 = 0^4 = 0
+\end{aligned}
+++++
+====
+
+.Limit sandwich
+[%collapsible]
+====
+
+
+[stem]
+++++
+\begin{aligned}
+    & \text {Find } \lim_{x \to 0}(x \sin \frac{1}{x})
+\end{aligned}
+++++
+
+Because the range of the sine funciton is from negaitve 1 to positive 1,
+whenever you multiply a number by the sine of anything, the result either stays
+the same distance from zero or gets closer to zero. Thus 
+stem:[x \sin \frac{1}{x}] will never get above stem:[|x|] or below stem:[-|x|].
+
+[.center]
+image::limit-sandwich.png[]
+
+[stem]
+++++
+\begin{aligned}
+    & f(x) \ge g(x) \ge h(x) \\[10pt]
+    & \text{Because } \lim_{x \to 0} f(x) = \lim_{x \to 0} h(x) = 0 \\[10pt]
+    & \text{it follows that } \\[10pt]
+    & \lim_{x \to 0} g(x) = 0
+\end{aligned}
+++++
+
+====
+
+.Trig Example 1
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Find the limit} \\
+    & \lim_{x \to 0} \frac{\sin(3x)}{2x} \qquad \boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1} \\[15pt]
+    & \underline{\text{Sol}} \quad \frac{3}{1} \cdot \frac{1}{2} \left( \lim_{x \to 0} \frac{\sin(3x)}{3x} \right) = \frac{3}{2} \cdot 1 = \frac{3}{2}
+\end{aligned}
+++++
+====
+
+.Trig Example 2
+[%collapsible]
+====
+[stem]
+++++
+\text{Find } \lim_{x \to 0} \frac{\sin(4x)}{x} \qquad \boxed{\begin{aligned} \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \\ \text{match!} \end{aligned}} \\[15pt]
+++++
+
+You can solve this problem in two different ways
+
+.way 1
+[stem]
+++++
+\begin{aligned}
+    & \underline{\text{Sol}} \quad 4 \left( \lim_{x \to 0} \frac{\sin(4x)}{4x} \right) \\[10pt]
+    & = 4 \cdot 1 = \boxed{4}
+\end{aligned}
+++++
+
+.way 2
+[stem]
+++++
+\begin{aligned}
+    & 4 \cdot \lim_{x \to 0} \frac{\sin(4x)}{4x} \qquad , \quad \begin{aligned} &\text{let } t = 4x \\ &\text{Note as } x \to 0, t \to 0 \end{aligned} \\[15pt]
+    & 4 \left( \lim_{t \to 0} \frac{\sin t}{t} \right) = 4 \cdot 1 = \boxed{4}
+\end{aligned}
+++++
+====
+
+.Trig Example 3
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \text{Find } \lim_{x \to 0} \frac{\sin(3x)}{\sin(2x)} \qquad \boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1} \\[15pt]
+    & \text{Sol} \\
+    & \lim_{x \to 0} \frac{3x \cdot \frac{\sin(3x)}{3x}}{2x \cdot \frac{\sin(2x)}{2x}} = \frac{3}{2} \lim_{x \to 0} \frac{\left( \frac{\sin(3x)}{3x} \right)}{\left( \frac{\sin(2x)}{2x} \right)} = \frac{3}{2} \cdot \frac{1}{1} = \boxed{\frac{3}{2}}
+\end{aligned}
+++++
+====
+
+.Trig Example 7
+[%collapsible]
+====
+[stem]
+++++
+\begin{aligned}
+    & \lim_{x \to 1} \frac{\sin(x^2 - 1)}{x - 1} \cdot \frac{x + 1}{x + 1} \\[15pt]
+    & \lim_{x \to 1} \frac{(x + 1) \sin(x^2 - 1)}{x^2 - 1} \\[10pt]
+    & \lim_{x \to 1} (x + 1) \cdot \frac{\sin(x^2 - 1)}{x^2 - 1} \\[10pt]
+    & \left( \lim_{x \to 1} (x + 1) \right) \left( \lim_{x \to 1} \frac{\sin(x^2 - 1)}{x^2 - 1} \right) \\[10pt]
+    & = (1 + 1) \cdot 1 = 2 \cdot 1 = 2
+\end{aligned}
+++++
+====
+
+== Links
+
+* https://www.youtube.com/watch?v=kfF40MiS7zA&t=292s[3Blue1Brown Limits | Essence of calculus]
diff --git a/Calculus/Limits.html b/Calculus/Limits.html
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+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Limits</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_delta_epsilon_definition_of_a_limit">Delta, Epsilon Definition of a Limit</a></li>
+<li><a href="#_limits_to_memorize">Limits to Memorize</a></li>
+<li><a href="#_when_limits_exist">When Limits exist</a></li>
+<li><a href="#_bounded_function_and_limit_proof_with_squeeze_theorom">Bounded Function and Limit Proof with Squeeze Theorom</a></li>
+<li><a href="#_solving_limits">Solving Limits</a>
+<ul class="sectlevel2">
+<li><a href="#_plug_in_the_number">Plug in the number</a></li>
+<li><a href="#_other_methods">Other Methods</a></li>
+</ul>
+</li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_delta_epsilon_definition_of_a_limit">Delta, Epsilon Definition of a Limit</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\forall \epsilon &gt; 0, \exists \delta &gt; 0 \text{ s.t. } \forall x \in \mathbb{R}
+\text{ with } 0 &lt; |x - a| &lt; \delta, \text{ we have } |f(x) - L| &lt; \epsilon\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Epsilon-Delta-Definition-of-a-Limit.png" alt="Epsilon Delta Definition of a Limit">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_limits_to_memorize">Limits to Memorize</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to 0^+} \frac{1}{x} = \infty\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to 0^-} \frac{1}{x} = -\infty\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to \infty} \frac{1}{x} = 0\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{x}{\sin x} = 1\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to 0} \frac{\cos x - 1}{x} = 0\)</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\(\lim_{x \to \infty} (\frac{1 + 1}{x})^x = e \approx 2.718\)</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+1 is the placeholder for any constant (any number)
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_when_limits_exist">When Limits exist</h2>
+<div class="sectionbody">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/when-limits-exist.png" alt="when limits exist">
+</div>
+</div>
+<details>
+<summary class="title">Left Side Limit</summary>
+<div class="content">
+<div class="paragraph">
+<p>When approaching the limit from the left</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to 3^{-}}\]
+</div>
+</div>
+<div class="paragraph">
+<p>you get \(1\), even though there is a hole their. That is because it is only
+"as it is approaching 3".</p>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Right side limit</summary>
+<div class="content">
+<div class="paragraph">
+<p>When approaching the limit from the Right</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to 3^{+}}\]
+</div>
+</div>
+<div class="paragraph">
+<p>you get \(1\),</p>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Regular Limit</summary>
+<div class="content">
+<div class="paragraph">
+<p>When approaching the limit from both sides</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to 3}\]
+</div>
+</div>
+<div class="paragraph">
+<p>the answer is
+DNE (Does not exist). That is because you cannot have two answers for a limit</p>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_bounded_function_and_limit_proof_with_squeeze_theorom">Bounded Function and Limit Proof with Squeeze Theorom</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Prove that if } g \text{ is bounded and } \lim_{x \to c} f(x) = 0, \text{ then } \\
+    &amp; \lim_{x \to c} (f(x)g(x)) = 0. \\[10pt]
+    &amp; \underline{\text{Proof}} \text{ Spse. } g \text{ is bounded by say } M \text{ and spse. } \\
+    &amp; \lim_{x \to c} f(x) = 0. \qquad (|g(x)| \leq M) \\[10pt]
+    &amp; \text{Then } 0 \leq |f(x)g(x)| = |f(x)| \cdot |g(x)| \leq |f(x)| \cdot M \\[10pt]
+    &amp; \text{Then } \lim_{x \to c} 0 = 0, \quad \lim_{x \to c} |f(x)| \cdot M = |0| \cdot M = 0 \\[10pt]
+    &amp; \text{Thus by squeeze, } \lim_{x \to c} |f(x)g(x)| = 0 \\[10pt]
+    &amp; \text{So } \lim_{x \to c} f(x)g(x) = 0. \quad \square
+\end{aligned}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Whenever you&#8217;re trying to show something is equal to zero if you take the
+      absolute value and show that&#8217;s zero, then whatever is inside is also zero
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_solving_limits">Solving Limits</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_plug_in_the_number">Plug in the number</h3>
+<div class="paragraph">
+<p>Try to plug in the number into the limit. If you can solve it than that&#8217;s
+you&#8217;re answer. Always try this first.</p>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="paragraph">
+<p>we plug 1 into the limit</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to 1} (3x + 2) = 3 \cdot 1 + 2 = 5\]
+</div>
+</div>
+<div class="paragraph">
+<p>In this case we can solve so the solution is \(5\)</p>
+</div>
+</div>
+</details>
+<div class="paragraph">
+<p>If this doen&#8217;t work then move on to the other steps</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_other_methods">Other Methods</h3>
+<details>
+<summary class="title">Difference of squares</summary>
+<div class="content">
+<div class="paragraph">
+<p>Using the difference of squares</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \\
+    &amp; \lim_{x \to 3} \frac{\cancel{(x - 3)}(x + 3)}{\cancel{x - 3}} \\
+    &amp; \lim_{x \to 3} (x + 3) = 3 + 3 = 6
+\end{aligned}\]
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+You only get rid of the limit sign when you take the limit (plug in the
+      number).
+</td>
+</tr>
+</table>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Difference of cubes</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to 2} \frac{x^3 - 8}{x-2}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; a^3 - b^3 = (a - b)(a^2 + ab + b^2) \\[10pt]
+    &amp; \lim_{x \to 2} \frac{\cancel{(x - 2)}(x^2 + 2x + 4)}{\cancel{x - 2}} \\[10pt]
+    &amp; \lim_{x \to 2} (x^2 + 2x + 4) = 2^2 + 2(2) + 4 \\
+    &amp; \phantom{\lim_{x \to 2} (x^2 + 2x + 4)} = 4 + 4 + 4 \\
+    &amp; \phantom{\lim_{x \to 2} (x^2 + 2x + 4)} = 12
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Absolute Value 1</summary>
+<div class="content">
+<div class="paragraph">
+<p>This is actually a piecewise function</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+This piecewise function works is the absolute value is in the numerator or
+      denominator
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Find } \lim_{x \to 3^-} \frac{|x - 3|}{x - 3} \\[10pt]
+    &amp; \text{Sol. } |x| = \begin{cases} x, &amp; \text{if } x \geq 0 \\ -x, &amp; \text{if } x &lt; 0 \end{cases}
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Because \(x\) is approaching from the left, \(x\) must be less then three,
+so \(x-3 \lt 0\). Looking at the piecewise function we see that we can
+substute the top of the fraction with \(-(x-3)\).</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    \lim_{x \to 3^-} \frac{-(x - 3)}{x - 3} &amp; = \lim_{x \to 3^-} \frac{-\cancel{(x - 3)}}{\cancel{x - 3}} \\
+    &amp; = \lim_{x \to 3^-} (-1) \\
+    &amp; = -1
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Absolute Value 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to -6} \frac{2x + 12}{|x + 6|} \\[10pt]\]
+</div>
+</div>
+<div class="paragraph">
+<p>We can use the distributive property</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to -6} \frac{2(x + 6)}{|x + 6|}\]
+</div>
+</div>
+<div class="paragraph">
+<p>The piecewise function</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    |x| &amp;= \begin{cases}
+        x, &amp; \text{if } x \geq 0 \\
+        -x, &amp; \text{if } x &lt; 0
+    \end{cases} \\[15pt]
+    |x + 6| &amp;= \begin{cases}
+        x + 6, &amp; \text{if } x + 6 \geq 0 \implies x \geq -6 \\
+        -(x + 6), &amp; \text{if } x + 6 &lt; 0 \implies x &lt; -6
+    \end{cases}
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>To solve this we are going to approach from the left and right. First we are
+going to to approach from the left, so we&#8217;ll use \(-(x + 6)\).</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Whenever you have an absoluate value and you&#8217;re solving for a normal limit
+      you want to use one sided limits, you take the limit from the left first
+      and then the limit from the right. If the limits are the same then that&#8217;ll
+      be the limit and the limit exists other with it does not exist (DNE).
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    \lim_{x \to -6^-} \frac{2(x + 6)}{-(x + 6)} &amp;= \lim_{x \to -6^-} \frac{2\cancel{(x + 6)}}{-\cancel{(x + 6)}} \\[10pt]
+    &amp;= \lim_{x \to -6^-} (-2) \\[10pt]
+    &amp;= -2
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Now we&#8217;ll solve the right side. Looking at are peicewise function we can use
+\(x + 6\) for our denominator because we are approaching form the right.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    \lim_{x \to -6^+} \frac{2(x + 6)}{x + 6} &amp;= \lim_{x \to -6^+} \frac{2\cancel{(x + 6)}}{\cancel{x + 6}} \\[10pt]
+    &amp;= \lim_{x \to -6^+} (2) \\[10pt]
+    &amp;= 2
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>The limit from the left is \(-2\) and the limimt from the right is \(2\)
+so the limit doesn not exist.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\lim_{x \to -6} \frac{2x + 12}{|x + 6|} \\[10pt] = DNE\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit of rational function example 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \lim_{x \to 0} \frac{x^2 + 3x}{x} \\
+    &amp; \lim_{x \to 0} \frac{\cancel{x}(x + 3)}{\cancel{x}} \\
+    &amp; \lim_{x \to 0} (x + 3) = 0 + 3 \\
+    &amp; \phantom{\lim_{x \to 0} (x + 3)} = 3
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit of rational function example 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \lim_{x \to 2} \frac{2 - x}{x^2 - 4} \\
+    &amp; \lim_{x \to 2} \frac{-1(x - 2)}{(x - 2)(x + 2)} \\
+    &amp; \lim_{x \to 2} \frac{-1\cancel{(x - 2)}}{\cancel{(x - 2)}(x + 2)} \\
+    &amp; \lim_{x \to 2} \frac{-1}{x + 2} = \frac{-1}{2 + 2} \\
+    &amp; \phantom{\lim_{x \to 2} \frac{-1}{x + 2}} = -\frac{1}{4}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit by rationalizing</summary>
+<div class="content">
+<div class="paragraph">
+<p>We Multiple by the conjugate to get the radical on the bottom.
+\(\sqrt{x + 3} - \sqrt{3}\) is the difference of squares.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>\(x + 3\) is a</p>
+</li>
+<li>
+<p>\(3\) is b</p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \text{Compute } \lim_{x \to 0} \frac{\sqrt{x+3} - \sqrt{3}}{x} \\[10pt]
+    &amp; \text{Sol. } \lim_{x \to 0} \frac{\sqrt{x+3} - \sqrt{3}}{x} \cdot \left( \frac{\sqrt{x+3} + \sqrt{3}}{\sqrt{x+3} + \sqrt{3}} \right) \\[10pt]
+    &amp; = \lim_{x \to 0} \frac{(x + 3) - 3}{x(\sqrt{x+3} + \sqrt{3})} \\
+    &amp; = \lim_{x \to 0} \frac{\cancel{x}}{\cancel{x}(\sqrt{x+3} + \sqrt{3})} \\
+    &amp; = \lim_{x \to 0} \frac{1}{\sqrt{x+3} + \sqrt{3}} = \frac{1}{\sqrt{0+3} + \sqrt{3}} \\
+    &amp; = \frac{1}{2\sqrt{3}}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit by rationalizing 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \lim_{u \to 2} \frac{\sqrt{4u+1} - 3}{u - 2} \\[10pt]
+    &amp; \text{Sol. } \lim_{u \to 2} \frac{\sqrt{4u+1} - 3}{u - 2} \cdot \left( \frac{\sqrt{4u+1} + 3}{\sqrt{4u+1} + 3} \right) \\[10pt]
+    &amp; = \lim_{u \to 2} \frac{(4u + 1) - 9}{(u - 2)(\sqrt{4u+1} + 3)} \\
+    &amp; = \lim_{u \to 2} \frac{4u - 8}{(u - 2)(\sqrt{4u+1} + 3)} \\
+    &amp; = \lim_{u \to 2} \frac{4\cancel{(u - 2)}}{\cancel{(u - 2)}(\sqrt{4u+1} + 3)} \\
+    &amp; = \lim_{u \to 2} \frac{4}{\sqrt{4u+1} + 3} = \frac{4}{\sqrt{4(2)+1} + 3} \\
+    &amp; = \frac{4}{3 + 3} = \frac{4}{6} = \frac{2}{3}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit of a piecewise function</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; f(x) = \begin{cases} x^2 - 4x + 6, &amp; \text{if } x &lt; 2 \\ -x^2 + 4x - 2, &amp; \text{if } x \geq 2 \end{cases} \\[15pt]
+    &amp; \text{Find } \lim_{x \to 2} f(x)
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>We solve for the left and right.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 - 4x + 6) = 2^2 - 4(2) + 6 = 2 \\
+    &amp; \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (-x^2 + 4x - 2) = -(2^2) + 4(2) - 2 = 2 \\
+    &amp; \therefore \lim_{x \to 2} f(x) = 2
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>The left and right are equal so we get a solution.</p>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit of a Difference Quotient with Delta x</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\lim_{\Delta x \to 0^+} \frac{\frac{2}{x + \Delta x} - \frac{2}{x}}{\Delta x}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Get the LCD</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \frac{x \cdot 2}{x \cdot (x + \Delta x)} - \frac{2 \cdot (x + \Delta x)}{x \cdot (x + \Delta x)} \\[15pt]
+    &amp; = \frac{2x - 2(x + \Delta x)}{x(x + \Delta x)}
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Distribute.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Dividing fractions is the same as multiplying by the reciprical
+</td>
+</tr>
+</table>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\require{cancel}
+\begin{aligned}
+    &amp; \lim_{\Delta x \to 0^+} \frac{\frac{2x - 2(x + \Delta x)}{x(x + \Delta x)}}{\Delta x} \\
+    &amp; = \lim_{\Delta x \to 0^+} \frac{\frac{\cancel{2x - 2x} - 2\Delta x}{x(x + \Delta x)}}{\Delta x} \\
+    &amp; = \lim_{\Delta x \to 0^+} \frac{\frac{-2\Delta x}{x(x + \Delta x)}}{\Delta x} \\
+    &amp; = \lim_{\Delta x \to 0^+} \frac{-2 \cancel{\Delta x}}{(x + \Delta x)x} \cdot \frac{1}{\cancel{\Delta x}} \\
+    &amp; = \lim_{\Delta x \to 0^+} \frac{-2}{(x + \Delta x)x} \\
+    &amp; = \frac{-2}{(x + 0)x} = \frac{-2}{x^2}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Squeeze Theorem Example 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Find } \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) \\[10pt]
+    &amp; -1 \leq \cos\left(\frac{1}{x}\right) \leq 1 \\
+    &amp; -x^2 \leq x^2 \cos\left(\frac{1}{x}\right) \leq x^2 \\[10pt]
+    &amp; \lim_{x \to 0} (-x^2) = 0, \quad \lim_{x \to 0} x^2 = 0. \text{ So by the} \\
+    &amp; \text{squeeze thm, } \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) = 0.
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Squeeze Theorem Example 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Prove } \lim_{x \to 0} x^4 \cos\left(\frac{17}{x}\right) = 0. \\[10pt]
+    &amp; \underline{\text{Proof}} \qquad (-1 \leq \cos x \leq 1) \\[10pt]
+    &amp; -x^4 \leq x^4 \cos\left(\frac{17}{x}\right) \leq x^4 \\[10pt]
+    &amp; \lim_{x \to 0} (-x^4) = 0, \quad \lim_{x \to 0} x^4 = 0 \\[10pt]
+    &amp; \text{Thus by the squeeze thm,} \\
+    &amp; \lim_{x \to 0} x^4 \cos\left(\frac{17}{x}\right) = 0
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Squeeze Theorem Example 3</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Prove } \lim_{x \to 1} (x-1)^4 \sin\left(\frac{1}{x-1}\right) = 0 \\[10pt]
+    &amp; \underline{\text{Proof}} \\[10pt]
+    &amp; -1 \leq \sin\left(\frac{1}{x-1}\right) \leq 1 \\
+    &amp; -(x-1)^4 \leq (x-1)^4 \sin\left(\frac{1}{x-1}\right) \leq (x-1)^4 \\[10pt]
+    &amp; \lim_{x \to 1} -(x-1)^4 = -(1-1)^4 = -0^4 = 0 \\
+    &amp; \lim_{x \to 1} (x-1)^4 = (1-1)^4 = 0^4 = 0
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Limit sandwich</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text {Find } \lim_{x \to 0}(x \sin \frac{1}{x})
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Because the range of the sine funciton is from negaitve 1 to positive 1,
+whenever you multiply a number by the sine of anything, the result either stays
+the same distance from zero or gets closer to zero. Thus
+\(x \sin \frac{1}{x}\) will never get above \(|x|\) or below \(-|x|\).</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/limit-sandwich.png" alt="limit sandwich">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; f(x) \ge g(x) \ge h(x) \\[10pt]
+    &amp; \text{Because } \lim_{x \to 0} f(x) = \lim_{x \to 0} h(x) = 0 \\[10pt]
+    &amp; \text{it follows that } \\[10pt]
+    &amp; \lim_{x \to 0} g(x) = 0
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Trig Example 1</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Find the limit} \\
+    &amp; \lim_{x \to 0} \frac{\sin(3x)}{2x} \qquad \boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1} \\[15pt]
+    &amp; \underline{\text{Sol}} \quad \frac{3}{1} \cdot \frac{1}{2} \left( \lim_{x \to 0} \frac{\sin(3x)}{3x} \right) = \frac{3}{2} \cdot 1 = \frac{3}{2}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Trig Example 2</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\text{Find } \lim_{x \to 0} \frac{\sin(4x)}{x} \qquad \boxed{\begin{aligned} \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \\ \text{match!} \end{aligned}} \\[15pt]\]
+</div>
+</div>
+<div class="paragraph">
+<p>You can solve this problem in two different ways</p>
+</div>
+<div class="stemblock">
+<div class="title">way 1</div>
+<div class="content">
+\[\begin{aligned}
+    &amp; \underline{\text{Sol}} \quad 4 \left( \lim_{x \to 0} \frac{\sin(4x)}{4x} \right) \\[10pt]
+    &amp; = 4 \cdot 1 = \boxed{4}
+\end{aligned}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">way 2</div>
+<div class="content">
+\[\begin{aligned}
+    &amp; 4 \cdot \lim_{x \to 0} \frac{\sin(4x)}{4x} \qquad , \quad \begin{aligned} &amp;\text{let } t = 4x \\ &amp;\text{Note as } x \to 0, t \to 0 \end{aligned} \\[15pt]
+    &amp; 4 \left( \lim_{t \to 0} \frac{\sin t}{t} \right) = 4 \cdot 1 = \boxed{4}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Trig Example 3</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \text{Find } \lim_{x \to 0} \frac{\sin(3x)}{\sin(2x)} \qquad \boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1} \\[15pt]
+    &amp; \text{Sol} \\
+    &amp; \lim_{x \to 0} \frac{3x \cdot \frac{\sin(3x)}{3x}}{2x \cdot \frac{\sin(2x)}{2x}} = \frac{3}{2} \lim_{x \to 0} \frac{\left( \frac{\sin(3x)}{3x} \right)}{\left( \frac{\sin(2x)}{2x} \right)} = \frac{3}{2} \cdot \frac{1}{1} = \boxed{\frac{3}{2}}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Trig Example 7</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    &amp; \lim_{x \to 1} \frac{\sin(x^2 - 1)}{x - 1} \cdot \frac{x + 1}{x + 1} \\[15pt]
+    &amp; \lim_{x \to 1} \frac{(x + 1) \sin(x^2 - 1)}{x^2 - 1} \\[10pt]
+    &amp; \lim_{x \to 1} (x + 1) \cdot \frac{\sin(x^2 - 1)}{x^2 - 1} \\[10pt]
+    &amp; \left( \lim_{x \to 1} (x + 1) \right) \left( \lim_{x \to 1} \frac{\sin(x^2 - 1)}{x^2 - 1} \right) \\[10pt]
+    &amp; = (1 + 1) \cdot 1 = 2 \cdot 1 = 2
+\end{aligned}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=kfF40MiS7zA&amp;t=292s">3Blue1Brown Limits | Essence of calculus</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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\ No newline at end of file
diff --git a/Calculus/Mean-Value-Theorem.adoc b/Calculus/Mean-Value-Theorem.adoc
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+= Calculus | Mean Value Theorem
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+[.center]
+image::Mean-Value-Theorem.png[]
+
+== Formal Definition
+
+If f is continuous on the closed interval [a, b] and differentiable on the open
+interval (a,b) then tehre exists at least on number c in (a,b) such that
+
+[stem]
+++++
+f\prime(c) = \frac{f(b) - f(a)}{b - a}
+++++
+
+NOTE: you can look at page 168 of calculus for dummies for a good explaination
+
+== Plain-English Definition
+
+There must be at least one point between a and b where the slope of the tangent
+is the same as the slope of the secant line from a to be.
+
+Imagine that you grab the secant line connecting (a, f(a)) and (b, f(b)), and
+then you slide it up, keeping it parallel to the original secant line. Can you
+see that the two points of intersection between this sliding line and the
+function -- the two points that begin at (a, f(a)) and (b, f(b)) -- will
+gradually get closer and closer to each other until they come together at (c,
+f(c))? If you raise the line any further, you break away from the function
+entirely. At this last point of intersection, (c, f(c)), the sliding line
+touches the function at a single point and is thus tangent to the function
+there, and it has the same slope as the original secant line.
+
+[.center]
+image::mean-value-theorem-visual.png[]
+
diff --git a/Calculus/Mean-Value-Theorem.html b/Calculus/Mean-Value-Theorem.html
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+<title>Calculus | Mean Value Theorem</title>
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+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
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+    window.parent.postMessage({
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+        scrollPosition: scrollPosition
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+window.addEventListener("message", (e) => {
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Mean Value Theorem</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_formal_definition">Formal Definition</a></li>
+<li><a href="#_plain_english_definition">Plain-English Definition</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Mean-Value-Theorem.png" alt="Mean Value Theorem">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_formal_definition">Formal Definition</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>If f is continuous on the closed interval [a, b] and differentiable on the open
+interval (a,b) then tehre exists at least on number c in (a,b) such that</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\$f\prime(c) = \frac{f(b) - f(a)}{b - a}\$
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+you can look at page 168 of calculus for dummies for a good explaination
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_plain_english_definition">Plain-English Definition</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>There must be at least one point between a and b where the slope of the tangent
+is the same as the slope of the secant line from a to be.</p>
+</div>
+<div class="paragraph">
+<p>Imagine that you grab the secant line connecting (a, f(a)) and (b, f(b)), and
+then you slide it up, keeping it parallel to the original secant line. Can you
+see that the two points of intersection between this sliding line and the
+function&#8201;&#8212;&#8201;the two points that begin at (a, f(a)) and (b, f(b))&#8201;&#8212;&#8201;will
+gradually get closer and closer to each other until they come together at (c,
+f(c))? If you raise the line any further, you break away from the function
+entirely. At this last point of intersection, (c, f(c)), the sliding line
+touches the function at a single point and is thus tangent to the function
+there, and it has the same slope as the original secant line.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/mean-value-theorem-visual.png" alt="mean value theorem visual">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+    inlineMath: [["\\(", "\\)"]],
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+</html>
\ No newline at end of file
diff --git a/Calculus/Spherical-Coordinates.adoc b/Calculus/Spherical-Coordinates.adoc
new file mode 100644 (file)
index 0000000..2eb91c5
--- /dev/null
+++ b/Calculus/Spherical-Coordinates.adoc
@@ -0,0 +1,17 @@
+= Calculus | Spherical Coordinates
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Links
+
+* https://www.youtube.com/watch?v=jBcNOkwiS6k[YouTube | inegrating spherical coordinates]
+* https://www.youtube.com/watch?v=w79nZGzWMyI[YouTube | deriving spherical coordinates]
+* https://mathworld.wolfram.com/SphericalCoordinates.html[Spherical coordinates]
+* https://www.youtube.com/watch?v=R50_LFbJ3aM
+* https://www.youtube.com/watch?v=Ex_g2w4E5lQ[spherical coordinates animation]
diff --git a/Calculus/Spherical-Coordinates.html b/Calculus/Spherical-Coordinates.html
new file mode 100644 (file)
index 0000000..11b8a72
--- /dev/null
+++ b/Calculus/Spherical-Coordinates.html
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+html{font-size:80%}
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+abbr[title]{border-bottom:1px dotted}
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+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
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+body.book #header{text-align:center}
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+body.book #header .details{border:0!important;display:block;padding:0!important}
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+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
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+:root {
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+
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+    background-attachment: fixed;
+    background-size: cover;
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+    color: #ebdbb2;
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+    /*opacity: .9;*/
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+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
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+h4 {
+  color: var(--gruvbox-yellow) !important;
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+h5 {
+  color: #83a598 !important;
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+    white-space: pre;
+    overflow-x: scroll !important;
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+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
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+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
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+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
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+}
+h3 ~ *:not(h2):not(h3) {
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+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
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+    /*opacity: .9;*/
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+  color: var(--gruvbox-white) !important;
+}
+
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+    text-align: center;
+}
+
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+    margin-left: auto !important;
+    margin-right: auto !important;
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+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
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+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
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+details[open] > summary::before {
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+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
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+    max-width: 40%;
+    margin: auto;
+}
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+.center {
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+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
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+
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+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
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+}
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+}
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+}
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+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
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+li.started .checkbox {
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+li.completed .checkbox {
+  color: var(--gruvbox-green);
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+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
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+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
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+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
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+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
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+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
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+});
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+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
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+function scrollStep() {
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+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
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+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
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+        scrollKeys[e.key] = true;
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+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
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+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
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+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
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+});
+
+
+
+window.onload = function () {
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+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
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+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Calculus | Spherical Coordinates</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=jBcNOkwiS6k">YouTube | inegrating spherical coordinates</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=w79nZGzWMyI">YouTube | deriving spherical coordinates</a></p>
+</li>
+<li>
+<p><a href="https://mathworld.wolfram.com/SphericalCoordinates.html">Spherical coordinates</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=R50_LFbJ3aM" class="bare">https://www.youtube.com/watch?v=R50_LFbJ3aM</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=Ex_g2w4E5lQ">spherical coordinates animation</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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\ No newline at end of file
diff --git a/Calculus/index.adoc b/Calculus/index.adoc
new file mode 100644 (file)
index 0000000..c43a138
--- /dev/null
+++ b/Calculus/index.adoc
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+= ./Calculus
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Derivatives.html;hb=HEAD[Derivatives]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Intermediate-Value-Theorem.html;hb=HEAD[Intermediate-Value-Theorem]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Spherical-Coordinates.html;hb=HEAD[Spherical-Coordinates]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Integrals.html;hb=HEAD[Integrals]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Asymptotes.html;hb=HEAD[Asymptotes]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Limits.html;hb=HEAD[Limits]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/Mean-Value-Theorem.html;hb=HEAD[Mean-Value-Theorem]
diff --git a/Calculus/index.html b/Calculus/index.html
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+.literalblock pre,.listingblock>.content>pre:not(.highlight),.listingblock>.content>pre[class=highlight],.listingblock>.content>pre[class^="highlight "]{background:#f7f7f8}
+.literalblock.output pre{color:#f7f7f8;background:rgba(0,0,0,.9)}
+.listingblock>.content{position:relative}
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+.listingblock:hover code[data-lang]::before{display:block}
+.listingblock.terminal pre .command::before{content:attr(data-prompt);padding-right:.5em;color:inherit;opacity:.5}
+.listingblock.terminal pre .command:not([data-prompt])::before{content:"$"}
+.listingblock pre.highlightjs{padding:0}
+.listingblock pre.highlightjs>code{padding:1em;border-radius:4px}
+.listingblock pre.prettyprint{border-width:0}
+.prettyprint{background:#f7f7f8}
+pre.prettyprint .linenums{line-height:1.45;margin-left:2em}
+pre.prettyprint li{background:none;list-style-type:inherit;padding-left:0}
+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
+table.linenotable td.code{padding-left:.75em}
+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
+pre.pygments span.linenos{display:inline-block;margin-right:.75em}
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+.quoteblock:not(.excerpt)>.title{margin-left:-1.5em;margin-bottom:.75em}
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+.quoteblock blockquote{margin:0;padding:0;border:0}
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+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
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+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
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+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
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+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
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+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
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+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
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+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Calculus</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Derivatives.html;hb=HEAD">Derivatives</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Intermediate-Value-Theorem.html;hb=HEAD">Intermediate-Value-Theorem</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Spherical-Coordinates.html;hb=HEAD">Spherical-Coordinates</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Integrals.html;hb=HEAD">Integrals</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Asymptotes.html;hb=HEAD">Asymptotes</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Limits.html;hb=HEAD">Limits</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/Mean-Value-Theorem.html;hb=HEAD">Mean-Value-Theorem</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Calculus/scarp b/Calculus/scarp
new file mode 100644 (file)
index 0000000..b40828d
--- /dev/null
+++ b/Calculus/scarp
@@ -0,0 +1,4 @@
+math sorcerer Calculus 1
+Ended at Calculus 1 with the Math Sorcerer Section 1.13
+Ended at Calculus 1 with the Math Sorcerer Section 3.21
+on section 5
diff --git a/Geometry/Angles-And-Sides.adoc b/Geometry/Angles-And-Sides.adoc
new file mode 100644 (file)
index 0000000..f988c57
--- /dev/null
+++ b/Geometry/Angles-And-Sides.adoc
@@ -0,0 +1,53 @@
+= Algrebra | Angles and Sides
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Transversals
+When two parallel lines are cut by a transveresal, alternate interior angles are
+formed.
+
+If two side of a triangle are congruent, the angles opposite those sides are
+also congruent.
+
+== Triangles
+All of the angles of a triangle equal 180 degrees. This image is a simple proof
+of that.
+
+image::180-degree-triangle-proof.png[]
+
+In an equalateral triangle the angle bisector is always perpendicular to the
+opposite angle
+
+There *cannot* be two abtuse angles in a triangle.
+
+
+== Definitions
+
+bisector::
+    divides an angle into two congruent angles.
+
+subtraction postulate::
+    if equal quantities are subtracted from equal quantities, the differences
+    are equal.
+
+transversal::
+    a line that passes through two lines in the same plane at two
+    distinct points in the geometry.
+
+circumscribed::
+    a figure within another so that their boundries touch but do not intersect
+
+inscribed::
+    a figure within another so that their boundries touch but do not intersect.
+
+circumcenter::
+    a center of a circle when the circle is circumscribed around the figure.
+
+incenter::
+    a center of a circle when the circle is inscribed in the figure.
diff --git a/Geometry/Angles-And-Sides.html b/Geometry/Angles-And-Sides.html
new file mode 100644 (file)
index 0000000..94b8cce
--- /dev/null
+++ b/Geometry/Angles-And-Sides.html
@@ -0,0 +1,1145 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Algrebra | Angles and Sides</title>
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+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
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+
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+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
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+});
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+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
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+        window.scrollTo(0, e.data.scrollPosition);
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+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | Angles and Sides</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_transversals">Transversals</a></li>
+<li><a href="#_triangles">Triangles</a></li>
+<li><a href="#_definitions">Definitions</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_transversals">Transversals</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When two parallel lines are cut by a transveresal, alternate interior angles are
+formed.</p>
+</div>
+<div class="paragraph">
+<p>If two side of a triangle are congruent, the angles opposite those sides are
+also congruent.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_triangles">Triangles</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>All of the angles of a triangle equal 180 degrees. This image is a simple proof
+of that.</p>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/180-degree-triangle-proof.png" alt="180 degree triangle proof">
+</div>
+</div>
+<div class="paragraph">
+<p>In an equalateral triangle the angle bisector is always perpendicular to the
+opposite angle</p>
+</div>
+<div class="paragraph">
+<p>There <strong>cannot</strong> be two abtuse angles in a triangle.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_definitions">Definitions</h2>
+<div class="sectionbody">
+<div class="dlist">
+<dl>
+<dt class="hdlist1">bisector</dt>
+<dd>
+<p>divides an angle into two congruent angles.</p>
+</dd>
+<dt class="hdlist1">subtraction postulate</dt>
+<dd>
+<p>if equal quantities are subtracted from equal quantities, the differences
+are equal.</p>
+</dd>
+<dt class="hdlist1">transversal</dt>
+<dd>
+<p>a line that passes through two lines in the same plane at two
+distinct points in the geometry.</p>
+</dd>
+<dt class="hdlist1">circumscribed</dt>
+<dd>
+<p>a figure within another so that their boundries touch but do not intersect</p>
+</dd>
+<dt class="hdlist1">inscribed</dt>
+<dd>
+<p>a figure within another so that their boundries touch but do not intersect.</p>
+</dd>
+<dt class="hdlist1">circumcenter</dt>
+<dd>
+<p>a center of a circle when the circle is circumscribed around the figure.</p>
+</dd>
+<dt class="hdlist1">incenter</dt>
+<dd>
+<p>a center of a circle when the circle is inscribed in the figure.</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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\ No newline at end of file
diff --git a/Geometry/Circle-Theorems.adoc b/Geometry/Circle-Theorems.adoc
new file mode 100644 (file)
index 0000000..738e0f3
--- /dev/null
+++ b/Geometry/Circle-Theorems.adoc
@@ -0,0 +1,114 @@
+= Geometry | Circle Theorems
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+
+== Formulas
+
+* Circumfrerence = stem:[2\pir]
+* Diameter = stem:[2r]
+* Area = stem:[\pir^2]
+
+== Lines
+
+[.center]
+image::lines-of-circle.png[]
+
+Chords::
+    Line segment that connects two endpoints on a circle. If a chord passes
+    through the center of a circle its known as a diameter
+
+Tangent Line::
+    Touches just one point at a curce.
+
+Secant Line::
+    A line the passes through two points of a curve.
+
+Radius::
+    half of diameter
+
+== Central Angle
+[stem]
+++++
+m\angle ACB = mACB
+++++
+// [stem]
+// ++++
+// \angle ACB = 50^\circ
+// \
+// mACB = 50^\circ
+// ++++
+
+[.center]
+image::Central-Angle.png[]
+
+== Inscribed Angle
+[stem]
+++++
+m\angle ABC = \frac{1}{2}m \overarc{AC}
+++++
+
+[.center]
+image::Inscribed-Angle.png[]
+
+== Tangent-Chord Angle
+[stem]
+++++
+m\angle ABC = \frac{1}{2}(m\overarc{AB})
+++++
+
+[.center]
+image::Tangent-Chord-Angle.png[]
+
+== Unit Circle
+y:: sine
+x:: cosine
+image::Unit-Circle.png[]
+
+== Standard Circle Formula
+
+[stem]
+++++
+(x -h)^2 + (y - k)^2 =r^2
+++++
+
+`h` and `k` are the x and y coordinates of the center of the circle.
+
+== Drawing a Circle in C
+
+Without Standard Circle Formula:
+[source, c]
+----
+void drawCircle(SDL_Renderer *ren, float radius, SDL_Point center, uint32_t color)
+{
+    
+    float x = 0;
+    float y = 0;
+    for(float i = 0; i < 2 * M_PI; i += 0.001f)
+    {
+        x = cos(2.0f * M_PI * i);
+        y = sin(2.0f * M_PI * i);
+        SDL_Rect point = {
+            (x * radius) + center.x, 
+            (y * radius) + center.y,
+            1, 
+            1
+        };
+        SDL_SetRenderDrawColor(ren, 0, 0, 0, 255);
+        SDL_RenderFillRect(ren, &point);
+    }
+}
+----
+
+== Rotation
+
+image:Rotation.png[]
+
+== Links
+- https://www.desmos.com/geometry[Desmos Geometry]
diff --git a/Geometry/Circle-Theorems.html b/Geometry/Circle-Theorems.html
new file mode 100644 (file)
index 0000000..0500e5c
--- /dev/null
+++ b/Geometry/Circle-Theorems.html
@@ -0,0 +1,1341 @@
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+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<style>
+pre.pygments .hll { background-color: #49483e }
+pre.pygments { background: #272822; color: #F8F8F2 }
+pre.pygments .tok-c { color: #959077 } /* Comment */
+pre.pygments .tok-err { color: #ED007E; background-color: #1E0010 } /* Error */
+pre.pygments .tok-esc { color: #F8F8F2 } /* Escape */
+pre.pygments .tok-g { color: #F8F8F2 } /* Generic */
+pre.pygments .tok-k { color: #66D9EF } /* Keyword */
+pre.pygments .tok-l { color: #AE81FF } /* Literal */
+pre.pygments .tok-n { color: #F8F8F2 } /* Name */
+pre.pygments .tok-o { color: #FF4689 } /* Operator */
+pre.pygments .tok-x { color: #F8F8F2 } /* Other */
+pre.pygments .tok-p { color: #F8F8F2 } /* Punctuation */
+pre.pygments .tok-ch { color: #959077 } /* Comment.Hashbang */
+pre.pygments .tok-cm { color: #959077 } /* Comment.Multiline */
+pre.pygments .tok-cp { color: #959077 } /* Comment.Preproc */
+pre.pygments .tok-cpf { color: #959077 } /* Comment.PreprocFile */
+pre.pygments .tok-c1 { color: #959077 } /* Comment.Single */
+pre.pygments .tok-cs { color: #959077 } /* Comment.Special */
+pre.pygments .tok-gd { color: #FF4689 } /* Generic.Deleted */
+pre.pygments .tok-ge { color: #F8F8F2; font-style: italic } /* Generic.Emph */
+pre.pygments .tok-ges { color: #F8F8F2; font-weight: bold; font-style: italic } /* Generic.EmphStrong */
+pre.pygments .tok-gr { color: #F8F8F2 } /* Generic.Error */
+pre.pygments .tok-gh { color: #F8F8F2 } /* Generic.Heading */
+pre.pygments .tok-gi { color: #A6E22E } /* Generic.Inserted */
+pre.pygments .tok-go { color: #66D9EF } /* Generic.Output */
+pre.pygments .tok-gp { color: #FF4689; font-weight: bold } /* Generic.Prompt */
+pre.pygments .tok-gs { color: #F8F8F2; font-weight: bold } /* Generic.Strong */
+pre.pygments .tok-gu { color: #959077 } /* Generic.Subheading */
+pre.pygments .tok-gt { color: #F8F8F2 } /* Generic.Traceback */
+pre.pygments .tok-kc { color: #66D9EF } /* Keyword.Constant */
+pre.pygments .tok-kd { color: #66D9EF } /* Keyword.Declaration */
+pre.pygments .tok-kn { color: #FF4689 } /* Keyword.Namespace */
+pre.pygments .tok-kp { color: #66D9EF } /* Keyword.Pseudo */
+pre.pygments .tok-kr { color: #66D9EF } /* Keyword.Reserved */
+pre.pygments .tok-kt { color: #66D9EF } /* Keyword.Type */
+pre.pygments .tok-ld { color: #E6DB74 } /* Literal.Date */
+pre.pygments .tok-m { color: #AE81FF } /* Literal.Number */
+pre.pygments .tok-s { color: #E6DB74 } /* Literal.String */
+pre.pygments .tok-na { color: #A6E22E } /* Name.Attribute */
+pre.pygments .tok-nb { color: #F8F8F2 } /* Name.Builtin */
+pre.pygments .tok-nc { color: #A6E22E } /* Name.Class */
+pre.pygments .tok-no { color: #66D9EF } /* Name.Constant */
+pre.pygments .tok-nd { color: #A6E22E } /* Name.Decorator */
+pre.pygments .tok-ni { color: #F8F8F2 } /* Name.Entity */
+pre.pygments .tok-ne { color: #A6E22E } /* Name.Exception */
+pre.pygments .tok-nf { color: #A6E22E } /* Name.Function */
+pre.pygments .tok-nl { color: #F8F8F2 } /* Name.Label */
+pre.pygments .tok-nn { color: #F8F8F2 } /* Name.Namespace */
+pre.pygments .tok-nx { color: #A6E22E } /* Name.Other */
+pre.pygments .tok-py { color: #F8F8F2 } /* Name.Property */
+pre.pygments .tok-nt { color: #FF4689 } /* Name.Tag */
+pre.pygments .tok-nv { color: #F8F8F2 } /* Name.Variable */
+pre.pygments .tok-ow { color: #FF4689 } /* Operator.Word */
+pre.pygments .tok-pm { color: #F8F8F2 } /* Punctuation.Marker */
+pre.pygments .tok-w { color: #F8F8F2 } /* Text.Whitespace */
+pre.pygments .tok-mb { color: #AE81FF } /* Literal.Number.Bin */
+pre.pygments .tok-mf { color: #AE81FF } /* Literal.Number.Float */
+pre.pygments .tok-mh { color: #AE81FF } /* Literal.Number.Hex */
+pre.pygments .tok-mi { color: #AE81FF } /* Literal.Number.Integer */
+pre.pygments .tok-mo { color: #AE81FF } /* Literal.Number.Oct */
+pre.pygments .tok-sa { color: #E6DB74 } /* Literal.String.Affix */
+pre.pygments .tok-sb { color: #E6DB74 } /* Literal.String.Backtick */
+pre.pygments .tok-sc { color: #E6DB74 } /* Literal.String.Char */
+pre.pygments .tok-dl { color: #E6DB74 } /* Literal.String.Delimiter */
+pre.pygments .tok-sd { color: #E6DB74 } /* Literal.String.Doc */
+pre.pygments .tok-s2 { color: #E6DB74 } /* Literal.String.Double */
+pre.pygments .tok-se { color: #AE81FF } /* Literal.String.Escape */
+pre.pygments .tok-sh { color: #E6DB74 } /* Literal.String.Heredoc */
+pre.pygments .tok-si { color: #E6DB74 } /* Literal.String.Interpol */
+pre.pygments .tok-sx { color: #E6DB74 } /* Literal.String.Other */
+pre.pygments .tok-sr { color: #E6DB74 } /* Literal.String.Regex */
+pre.pygments .tok-s1 { color: #E6DB74 } /* Literal.String.Single */
+pre.pygments .tok-ss { color: #E6DB74 } /* Literal.String.Symbol */
+pre.pygments .tok-bp { color: #F8F8F2 } /* Name.Builtin.Pseudo */
+pre.pygments .tok-fm { color: #A6E22E } /* Name.Function.Magic */
+pre.pygments .tok-vc { color: #F8F8F2 } /* Name.Variable.Class */
+pre.pygments .tok-vg { color: #F8F8F2 } /* Name.Variable.Global */
+pre.pygments .tok-vi { color: #F8F8F2 } /* Name.Variable.Instance */
+pre.pygments .tok-vm { color: #F8F8F2 } /* Name.Variable.Magic */
+pre.pygments .tok-il { color: #AE81FF } /* Literal.Number.Integer.Long */
+</style>
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
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+});
+
+// Capture scroll position while scrolling
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+    window.parent.postMessage({
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Geometry | Circle Theorems</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_formulas">Formulas</a></li>
+<li><a href="#_lines">Lines</a></li>
+<li><a href="#_central_angle">Central Angle</a></li>
+<li><a href="#_inscribed_angle">Inscribed Angle</a></li>
+<li><a href="#_tangent_chord_angle">Tangent-Chord Angle</a></li>
+<li><a href="#_unit_circle">Unit Circle</a></li>
+<li><a href="#_standard_circle_formula">Standard Circle Formula</a></li>
+<li><a href="#_drawing_a_circle_in_c">Drawing a Circle in C</a></li>
+<li><a href="#_rotation">Rotation</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_formulas">Formulas</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>Circumfrerence = \$2\pir\$</p>
+</li>
+<li>
+<p>Diameter = \$2r\$</p>
+</li>
+<li>
+<p>Area = \$\pir^2\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_lines">Lines</h2>
+<div class="sectionbody">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/lines-of-circle.png" alt="lines of circle">
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Chords</dt>
+<dd>
+<p>Line segment that connects two endpoints on a circle. If a chord passes
+through the center of a circle its known as a diameter</p>
+</dd>
+<dt class="hdlist1">Tangent Line</dt>
+<dd>
+<p>Touches just one point at a curce.</p>
+</dd>
+<dt class="hdlist1">Secant Line</dt>
+<dd>
+<p>A line the passes through two points of a curve.</p>
+</dd>
+<dt class="hdlist1">Radius</dt>
+<dd>
+<p>half of diameter</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_central_angle">Central Angle</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$m\angle ACB = mACB\$
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Central-Angle.png" alt="Central Angle">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_inscribed_angle">Inscribed Angle</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$m\angle ABC = \frac{1}{2}m \overarc{AC}\$
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Inscribed-Angle.png" alt="Inscribed Angle">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_tangent_chord_angle">Tangent-Chord Angle</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$m\angle ABC = \frac{1}{2}(m\overarc{AB})\$
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Tangent-Chord-Angle.png" alt="Tangent Chord Angle">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_unit_circle">Unit Circle</h2>
+<div class="sectionbody">
+<div class="dlist">
+<dl>
+<dt class="hdlist1">y</dt>
+<dd>
+<p>sine</p>
+</dd>
+<dt class="hdlist1">x</dt>
+<dd>
+<p>cosine</p>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Unit-Circle.png" alt="Unit Circle">
+</div>
+</div>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_standard_circle_formula">Standard Circle Formula</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$(x -h)^2 + (y - k)^2 =r^2\$
+</div>
+</div>
+<div class="paragraph">
+<p><code>h</code> and <code>k</code> are the x and y coordinates of the center of the circle.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_drawing_a_circle_in_c">Drawing a Circle in C</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Without Standard Circle Formula:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-kt">void</span><span class="tok-w"> </span><span class="tok-nf">drawCircle</span><span class="tok-p">(</span><span class="tok-n">SDL_Renderer</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-n">ren</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">radius</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">SDL_Point</span><span class="tok-w"> </span><span class="tok-n">center</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-kt">uint32_t</span><span class="tok-w"> </span><span class="tok-n">color</span><span class="tok-p">)</span>
+<span class="tok-p">{</span>
+
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-k">for</span><span class="tok-p">(</span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">i</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">;</span><span class="tok-w"> </span><span class="tok-n">i</span><span class="tok-w"> </span><span class="tok-o">&lt;</span><span class="tok-w"> </span><span class="tok-mi">2</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">M_PI</span><span class="tok-p">;</span><span class="tok-w"> </span><span class="tok-n">i</span><span class="tok-w"> </span><span class="tok-o">+=</span><span class="tok-w"> </span><span class="tok-mf">0.001f</span><span class="tok-p">)</span>
+<span class="tok-w">    </span><span class="tok-p">{</span>
+<span class="tok-w">        </span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">cos</span><span class="tok-p">(</span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">M_PI</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">i</span><span class="tok-p">);</span>
+<span class="tok-w">        </span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sin</span><span class="tok-p">(</span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">M_PI</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">i</span><span class="tok-p">);</span>
+<span class="tok-w">        </span><span class="tok-n">SDL_Rect</span><span class="tok-w"> </span><span class="tok-n">point</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">{</span>
+<span class="tok-w">            </span><span class="tok-p">(</span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">radius</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">center</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-p">,</span>
+<span class="tok-w">            </span><span class="tok-p">(</span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">radius</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">center</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-p">,</span>
+<span class="tok-w">            </span><span class="tok-mi">1</span><span class="tok-p">,</span>
+<span class="tok-w">            </span><span class="tok-mi">1</span>
+<span class="tok-w">        </span><span class="tok-p">};</span>
+<span class="tok-w">        </span><span class="tok-n">SDL_SetRenderDrawColor</span><span class="tok-p">(</span><span class="tok-n">ren</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">0</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">255</span><span class="tok-p">);</span>
+<span class="tok-w">        </span><span class="tok-n">SDL_RenderFillRect</span><span class="tok-p">(</span><span class="tok-n">ren</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-o">&amp;</span><span class="tok-n">point</span><span class="tok-p">);</span>
+<span class="tok-w">    </span><span class="tok-p">}</span>
+<span class="tok-p">}</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_rotation">Rotation</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p><span class="image"><img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Rotation.png" alt="Rotation"></span></p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.desmos.com/geometry">Desmos Geometry</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
+  },
+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
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+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
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diff --git a/Geometry/Geometric-Proportions.adoc b/Geometry/Geometric-Proportions.adoc
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+= Geometry | Geometric Proportions
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+
+In geometry, proportional relationships appear whenever shapes are "scaled"
+versions of each other. In Calculus—specifically Related Rates—these are used to
+eliminate variables before differentiating.
+
+== 1. Similar Triangles (The "Trough" or "Shadow" Rule)
+This applies when one triangle is nested inside another or when triangles are
+created by a single line of sight.
+
+* **The Proportion:**
+
+[stem]
+++++
+\frac{\text{base}_1}{\text{height}_1} = \frac{\text{base}_2}{\text{height}_2}
+++++
+
+* **Common Use:** Calculating the length of a shadow cast by a person walking
+              away from a street lamp, or water levels in a triangular trough.
+
+== 2. Conical Proportions (The "Water Tank" Rule)
+Because a cone has a uniform slope, any cross-section of the liquid forms a
+triangle similar to the triangle of the full cone.
+
+* **The Proportion:**
+
+[stem]
+++++
+\frac{r}{h} = \frac{R}{H}
+++++
+
+**Note:** _r_ and _h_ are the liquid's radius/height; _R_ and _H_ are the tank's
+     total dimensions.
+
+== 3. Circles and Sectors (The "Pizza" Rule)
+The relationship between a slice's angle, its arc length, and its area is
+proportional to the whole circle.
+
+* **The Proportion:**
+
+[stem]
+++++
+\frac{\text{Arc Length}}{2\pi r} = \frac{\text{Sector Area}}{\pi r^2} = \frac{\text{Central Angle}}{360^\circ}
+++++
+
+== 4. Direct Square/Cube Law (The "Scaling" Rule)
+This describes how dimensions relate to surface area and volume as an object scales.
+
+* **Area Proportion:** 
+
+[stem]
+++++
+\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2
+++++
+
+* **Volume Proportion:**
+
+[stem]
+++++
+\frac{V_1}{V_2} = \left(\frac{s_1}{s_2}\right)^3
+++++
+
+== 5. Chord-Segment Proportion
+When two chords intersect inside a circle, the products of their segments are
+proportional.
+
+* **The Proportion:**
+
+[stem]
+++++
+a \cdot b = c \cdot d
+++++
+
+== Summary Reference Table
+
+[cols="1,2,2", options="header"]
+|===
+| Shape | Relationship | Common Application
+| **Triangle** | base / height = constant | Troughs, Shadows, Ladders
+| **Cone** | radius / height = constant | Funnels, Sand piles, Water tanks
+| **Circle** | arc length / radius = theta | Rotational motion, Gears
+| **Sphere** | Volume is proportional to r^3 | Balloons, Melting snowballs
+|===
diff --git a/Geometry/Geometric-Proportions.html b/Geometry/Geometric-Proportions.html
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+.quoteblock blockquote{margin:0;padding:0;border:0}
+.quoteblock blockquote::before{content:"\201c";float:left;font-size:2.75em;font-weight:bold;line-height:.6em;margin-left:-.6em;color:#7a2518;text-shadow:0 1px 2px rgba(0,0,0,.1)}
+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
+.verseblock{margin:0 1em 1.25em}
+.verseblock pre{font-family:"Open Sans","DejaVu Sans",sans-serif;font-size:1.15rem;color:rgba(0,0,0,.85);font-weight:300;text-rendering:optimizeLegibility}
+.verseblock pre strong{font-weight:400}
+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
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+    else if (state === 'completed') checkbox.textContent = '▣';
+
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+    p.appendChild(document.createTextNode(' ' + content));
+  });
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+};
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+    // Request the scroll position as soon as the iframe is loaded
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+        type: "SCROLL_REQUEST_GET"
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+// Capture scroll position while scrolling
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+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Geometry | Geometric Proportions</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_1_similar_triangles_the_trough_or_shadow_rule">1. Similar Triangles (The "Trough" or "Shadow" Rule)</a></li>
+<li><a href="#_2_conical_proportions_the_water_tank_rule">2. Conical Proportions (The "Water Tank" Rule)</a></li>
+<li><a href="#_3_circles_and_sectors_the_pizza_rule">3. Circles and Sectors (The "Pizza" Rule)</a></li>
+<li><a href="#_4_direct_squarecube_law_the_scaling_rule">4. Direct Square/Cube Law (The "Scaling" Rule)</a></li>
+<li><a href="#_5_chord_segment_proportion">5. Chord-Segment Proportion</a></li>
+<li><a href="#_summary_reference_table">Summary Reference Table</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div class="paragraph">
+<p>In geometry, proportional relationships appear whenever shapes are "scaled"
+versions of each other. In Calculus—specifically Related Rates—these are used to
+eliminate variables before differentiating.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_1_similar_triangles_the_trough_or_shadow_rule">1. Similar Triangles (The "Trough" or "Shadow" Rule)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This applies when one triangle is nested inside another or when triangles are
+created by a single line of sight.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>The Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{\text{base}_1}{\text{height}_1} = \frac{\text{base}_2}{\text{height}_2}\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Common Use:</strong> Calculating the length of a shadow cast by a person walking
+away from a street lamp, or water levels in a triangular trough.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_2_conical_proportions_the_water_tank_rule">2. Conical Proportions (The "Water Tank" Rule)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Because a cone has a uniform slope, any cross-section of the liquid forms a
+triangle similar to the triangle of the full cone.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>The Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{r}{h} = \frac{R}{H}\]
+</div>
+</div>
+<div class="paragraph">
+<p><strong>Note:</strong> <em>r</em> and <em>h</em> are the liquid&#8217;s radius/height; <em>R</em> and <em>H</em> are the tank&#8217;s
+     total dimensions.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_3_circles_and_sectors_the_pizza_rule">3. Circles and Sectors (The "Pizza" Rule)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The relationship between a slice&#8217;s angle, its arc length, and its area is
+proportional to the whole circle.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>The Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{\text{Arc Length}}{2\pi r} = \frac{\text{Sector Area}}{\pi r^2} = \frac{\text{Central Angle}}{360^\circ}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_4_direct_squarecube_law_the_scaling_rule">4. Direct Square/Cube Law (The "Scaling" Rule)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This describes how dimensions relate to surface area and volume as an object scales.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Area Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Volume Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\frac{V_1}{V_2} = \left(\frac{s_1}{s_2}\right)^3\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_5_chord_segment_proportion">5. Chord-Segment Proportion</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When two chords intersect inside a circle, the products of their segments are
+proportional.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>The Proportion:</strong></p>
+</li>
+</ul>
+</div>
+<div class="stemblock">
+<div class="content">
+\[a \cdot b = c \cdot d\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_summary_reference_table">Summary Reference Table</h2>
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 20%;">
+<col style="width: 40%;">
+<col style="width: 40%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Shape</th>
+<th class="tableblock halign-left valign-top">Relationship</th>
+<th class="tableblock halign-left valign-top">Common Application</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Triangle</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">base / height = constant</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Troughs, Shadows, Ladders</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Cone</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">radius / height = constant</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Funnels, Sand piles, Water tanks</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Circle</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">arc length / radius = theta</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Rotational motion, Gears</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Sphere</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Volume is proportional to r^3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Balloons, Melting snowballs</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+</html>
\ No newline at end of file
diff --git a/Geometry/Lines.adoc b/Geometry/Lines.adoc
new file mode 100644 (file)
index 0000000..35e28d1
--- /dev/null
+++ b/Geometry/Lines.adoc
@@ -0,0 +1,17 @@
+= Geometry | Lines
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Perpendicular Lines
+Perpendicular lines have negative reciprocal slopes
+
+[.center]
+image::Perpendicular-Angle.jpg[]
+
+The product of the two slopes should be equal to stem:[-1].
diff --git a/Geometry/Lines.html b/Geometry/Lines.html
new file mode 100644 (file)
index 0000000..d36515e
--- /dev/null
+++ b/Geometry/Lines.html
@@ -0,0 +1,1086 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Geometry | Lines</title>
+<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700">
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+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
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+        scrollPosition: scrollPosition
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Geometry | Lines</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_perpendicular_lines">Perpendicular Lines</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_perpendicular_lines">Perpendicular Lines</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Perpendicular lines have negative reciprocal slopes</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Perpendicular-Angle.jpg" alt="Perpendicular Angle">
+</div>
+</div>
+<div class="paragraph">
+<p>The product of the two slopes should be equal to \$-1\$.</p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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diff --git a/Geometry/Triangles.adoc b/Geometry/Triangles.adoc
new file mode 100644 (file)
index 0000000..8b82a83
--- /dev/null
+++ b/Geometry/Triangles.adoc
@@ -0,0 +1,27 @@
+= Geometry | Triangles
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Types of Triangles
+
+Scalene Triangle::
+    A triangle whose all sides are unequal and all angles have different
+    measures.
+
+Scalene Right Triangle::
+    Same as above but with a right angle.
+
+
+== Similar Triangles
+
+image::similar-triangles.png[]
+
+== Links
+* https://www.youtube.com/watch?v=bGe-d09Nc_M
+* https://www.cuemath.com/geometry/similar-triangles/[cuemath | similar triangles]
diff --git a/Geometry/Triangles.html b/Geometry/Triangles.html
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+    // Request the scroll position as soon as the iframe is loaded
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+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
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+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+window.addEventListener("message", (e) => {
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+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Geometry | Triangles</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_types_of_triangles">Types of Triangles</a></li>
+<li><a href="#_similar_triangles">Similar Triangles</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_types_of_triangles">Types of Triangles</h2>
+<div class="sectionbody">
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Scalene Triangle</dt>
+<dd>
+<p>A triangle whose all sides are unequal and all angles have different
+measures.</p>
+</dd>
+<dt class="hdlist1">Scalene Right Triangle</dt>
+<dd>
+<p>Same as above but with a right angle.</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_similar_triangles">Similar Triangles</h2>
+<div class="sectionbody">
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/similar-triangles.png" alt="similar triangles">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=bGe-d09Nc_M" class="bare">https://www.youtube.com/watch?v=bGe-d09Nc_M</a></p>
+</li>
+<li>
+<p><a href="https://www.cuemath.com/geometry/similar-triangles/">cuemath | similar triangles</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
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+</html>
\ No newline at end of file
diff --git a/Geometry/index.adoc b/Geometry/index.adoc
new file mode 100644 (file)
index 0000000..5ac7919
--- /dev/null
+++ b/Geometry/index.adoc
@@ -0,0 +1,14 @@
+= ./Geometry
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/Circle-Theorems.html;hb=HEAD[Circle-Theorems]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/Triangles.html;hb=HEAD[Triangles]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/Angles-And-Sides.html;hb=HEAD[Angles-And-Sides]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/Lines.html;hb=HEAD[Lines]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/Geometric-Proportions.html;hb=HEAD[Geometric-Proportions]
diff --git a/Geometry/index.html b/Geometry/index.html
new file mode 100644 (file)
index 0000000..792522a
--- /dev/null
+++ b/Geometry/index.html
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+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
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+details[open] > summary::before {
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+    margin-top: -12px;
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+    justify-content: center;
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+    border-radius: 4px;
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+div.ulist {
+    max-width: 95%;
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+    max-width: 98%;
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+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
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+ul.checklist li p {
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+
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+div#header {
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+    height: 100px;
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+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
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+#content {
+    /*margin-top: 105px;*/
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+details > .content {
+    border: solid;
+    padding: 10px;
+}
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+</style>
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+</head>
+
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+    */
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+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Geometry</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/Circle-Theorems.html;hb=HEAD">Circle-Theorems</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/Triangles.html;hb=HEAD">Triangles</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/Angles-And-Sides.html;hb=HEAD">Angles-And-Sides</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/Lines.html;hb=HEAD">Lines</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/Geometric-Proportions.html;hb=HEAD">Geometric-Proportions</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Graphics/3D-Objects.adoc b/Graphics/3D-Objects.adoc
new file mode 100644 (file)
index 0000000..30d0b32
--- /dev/null
+++ b/Graphics/3D-Objects.adoc
@@ -0,0 +1,18 @@
+= Algrebra | 3D Objects
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Plane
+
+=== Formula
+
+[stem]
+++++
+ax + by + cz = 0
+++++
diff --git a/Graphics/3D-Objects.html b/Graphics/3D-Objects.html
new file mode 100644 (file)
index 0000000..03e853c
--- /dev/null
+++ b/Graphics/3D-Objects.html
@@ -0,0 +1,1087 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Algrebra | 3D Objects</title>
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+pre.nowrap,pre.nowrap pre{white-space:pre;word-wrap:normal}
+em em{font-style:normal}
+strong strong{font-weight:400}
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+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Algrebra | 3D Objects</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_plane">Plane</a>
+<ul class="sectlevel2">
+<li><a href="#_formula">Formula</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_plane">Plane</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_formula">Formula</h3>
+<div class="stemblock">
+<div class="content">
+\$ax + by + cz = 0\$
+</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
+  },
+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Graphics/Quanternions.adoc b/Graphics/Quanternions.adoc
new file mode 100644 (file)
index 0000000..e7134dd
--- /dev/null
+++ b/Graphics/Quanternions.adoc
@@ -0,0 +1,13 @@
+= Quanternions | Graphics
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Links
+* https://www.youtube.com/watch?v=xoTqBqQtrY4[Covenant Turtle | Basic Intro for Quanternions for 3D rotations]
+* https://youtu.be/-zsnHbQyRnc?si=xPlKN15Kt6GyioXz[Wumbo | how quanternions visualize 3d space]
diff --git a/Graphics/Quanternions.html b/Graphics/Quanternions.html
new file mode 100644 (file)
index 0000000..f379967
--- /dev/null
+++ b/Graphics/Quanternions.html
@@ -0,0 +1,1085 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
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+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
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+    margin-right: 5%;
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+img {
+    margin: auto;
+    width: 100%;
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+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
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+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
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+::-webkit-scrollbar-track {
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+div.ulist {
+    max-width: 95%;
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+  display: inline-block;   /* already set */
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+  font-family: "inherit";      /* inherits whatever body is using */
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+  font-size: 1.2em;   /* same as body text */
+}
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+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
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+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
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+    else if (state === 'started') checkbox.textContent = '◪';
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+
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+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Quanternions | Graphics</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=xoTqBqQtrY4">Covenant Turtle | Basic Intro for Quanternions for 3D rotations</a></p>
+</li>
+<li>
+<p><a href="https://youtu.be/-zsnHbQyRnc?si=xPlKN15Kt6GyioXz">Wumbo | how quanternions visualize 3d space</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:55 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
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+  },
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+    ignoreClass: "nostem|noasciimath"
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+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Graphics/Rasterization.adoc b/Graphics/Rasterization.adoc
new file mode 100644 (file)
index 0000000..30aa289
--- /dev/null
+++ b/Graphics/Rasterization.adoc
@@ -0,0 +1,18 @@
+= Graphics | Rasterization
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+
+== Interpolation
+NOTE: This is helpful outside of rasterization
+
+image::interpolation.png[]
+
+== Links
+- https://www.youtube.com/watch?v=PahbNFypubE&t=398s[Bisqwit | Texture mapping.. | interpolation]
diff --git a/Graphics/Rasterization.html b/Graphics/Rasterization.html
new file mode 100644 (file)
index 0000000..3e4f84a
--- /dev/null
+++ b/Graphics/Rasterization.html
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+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
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+
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+
+:root {
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+    --gruvbox-green: #b8bb26;
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+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
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+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
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+    width: fit-content;
+}
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+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
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+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
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+h5 {
+  color: #83a598 !important;
+}
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+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
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+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
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+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
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+        margin-bottom: 10px !important;
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+}
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+}
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+}
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+
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+    /*
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+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
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+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
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+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
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+    max-width: 40%;
+    margin: auto;
+}
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+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
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+}
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+}
+
+.inline {
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+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
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+    border-radius: 4px;
+}
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+}
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+div.ulist {
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+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
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+}
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+}
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+}
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+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
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+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
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+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
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+/*
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+    if (e.key === "o") {
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+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
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+});
+*/
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+
+function scrollStep() {
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+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
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+        requestAnimationFrame(scrollStep);
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+    }
+}
+
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+
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+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
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+
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+});
+
+
+
+window.onload = function () {
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+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
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+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
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+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
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+
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+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
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+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Graphics | Rasterization</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_interpolation">Interpolation</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_interpolation">Interpolation</h2>
+<div class="sectionbody">
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+This is helpful outside of rasterization
+</td>
+</tr>
+</table>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/interpolation.png" alt="interpolation">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=PahbNFypubE&amp;t=398s">Bisqwit | Texture mapping.. | interpolation</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Graphics/index.adoc b/Graphics/index.adoc
new file mode 100644 (file)
index 0000000..95ab256
--- /dev/null
+++ b/Graphics/index.adoc
@@ -0,0 +1,12 @@
+= ./Graphics
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphics/Rasterization.html;hb=HEAD[Rasterization]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphics/3D-Objects.html;hb=HEAD[3D-Objects]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphics/Quanternions.html;hb=HEAD[Quanternions]
diff --git a/Graphics/index.html b/Graphics/index.html
new file mode 100644 (file)
index 0000000..61d0286
--- /dev/null
+++ b/Graphics/index.html
@@ -0,0 +1,1058 @@
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+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>./Graphics</title>
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+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
+pre.pygments span.linenos{display:inline-block;margin-right:.75em}
+.quoteblock{margin:0 1em 1.25em 1.5em;display:table}
+.quoteblock:not(.excerpt)>.title{margin-left:-1.5em;margin-bottom:.75em}
+.quoteblock blockquote,.quoteblock p{color:rgba(0,0,0,.85);font-size:1.15rem;line-height:1.75;word-spacing:.1em;letter-spacing:0;font-style:italic;text-align:justify}
+.quoteblock blockquote{margin:0;padding:0;border:0}
+.quoteblock blockquote::before{content:"\201c";float:left;font-size:2.75em;font-weight:bold;line-height:.6em;margin-left:-.6em;color:#7a2518;text-shadow:0 1px 2px rgba(0,0,0,.1)}
+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
+.verseblock{margin:0 1em 1.25em}
+.verseblock pre{font-family:"Open Sans","DejaVu Sans",sans-serif;font-size:1.15rem;color:rgba(0,0,0,.85);font-weight:300;text-rendering:optimizeLegibility}
+.verseblock pre strong{font-weight:400}
+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Graphics</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphics/Rasterization.html;hb=HEAD">Rasterization</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphics/3D-Objects.html;hb=HEAD">3D-Objects</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphics/Quanternions.html;hb=HEAD">Quanternions</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Graphing/Parent-Functions.adoc b/Graphing/Parent-Functions.adoc
new file mode 100644 (file)
index 0000000..9c5539c
--- /dev/null
+++ b/Graphing/Parent-Functions.adoc
@@ -0,0 +1,144 @@
+= Parent Functions
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Constant Functions
+[.example]
+[stem]
+++++
+y = c
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[C] (constant)
+
+[.center]
+image::Constant-Functions.png[]
+
+
+== Linear Functions
+[.example]
+[stem]
+++++
+y = x
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[(-\infty,\infty)]
+
+[.center]
+image::Linear-Functions.png[]
+
+== Quadradic Functions
+[.example]
+[stem]
+++++
+y = x^2
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[[0,\infty)]
+
+[.center]
+image::Quadradic-Functions.png[]
+
+== Cubic Functions
+[.example]
+[stem]
+++++
+y = x^3
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[(-\infty,\infty)]
+
+[.center]
+image::Cubic-Functions.png[]
+
+== Absolute Value Functions
+[.example]
+[stem]
+++++
+y = |x|
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[[0,\infty)]
+
+[.center]
+image::Absolute-Value-Functions.png[]
+
+== Radical Functions
+=== Square Root
+[.example]
+[stem]
+++++
+y = \sqrt{x}
+++++
+
+Domain:: stem:[(0,\infty)]
+Range:: stem:[[0,\infty)]
+
+[.center]
+image::Radical-Square-Root-Functions.png[]
+
+=== Cubed Root
+[.example]
+[stem]
+++++
+y = \sqrt[3]{x}
+++++
+
+Domain:: stem:[(-\infty,\infty)]
+Range:: stem:[(-\infty),\infty)]
+
+[.center]
+image::Radical-Cubed-Root-Functions.png[]
+
+== Exponential Functions
+[.example]
+[stem]
+++++
+y = b^x
+++++
+
+Horizontal Asymbtote::: 0
+Domain:: stem:[(\infty,\infty)]
+Range:: stem:[(0,\infty)]
+
+[.center]
+image::Exponential-Functions.png[]
+
+== Logoritmic Functions
+[.example]
+[stem]
+++++
+y = log_{b} x
+++++
+
+Vertical Asymbtote:: 0
+Domain:: stem:[(0,\infty)]
+Range:: stem:[(-\infty,\infty)]
+
+[.center]
+image::Logorithmic-Functions.png[]
+
+== Reciprocal Functions
+[.example]
+[stem]
+++++
+y = \frac{1}{x}
+++++
+
+Vertical Asymptote:: 0
+Domain:: stem:[(-\infty, 0) \cup (0, \infty)]
+Range:: stem:[(-\infty, 0) \cup (0, \infty)]
+
+[.center]
+image::Reciprocal-Functions.png[]
diff --git a/Graphing/Parent-Functions.html b/Graphing/Parent-Functions.html
new file mode 100644 (file)
index 0000000..8cc4d29
--- /dev/null
+++ b/Graphing/Parent-Functions.html
@@ -0,0 +1,1370 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Parent Functions</title>
+<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700">
+<style>
+/*! Asciidoctor default stylesheet | MIT License | https://asciidoctor.org */
+/* Uncomment the following line when using as a custom stylesheet */
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+html{font-family:sans-serif;-webkit-text-size-adjust:100%}
+a{background:none}
+a:focus{outline:thin dotted}
+a:active,a:hover{outline:0}
+h1{font-size:2em;margin:.67em 0}
+b,strong{font-weight:bold}
+abbr{font-size:.9em}
+abbr[title]{cursor:help;border-bottom:1px dotted #dddddf;text-decoration:none}
+dfn{font-style:italic}
+hr{height:0}
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+code,kbd,pre,samp{font-family:monospace;font-size:1em}
+pre{white-space:pre-wrap}
+q{quotes:"\201C" "\201D" "\2018" "\2019"}
+small{font-size:80%}
+sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}
+sup{top:-.5em}
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+svg:not(:root){overflow:hidden}
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+button,input,select,textarea{font-family:inherit;font-size:100%;margin:0}
+button,input{line-height:normal}
+button,select{text-transform:none}
+button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}
+button[disabled],html input[disabled]{cursor:default}
+input[type=checkbox],input[type=radio]{padding:0}
+button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}
+textarea{overflow:auto;vertical-align:top}
+table{border-collapse:collapse;border-spacing:0}
+*,::before,::after{box-sizing:border-box}
+html,body{font-size:100%}
+body{background:#fff;color:rgba(0,0,0,.8);padding:0;margin:0;font-family:"Noto Serif","DejaVu Serif",serif;line-height:1;position:relative;cursor:auto;-moz-tab-size:4;-o-tab-size:4;tab-size:4;word-wrap:anywhere;-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased}
+a:hover{cursor:pointer}
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+object,embed{height:100%}
+img{-ms-interpolation-mode:bicubic}
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+.text-justify{text-align:justify!important}
+.hide{display:none}
+img,object,svg{display:inline-block;vertical-align:middle}
+textarea{height:auto;min-height:50px}
+select{width:100%}
+.subheader,.admonitionblock td.content>.title,.audioblock>.title,.exampleblock>.title,.imageblock>.title,.listingblock>.title,.literalblock>.title,.stemblock>.title,.openblock>.title,.paragraph>.title,.quoteblock>.title,table.tableblock>.title,.verseblock>.title,.videoblock>.title,.dlist>.title,.olist>.title,.ulist>.title,.qlist>.title,.hdlist>.title{line-height:1.45;color:#7a2518;font-weight:400;margin-top:0;margin-bottom:.25em}
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+a{color:#2156a5;text-decoration:underline;line-height:inherit}
+a:hover,a:focus{color:#1d4b8f}
+a img{border:0}
+p{line-height:1.6;margin-bottom:1.25em;text-rendering:optimizeLegibility}
+p aside{font-size:.875em;line-height:1.35;font-style:italic}
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+h1 small,h2 small,h3 small,#toctitle small,.sidebarblock>.content>.title small,h4 small,h5 small,h6 small{font-size:60%;color:#e99b8f;line-height:0}
+h1{font-size:2.125em}
+h2{font-size:1.6875em}
+h3,#toctitle,.sidebarblock>.content>.title{font-size:1.375em}
+h4,h5{font-size:1.125em}
+h6{font-size:1em}
+hr{border:solid #dddddf;border-width:1px 0 0;clear:both;margin:1.25em 0 1.1875em}
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+blockquote,blockquote p{line-height:1.6;color:rgba(0,0,0,.85)}
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+h1{font-size:2.75em}
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+table{background:#fff;margin-bottom:1.25em;border:1px solid #dedede;word-wrap:normal}
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+table tr th,table tr td{padding:.5625em .625em;font-size:inherit;color:rgba(0,0,0,.8)}
+table tr.even,table tr.alt{background:#f8f8f7}
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+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Parent Functions</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_constant_functions">Constant Functions</a></li>
+<li><a href="#_linear_functions">Linear Functions</a></li>
+<li><a href="#_quadradic_functions">Quadradic Functions</a></li>
+<li><a href="#_cubic_functions">Cubic Functions</a></li>
+<li><a href="#_absolute_value_functions">Absolute Value Functions</a></li>
+<li><a href="#_radical_functions">Radical Functions</a>
+<ul class="sectlevel2">
+<li><a href="#_square_root">Square Root</a></li>
+<li><a href="#_cubed_root">Cubed Root</a></li>
+</ul>
+</li>
+<li><a href="#_exponential_functions">Exponential Functions</a></li>
+<li><a href="#_logoritmic_functions">Logoritmic Functions</a></li>
+<li><a href="#_reciprocal_functions">Reciprocal Functions</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_constant_functions">Constant Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = c\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\(C\) (constant)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Constant-Functions.png" alt="Constant Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_linear_functions">Linear Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = x\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Linear-Functions.png" alt="Linear Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_quadradic_functions">Quadradic Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = x^2\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\([0,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Quadradic-Functions.png" alt="Quadradic Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_cubic_functions">Cubic Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = x^3\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Cubic-Functions.png" alt="Cubic Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_absolute_value_functions">Absolute Value Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = |x|\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\([0,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Absolute-Value-Functions.png" alt="Absolute Value Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_radical_functions">Radical Functions</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_square_root">Square Root</h3>
+<div class="stemblock example">
+<div class="content">
+\[y = \sqrt{x}\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((0,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\([0,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Radical-Square-Root-Functions.png" alt="Radical Square Root Functions">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_cubed_root">Cubed Root</h3>
+<div class="stemblock example">
+<div class="content">
+\[y = \sqrt[3]{x}\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((-\infty),\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Radical-Cubed-Root-Functions.png" alt="Radical Cubed Root Functions">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_exponential_functions">Exponential Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = b^x\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Horizontal Asymbtote</dt>
+<dd>
+<p>0</p>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((\infty,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((0,\infty)\)</p>
+</dd>
+</dl>
+</div>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Exponential-Functions.png" alt="Exponential Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_logoritmic_functions">Logoritmic Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = log_{b} x\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Vertical Asymbtote</dt>
+<dd>
+<p>0</p>
+</dd>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((0,\infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((-\infty,\infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Logorithmic-Functions.png" alt="Logorithmic Functions">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_reciprocal_functions">Reciprocal Functions</h2>
+<div class="sectionbody">
+<div class="stemblock example">
+<div class="content">
+\[y = \frac{1}{x}\]
+</div>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Vertical Asymptote</dt>
+<dd>
+<p>0</p>
+</dd>
+<dt class="hdlist1">Domain</dt>
+<dd>
+<p>\((-\infty, 0) \cup (0, \infty)\)</p>
+</dd>
+<dt class="hdlist1">Range</dt>
+<dd>
+<p>\((-\infty, 0) \cup (0, \infty)\)</p>
+</dd>
+</dl>
+</div>
+<div class="imageblock center center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Reciprocal-Functions.png" alt="Reciprocal Functions">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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--- /dev/null
+++ b/Graphing/index.adoc
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+= ./Graphing
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphing/Parent-Functions.html;hb=HEAD[Parent-Functions]
diff --git a/Graphing/index.html b/Graphing/index.html
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+
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+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Graphing</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphing/Parent-Functions.html;hb=HEAD">Parent-Functions</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Linear-Algrebra/Graphing.adoc b/Linear-Algrebra/Graphing.adoc
new file mode 100644 (file)
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+++ b/Linear-Algrebra/Graphing.adoc
@@ -0,0 +1,27 @@
+= Linear Algrebra | Graphing
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+=== Three Equations
+
+== Exactly One Soltion (Independent)
+
+In a System of three equations, if three planes intersect at a single unique
+point then there is only one soution. (like the corner of a room where two walls
+meet the ceiling).
+
+== Span
+
+In linear algebra, the span of a set of vectors is the collection of all
+possible vectors you can create by scaling and adding them together (their
+linear combinations).
+
+== Links
+
+* https://math.libretexts.org/Courses/University_of_California_Irvine/Linear_Algebra_(Math_3A_UCI)/01%3A_Systems_of_Linear_Equations/1.01%3A_Systems_of_Linear_Equations/1.1.01%3A_Lines_Planes_and_Space
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+.quoteblock blockquote::before{content:"\201c";float:left;font-size:2.75em;font-weight:bold;line-height:.6em;margin-left:-.6em;color:#7a2518;text-shadow:0 1px 2px rgba(0,0,0,.1)}
+.quoteblock blockquote>.paragraph:last-child p{margin-bottom:0}
+.quoteblock .attribution{margin-top:.75em;margin-right:.5ex;text-align:right}
+.verseblock{margin:0 1em 1.25em}
+.verseblock pre{font-family:"Open Sans","DejaVu Sans",sans-serif;font-size:1.15rem;color:rgba(0,0,0,.85);font-weight:300;text-rendering:optimizeLegibility}
+.verseblock pre strong{font-weight:400}
+.verseblock .attribution{margin-top:1.25rem;margin-left:.5ex}
+.quoteblock .attribution,.verseblock .attribution{font-size:.9375em;line-height:1.45;font-style:italic}
+.quoteblock .attribution br,.verseblock .attribution br{display:none}
+.quoteblock .attribution cite,.verseblock .attribution cite{display:block;letter-spacing:-.025em;color:rgba(0,0,0,.6)}
+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
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+        scrollPosition: scrollPosition
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+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Graphing</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel2">
+<li><a href="#_three_equations">Three Equations</a></li>
+<li><a href="#_exactly_one_soltion_independent">Exactly One Soltion (Independent)</a></li>
+<li><a href="#_span">Span</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect2">
+<h3 id="_three_equations">Three Equations</h3>
+
+</div>
+<div class="sect1">
+<h2 id="_exactly_one_soltion_independent">Exactly One Soltion (Independent)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In a System of three equations, if three planes intersect at a single unique
+point then there is only one soution. (like the corner of a room where two walls
+meet the ceiling).</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_span">Span</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In linear algebra, the span of a set of vectors is the collection of all
+possible vectors you can create by scaling and adding them together (their
+linear combinations).</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://math.libretexts.org/Courses/University_of_California_Irvine/Linear_Algebra_(Math_3A_UCI)/01%3A_Systems_of_Linear_Equations/1.01%3A_Systems_of_Linear_Equations/1.1.01%3A_Lines_Planes_and_Space" class="bare">https://math.libretexts.org/Courses/University_of_California_Irvine/Linear_Algebra_(Math_3A_UCI)/01%3A_Systems_of_Linear_Equations/1.01%3A_Systems_of_Linear_Equations/1.1.01%3A_Lines_Planes_and_Space</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+</html>
\ No newline at end of file
diff --git a/Linear-Algrebra/Matrix-Math.adoc b/Linear-Algrebra/Matrix-Math.adoc
new file mode 100644 (file)
index 0000000..ef106ac
--- /dev/null
+++ b/Linear-Algrebra/Matrix-Math.adoc
@@ -0,0 +1,110 @@
+= Linear Algrebra | Matrix Math
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+
+== Matrix Multiplication
+
+[stem]
+++++
+A \otimes B = C
+\\
+A = 
+\left[\begin{array}{a}
+a_{11} & a_{12} \\
+a_{21} & a_{22} \\
+\end{array}\right]
+
+B = 
+\left[\begin{array}{b}
+b_{11} & b_{12} & b_{13} \\
+b_{21} & b_{22} & b_{23} \\
+\end{array}\right]
+\\
+C =
+\\
+\left[\begin{array}{cc}
+(a_{11} * b_{11}) + (a_{12} * b_{21}), & (a_{11} * b_{12}) + (a_{12} * b_{22}), & (a_{11} * b_{13}) + (a_{12} * b_{23}) \\
+(a_{21} * b_{11}) + (a_{22} * b_{21}), & (a_{21} * b_{12}) + (a_{22} * b_{22}), & (a_{21} * b_{13}) + (a_{22} * b_{23})
+\end{array}\right]
+++++
+
+
+.Alternative Matrix Multiplication
+[%collapsible]
+====
+[.center]
+image::alternative-matrix-multiplication.png[]
+====
+
+=== Demension Rules
+
+[stem]
+++++
+B = p \times q
+\\
+A = m \times n
+++++
+
+stem:[n] has to be equal to stem:[p]. The demensions of stem:[C] is stem:[m \times q]
+
+== Matrix and Scalar Multiplication
+
+[stem]
+++++
+k \left[\begin{array}{cc}
+4 & 3 \\
+9 & 3 \\
+\end{array}\right] = 
+\left[\begin{array}{cc}
+k4 & k3 \\
+k9 & k3 \\
+\end{array}\right]
+++++
+
+== Matrix addition
+
+When you add or subtract matrices they need to be the same dimensions
+
+[stem]
+++++
+\left[\begin{array}{cc}
+4 & 3 &  5 \\
+2 & 1 & -4 \\
+1 & 2 &  2 \\
+\end{array}\right] +
+
+\left[\begin{array}{cc}
+1 & 2 &  3 \\
+4 & 5 & -6 \\
+8 & 4 &  2 \\
+\end{array}\right] =
+
+\left[\begin{array}{cc}
+4 + 1 & 3 + 2 &  5 +  3 \\
+2 + 4 & 1 + 4 & -4 + -6 \\
+1 + 8 & 2 + 4 &  2 +  2 \\
+\end{array}\right]
+++++
+
+== Three-Dimensional Transformations
+
+[.center]
+image::3d-vector-transformation.png[]
+
+[.center]
+image::3d-vecctor-transformation2.png[]
+
+== Links
+* https://www.youtube.com/watch?v=pJzmiywagfY[Khan Academy | Cross Product]
+* https://www.youtube.com/watch?v=eu6i7WJeinw[3Blue 1 Brown | Cross Product]
+* https://www.youtube.com/watch?v=Ip3X9LOh2dk[3Blue 1 Brown | Determinant]
+* https://youtu.be/XkY2DOUCWMU?si=GHdQZPG1qktUV1f2[3Blue 1Brown | Matrix Multiplication]
+* https://www.youtube.com/watch?v=rHLEWRxRGiM&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=6[3Blue 1Brown | Three-dimensional linear transofrmation]
+* https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html
diff --git a/Linear-Algrebra/Matrix-Math.html b/Linear-Algrebra/Matrix-Math.html
new file mode 100644 (file)
index 0000000..300184c
--- /dev/null
+++ b/Linear-Algrebra/Matrix-Math.html
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+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
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+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Matrix Math</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_matrix_multiplication">Matrix Multiplication</a>
+<ul class="sectlevel2">
+<li><a href="#_demension_rules">Demension Rules</a></li>
+</ul>
+</li>
+<li><a href="#_matrix_and_scalar_multiplication">Matrix and Scalar Multiplication</a></li>
+<li><a href="#_matrix_addition">Matrix addition</a></li>
+<li><a href="#_three_dimensional_transformations">Three-Dimensional Transformations</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_matrix_multiplication">Matrix Multiplication</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[A \otimes B = C
+\\
+A =
+\left[\begin{array}{a}
+a_{11} &amp; a_{12} \\
+a_{21} &amp; a_{22} \\
+\end{array}\right]
+
+B =
+\left[\begin{array}{b}
+b_{11} &amp; b_{12} &amp; b_{13} \\
+b_{21} &amp; b_{22} &amp; b_{23} \\
+\end{array}\right]
+\\
+C =
+\\
+\left[\begin{array}{cc}
+(a_{11} * b_{11}) + (a_{12} * b_{21}), &amp; (a_{11} * b_{12}) + (a_{12} * b_{22}), &amp; (a_{11} * b_{13}) + (a_{12} * b_{23}) \\
+(a_{21} * b_{11}) + (a_{22} * b_{21}), &amp; (a_{21} * b_{12}) + (a_{22} * b_{22}), &amp; (a_{21} * b_{13}) + (a_{22} * b_{23})
+\end{array}\right]\]
+</div>
+</div>
+<details>
+<summary class="title">Alternative Matrix Multiplication</summary>
+<div class="content">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/alternative-matrix-multiplication.png" alt="alternative matrix multiplication">
+</div>
+</div>
+</div>
+</details>
+<div class="sect2">
+<h3 id="_demension_rules">Demension Rules</h3>
+<div class="stemblock">
+<div class="content">
+\[B = p \times q
+\\
+A = m \times n\]
+</div>
+</div>
+<div class="paragraph">
+<p>\(n\) has to be equal to \(p\). The demensions of \(C\) is \(m \times q\)</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_matrix_and_scalar_multiplication">Matrix and Scalar Multiplication</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[k \left[\begin{array}{cc}
+4 &amp; 3 \\
+9 &amp; 3 \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+k4 &amp; k3 \\
+k9 &amp; k3 \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_matrix_addition">Matrix addition</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When you add or subtract matrices they need to be the same dimensions</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{cc}
+4 &amp; 3 &amp;  5 \\
+2 &amp; 1 &amp; -4 \\
+1 &amp; 2 &amp;  2 \\
+\end{array}\right] +
+
+\left[\begin{array}{cc}
+1 &amp; 2 &amp;  3 \\
+4 &amp; 5 &amp; -6 \\
+8 &amp; 4 &amp;  2 \\
+\end{array}\right] =
+
+\left[\begin{array}{cc}
+4 + 1 &amp; 3 + 2 &amp;  5 +  3 \\
+2 + 4 &amp; 1 + 4 &amp; -4 + -6 \\
+1 + 8 &amp; 2 + 4 &amp;  2 +  2 \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_three_dimensional_transformations">Three-Dimensional Transformations</h2>
+<div class="sectionbody">
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3d-vector-transformation.png" alt="3d vector transformation">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3d-vecctor-transformation2.png" alt="3d vecctor transformation2">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=pJzmiywagfY">Khan Academy | Cross Product</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=eu6i7WJeinw">3Blue 1 Brown | Cross Product</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=Ip3X9LOh2dk">3Blue 1 Brown | Determinant</a></p>
+</li>
+<li>
+<p><a href="https://youtu.be/XkY2DOUCWMU?si=GHdQZPG1qktUV1f2">3Blue 1Brown | Matrix Multiplication</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=rHLEWRxRGiM&amp;list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&amp;index=6">3Blue 1Brown | Three-dimensional linear transofrmation</a></p>
+</li>
+<li>
+<p><a href="https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html" class="bare">https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+    delimiters: [["\\$", "\\$"]],
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\ No newline at end of file
diff --git a/Linear-Algrebra/Matrix-Operations.adoc b/Linear-Algrebra/Matrix-Operations.adoc
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+= Linear Algrebra | Matrix Operations
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Determinants
+
+=== Solving 2x2 matrices
+
+[stem]
+++++
+\det(A) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc
+++++
+
+[.center]
+image::helpful-2x2-matrix-determinant-diagram.png[]
+
+=== Solving 3x3 matrices
+
+[stem]
+++++
+\det \begin{pmatrix} \begin{bmatrix}
+a & b & c \\
+d & e & f \\
+g & h & i
+\end{bmatrix} \end{pmatrix} = a \det \begin{pmatrix} \begin{bmatrix}
+e & f \\
+h & i
+\end{bmatrix} \end{pmatrix} - b \det \begin{pmatrix} \begin{bmatrix}
+d & f \\
+g & i
+\end{bmatrix} \end{pmatrix} + c \det \begin{pmatrix} \begin{bmatrix}
+d & e \\
+g & h
+\end{bmatrix} \end{pmatrix}
+++++
+
+[.center]
+image::rule-of-sarrus-3x3-matrices.png[]
+
+[.center]
+image::six-tetrahedra-parallepiped.png[]
+
+=== Two-Dimensions
+
+Determinants determine the scale of the area of how much the unit parallelogram is
+scaled.
+
+* *Positive Determinant*: stem:[\hat{i}] is right of stem:[\hat{j}].
+
+* *Negative Determinant*: stem:[\hat{i}] is left of stem:[\hat{j}].
+
+NOTE: You can think of negative determinants as inverting the oriantation of
+      space. (i.e. a reflection)
+
+=== Three-Dimensions
+
+Determinants determine the scale of the volume of how much the unit
+parallelepiped is scaled into. A parallelpiped is prism whose bases are
+parallelograms. (e.i. warped cubed).
+
+[.center]
+image::3d-determinant-area.png[]
+
+.Right hand | Positive determinant | unchanged orientation
+[.center]
+image::right-hand-rule2.png[]
+
+.Left hand | Negative determinant, changed orientation
+[.center]
+image::right-hand-rule-flipped.png[]
+
+== Eigenvalues and Eigenvectors
+
+== Transposing a matrix
+
+.Matrix Transpose Property
+[stem]
+++++
+A = \begin{bmatrix}
+a & b & c \\
+d & e & f
+\end{bmatrix}, \quad
+A^T = \begin{bmatrix}
+a & d \\
+b & e \\
+c & f
+\end{bmatrix}
+++++
+
+== Getting an Identity Matrix with Multiplication
+
+When a matrix is multiplied by an inverse matrix the result is an identity
+matrix
+
+== Row Operations to get a Row-Equivalent Matrix
+
+To change a Matrix to a Row-Equivalent Matrix, you can:
+
+* Interchange any two rows
+
+* Multiply all the elements in a row by a real number (not 0).
+
+* Add multiples of the elements of one row the the elements of another row.
+
+== Row Notation Examples
+
+[.inline]
+* *stem:[R_1 \leftrightarrow R_4]*: replace one row with the other
+
+[.inline]
+* *stem:[\frac{1}{5}R_3 \leftrightarrow R_3]*: One-fith of each element in row 3
+  becomes the new row 3.
+
+[.inline]
+* *stem:[-5R_1 + R_2 \rightarrow R_2]*: Negative 5 times each element in row 1
+  added to the elements in row 2 produces a new row 2.
+
+== Getting Inverse Matrix
+
+=== 2x2 Method
+
+[stem]
+++++
+M = \left[\begin{array}{cc}
+a & b \\
+c & d \\
+\end{array}\right]
+++++
+
+[stem]
+++++
+M^{-1} = 
+\left[\begin{array}{cc}
+\frac{d}{ab - bc} & \frac{-b}{ad - bc} \\
+\frac{-c}{ad - bc} & \frac{a}{ad - bc} \\
+\end{array}\right]
+++++
+
+If stem:[ad - bc] is zero, then the matrix has no inverse
+
+Many square matrices have inverses. When a matrix and its inverse are multiplied
+together, in either order, the result is an identity matrix. Matrix inverses are
+used to solve problems involving systems of equations and to perform matrix
+_division_.
+
+=== Row Reduction Method for Square Matrices
+
+Create an identity matrix on the right. We go throught the row operations to
+transform the left-hand matrix into an identity matix. When You're finished, the
+matrix on teh right is the original matrix's inverse.
+
+.example
+[example%collapsible]
+====
+
+.Row Operation #3
+[stem]
+++++
+D = \left[
+\begin{array}{ccc|ccc}
+ 1 & 2 & 0 & 1 & 0 & 0 \\
+-1 & 3 & 3 & 0 & 1 & 0 \\
+ 0 & 1 & 1 & 0 & 0 & 1 \\
+\end{array}
+\right]
+++++
+
+.Row Operation #2
+[stem]
+++++
+R_1 + R_2 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 & 2 & 0 & 1 & 0 & 0 \\
+ 0 & 5 & 3 & 1 & 1 & 0 \\
+ 0 & 1 & 1 & 0 & 0 & 1 \\
+\end{array}
+\right]
+++++
+
+.Row Operation #3
+[stem]
+++++
+-R_2 + R_1 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 & 0 & -\frac{6}{5} &  \frac{3}{5} & -\frac{2}{5} & 0 \\
+ 0 & 1 &  \frac{3}{5} &  \frac{1}{5} &  \frac{1}{5} & 0 \\
+ 0 & 0 &  \frac{2}{5} & -\frac{1}{5} & -\frac{1}{5} & 1 \\
+\end{array}
+\right]
+++++
+
+.Row Operation #2
+[stem]
+++++
+\frac{5}{2}R_3 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 & 0 & -\frac{6}{5} &  \frac{3}{5} & -\frac{2}{5} & 0 \\
+ 0 & 1 &  \frac{3}{5} &  \frac{1}{5} &  \frac{1}{5} & 0 \\
+ 0 & 0 &  1           & -\frac{1}{2} & -\frac{1}{2} & \frac{5}{2} \\
+\end{array}
+\right]
+++++
+
+.Row Operation #3
+[stem]
+++++
+\frac{6}{5}R_3 + R_1 \rightarrow R_1 \left[
+\begin{array}{ccc|ccc}
+ 1 & 0 &  0           &  0           & -1           & 0 \\
+ 0 & 1 &  \frac{3}{5} &  \frac{1}{5} &  \frac{1}{5} & 0 \\
+ 0 & 0 &  1           & -\frac{1}{2} & -\frac{1}{2} & \frac{5}{2} \\
+\end{array}
+\right]
+++++
+
+.Row Operation #3
+[stem]
+++++
+\frac{6}{5}R_3 + R_1 \rightarrow R_1 \left[
+\begin{array}{ccc|ccc}
+ 1 & 0 & 0 &  0           & -1           & 3 \\
+ 0 & 1 & 0 &  \frac{1}{2} &  \frac{1}{2} & -\frac{3}{2} \\
+ 0 & 0 & 1 & -\frac{1}{2} & -\frac{1}{2} & \frac{5}{2} \\
+\end{array}
+\right]
+++++
+
+You now have the identity matrix on the left and the inverse matrix
+(stem:[D^{-1}]) on the right. stem:[D * D^{-1}] and stem:[D^{-1} * D] both equal
+the identity matrix.
+
+====
+
+== Shear
+
+=== Horizontal Shear
+
+[.center]
+image::horizontal-shear.png[]
+
+[stem]
+++++
+\left[\begin{array}{cc}
+x'\\
+y'\\
+\end{array}\right] = 
+\left[\begin{array}{cc}
+1 & k\\
+0 & 1\\
+\end{array}\right]
+\left[\begin{array}{cc}
+x\\
+y\\
+\end{array}\right]
+++++
+
+* stem:[x' = x + ky]
+
+* stem:[y' = y]
+
+=== Vertical Shear
+
+[.center]
+image::vertical-shear.png[]
+
+[stem]
+++++
+\left[\begin{array}{cc}
+x'\\
+y'\\
+\end{array}\right] = 
+\left[\begin{array}{cc}
+1 & 0\\
+k & 1\\
+\end{array}\right]
+\left[\begin{array}{cc}
+x\\
+y\\
+\end{array}\right]
+++++
+
+* stem:[x' = x]
+
+* stem:[y' = y + kx]
diff --git a/Linear-Algrebra/Matrix-Operations.html b/Linear-Algrebra/Matrix-Operations.html
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+<!DOCTYPE html>
+<html lang="en">
+<head>
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+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
+.quoteblock.abstract{margin:0 1em 1.25em;display:block}
+.quoteblock.abstract>.title{margin:0 0 .375em;font-size:1.15em;text-align:center}
+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Matrix Operations</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_determinants">Determinants</a>
+<ul class="sectlevel2">
+<li><a href="#_solving_2x2_matrices">Solving 2x2 matrices</a></li>
+<li><a href="#_solving_3x3_matrices">Solving 3x3 matrices</a></li>
+<li><a href="#_two_dimensions">Two-Dimensions</a></li>
+<li><a href="#_three_dimensions">Three-Dimensions</a></li>
+</ul>
+</li>
+<li><a href="#_eigenvalues_and_eigenvectors">Eigenvalues and Eigenvectors</a></li>
+<li><a href="#_transposing_a_matrix">Transposing a matrix</a></li>
+<li><a href="#_getting_an_identity_matrix_with_multiplication">Getting an Identity Matrix with Multiplication</a></li>
+<li><a href="#_row_operations_to_get_a_row_equivalent_matrix">Row Operations to get a Row-Equivalent Matrix</a></li>
+<li><a href="#_row_notation_examples">Row Notation Examples</a></li>
+<li><a href="#_getting_inverse_matrix">Getting Inverse Matrix</a>
+<ul class="sectlevel2">
+<li><a href="#_2x2_method">2x2 Method</a></li>
+<li><a href="#_row_reduction_method_for_square_matrices">Row Reduction Method for Square Matrices</a></li>
+</ul>
+</li>
+<li><a href="#_shear">Shear</a>
+<ul class="sectlevel2">
+<li><a href="#_horizontal_shear">Horizontal Shear</a></li>
+<li><a href="#_vertical_shear">Vertical Shear</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_determinants">Determinants</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_solving_2x2_matrices">Solving 2x2 matrices</h3>
+<div class="stemblock">
+<div class="content">
+\[\det(A) = \begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix} = ad - bc\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/helpful-2x2-matrix-determinant-diagram.png" alt="helpful 2x2 matrix determinant diagram">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_solving_3x3_matrices">Solving 3x3 matrices</h3>
+<div class="stemblock">
+<div class="content">
+\[\det \begin{pmatrix} \begin{bmatrix}
+a &amp; b &amp; c \\
+d &amp; e &amp; f \\
+g &amp; h &amp; i
+\end{bmatrix} \end{pmatrix} = a \det \begin{pmatrix} \begin{bmatrix}
+e &amp; f \\
+h &amp; i
+\end{bmatrix} \end{pmatrix} - b \det \begin{pmatrix} \begin{bmatrix}
+d &amp; f \\
+g &amp; i
+\end{bmatrix} \end{pmatrix} + c \det \begin{pmatrix} \begin{bmatrix}
+d &amp; e \\
+g &amp; h
+\end{bmatrix} \end{pmatrix}\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/rule-of-sarrus-3x3-matrices.png" alt="rule of sarrus 3x3 matrices">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/six-tetrahedra-parallepiped.png" alt="six tetrahedra parallepiped">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_two_dimensions">Two-Dimensions</h3>
+<div class="paragraph">
+<p>Determinants determine the scale of the area of how much the unit parallelogram is
+scaled.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Positive Determinant</strong>: \(\hat{i}\) is right of \(\hat{j}\).</p>
+</li>
+<li>
+<p><strong>Negative Determinant</strong>: \(\hat{i}\) is left of \(\hat{j}\).</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+You can think of negative determinants as inverting the oriantation of
+      space. (i.e. a reflection)
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_three_dimensions">Three-Dimensions</h3>
+<div class="paragraph">
+<p>Determinants determine the scale of the volume of how much the unit
+parallelepiped is scaled into. A parallelpiped is prism whose bases are
+parallelograms. (e.i. warped cubed).</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3d-determinant-area.png" alt="3d determinant area">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/right-hand-rule2.png" alt="right hand rule2">
+</div>
+<div class="title">Figure 1. Right hand | Positive determinant | unchanged orientation</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/right-hand-rule-flipped.png" alt="right hand rule flipped">
+</div>
+<div class="title">Figure 2. Left hand | Negative determinant, changed orientation</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_eigenvalues_and_eigenvectors">Eigenvalues and Eigenvectors</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+<div class="sect1">
+<h2 id="_transposing_a_matrix">Transposing a matrix</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="title">Matrix Transpose Property</div>
+<div class="content">
+\[A = \begin{bmatrix}
+a &amp; b &amp; c \\
+d &amp; e &amp; f
+\end{bmatrix}, \quad
+A^T = \begin{bmatrix}
+a &amp; d \\
+b &amp; e \\
+c &amp; f
+\end{bmatrix}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_getting_an_identity_matrix_with_multiplication">Getting an Identity Matrix with Multiplication</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When a matrix is multiplied by an inverse matrix the result is an identity
+matrix</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_row_operations_to_get_a_row_equivalent_matrix">Row Operations to get a Row-Equivalent Matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>To change a Matrix to a Row-Equivalent Matrix, you can:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Interchange any two rows</p>
+</li>
+<li>
+<p>Multiply all the elements in a row by a real number (not 0).</p>
+</li>
+<li>
+<p>Add multiples of the elements of one row the the elements of another row.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_row_notation_examples">Row Notation Examples</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>\(R_1 \leftrightarrow R_4\)</strong>: replace one row with the other</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>\(\frac{1}{5}R_3 \leftrightarrow R_3\)</strong>: One-fith of each element in row 3
+becomes the new row 3.</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>\(-5R_1 + R_2 \rightarrow R_2\)</strong>: Negative 5 times each element in row 1
+added to the elements in row 2 produces a new row 2.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_getting_inverse_matrix">Getting Inverse Matrix</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_2x2_method">2x2 Method</h3>
+<div class="stemblock">
+<div class="content">
+\[M = \left[\begin{array}{cc}
+a &amp; b \\
+c &amp; d \\
+\end{array}\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[M^{-1} =
+\left[\begin{array}{cc}
+\frac{d}{ab - bc} &amp; \frac{-b}{ad - bc} \\
+\frac{-c}{ad - bc} &amp; \frac{a}{ad - bc} \\
+\end{array}\right]\]
+</div>
+</div>
+<div class="paragraph">
+<p>If \(ad - bc\) is zero, then the matrix has no inverse</p>
+</div>
+<div class="paragraph">
+<p>Many square matrices have inverses. When a matrix and its inverse are multiplied
+together, in either order, the result is an identity matrix. Matrix inverses are
+used to solve problems involving systems of equations and to perform matrix
+<em>division</em>.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_row_reduction_method_for_square_matrices">Row Reduction Method for Square Matrices</h3>
+<div class="paragraph">
+<p>Create an identity matrix on the right. We go throught the row operations to
+transform the left-hand matrix into an identity matix. When You&#8217;re finished, the
+matrix on teh right is the original matrix&#8217;s inverse.</p>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="stemblock">
+<div class="title">Row Operation #3</div>
+<div class="content">
+\[D = \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 2 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
+-1 &amp; 3 &amp; 3 &amp; 0 &amp; 1 &amp; 0 \\
+ 0 &amp; 1 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Row Operation #2</div>
+<div class="content">
+\[R_1 + R_2 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 2 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
+ 0 &amp; 5 &amp; 3 &amp; 1 &amp; 1 &amp; 0 \\
+ 0 &amp; 1 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Row Operation #3</div>
+<div class="content">
+\[-R_2 + R_1 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 0 &amp; -\frac{6}{5} &amp;  \frac{3}{5} &amp; -\frac{2}{5} &amp; 0 \\
+ 0 &amp; 1 &amp;  \frac{3}{5} &amp;  \frac{1}{5} &amp;  \frac{1}{5} &amp; 0 \\
+ 0 &amp; 0 &amp;  \frac{2}{5} &amp; -\frac{1}{5} &amp; -\frac{1}{5} &amp; 1 \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Row Operation #2</div>
+<div class="content">
+\[\frac{5}{2}R_3 \rightarrow R_2 \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 0 &amp; -\frac{6}{5} &amp;  \frac{3}{5} &amp; -\frac{2}{5} &amp; 0 \\
+ 0 &amp; 1 &amp;  \frac{3}{5} &amp;  \frac{1}{5} &amp;  \frac{1}{5} &amp; 0 \\
+ 0 &amp; 0 &amp;  1           &amp; -\frac{1}{2} &amp; -\frac{1}{2} &amp; \frac{5}{2} \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Row Operation #3</div>
+<div class="content">
+\[\frac{6}{5}R_3 + R_1 \rightarrow R_1 \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 0 &amp;  0           &amp;  0           &amp; -1           &amp; 0 \\
+ 0 &amp; 1 &amp;  \frac{3}{5} &amp;  \frac{1}{5} &amp;  \frac{1}{5} &amp; 0 \\
+ 0 &amp; 0 &amp;  1           &amp; -\frac{1}{2} &amp; -\frac{1}{2} &amp; \frac{5}{2} \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="title">Row Operation #3</div>
+<div class="content">
+\[\frac{6}{5}R_3 + R_1 \rightarrow R_1 \left[
+\begin{array}{ccc|ccc}
+ 1 &amp; 0 &amp; 0 &amp;  0           &amp; -1           &amp; 3 \\
+ 0 &amp; 1 &amp; 0 &amp;  \frac{1}{2} &amp;  \frac{1}{2} &amp; -\frac{3}{2} \\
+ 0 &amp; 0 &amp; 1 &amp; -\frac{1}{2} &amp; -\frac{1}{2} &amp; \frac{5}{2} \\
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="paragraph">
+<p>You now have the identity matrix on the left and the inverse matrix
+(\(D^{-1}\)) on the right. \(D * D^{-1}\) and \(D^{-1} * D\) both equal
+the identity matrix.</p>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_shear">Shear</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_horizontal_shear">Horizontal Shear</h3>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/horizontal-shear.png" alt="horizontal shear">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{cc}
+x'\\
+y'\\
+\end{array}\right] =
+\left[\begin{array}{cc}
+1 &amp; k\\
+0 &amp; 1\\
+\end{array}\right]
+\left[\begin{array}{cc}
+x\\
+y\\
+\end{array}\right]\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>\(x' = x + ky\)</p>
+</li>
+<li>
+<p>\(y' = y\)</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_vertical_shear">Vertical Shear</h3>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/vertical-shear.png" alt="vertical shear">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{cc}
+x'\\
+y'\\
+\end{array}\right] =
+\left[\begin{array}{cc}
+1 &amp; 0\\
+k &amp; 1\\
+\end{array}\right]
+\left[\begin{array}{cc}
+x\\
+y\\
+\end{array}\right]\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>\(x' = x\)</p>
+</li>
+<li>
+<p>\(y' = y + kx\)</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
+  },
+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Linear-Algrebra/Matrix-Types.adoc b/Linear-Algrebra/Matrix-Types.adoc
new file mode 100644 (file)
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+= Linear Algrebra | Matrix Types
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Square Matrices
+
+[cols="1,2a", options="header"]
+|===
+| Matrix Type | Definition & Example
+
+| **Upper Triangular**
+| All the elements below the main diagonal are 0s.
+
+[stem]
+++++
+\begin{bmatrix}
+4 & 3 & 5 \\
+0 & 1 & -4 \\
+0 & 0 & 2 
+\end{bmatrix}
+++++
+
+| **Lower Triangular**
+| All the elements above the main diagonal are 0s.
+
+[stem]
+++++
+\begin{bmatrix}
+7 & 0 & 0 & 0 \\
+7 & 8 & 0 & 0 \\
+5 & 2 & 3 & 0 \\
+5 & -4 & -9 & 1
+\end{bmatrix}
+++++
+
+| **Diagonal**
+| All entries above and below the main diagonal are 0s.
+
+[stem]
+++++
+\begin{bmatrix}
+4 & 0 & 0 & 0 \\
+0 & 3 & 0 & 0 \\
+0 & 0 & 1 & 0 \\
+0 & 0 & 0 & 2
+\end{bmatrix}
+++++
+|===
+
+== Addative Identity Matrix
+
+The identity matix needs to be the same dimensions as the original.
+
+[stem]
+++++
+\left[\begin{array}{cc}
+4 & 3 &  5 \\
+2 & 1 & -4 \\
+1 & 2 &  2 \\
+\end{array}\right] \to
+
+\left[\begin{array}{cc}
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0 \\
+\end{array}\right]
+++++
+
+== Multiplicative Identity Matrix 
+
+One's are on the diagonal from top left to bottom right. Every other element is
+zero
+
+[stem]
+++++
+\left[\begin{array}{a}
+1 & 0 & 0 \\
+0 & 1 & 0 \\
+0 & 0 & 1 \\
+\end{array}\right]
+++++
+
+== Reduced Row Echelon Form and Rown Echelon Form
+
+[stem]
+++++
+\begin{array}{cc}
+    \text{Reduced Row Echelon Form} & \text{Row Echelon Form} \\
+    \begin{bmatrix}
+    1 & 0 & 0 &  3 & 4 \\
+    0 & 1 & 0 &  0 & 2 \\
+    0 & 0 & 1 & -1 & 1 \\
+    0 & 0 & 0 & 0  & 0
+    \end{bmatrix} & 
+    \begin{bmatrix}
+    1 & 3 & 0 &  2 & -3 \\
+    0 & 1 & 4 &  0 &  3  \\
+    0 & 0 & 1 &  1 &  4  \\
+    0 & 0 & 0 & 0  & 0
+    \end{bmatrix}
+\end{array}
+++++
+
+=== Reduced Row Echelon Form
+
+When a matrix is in the _reduced row echelon form_, it has the following
+characteristics:
+
+[.inline]
+* If the matrix has any rows whose elements are all 0x, those rows are at the
+  bottom of the matrix
+
+[.inline]
+* Reading from left to right, the first element of a row that is _not_ a 0 is
+  always a 1. This first 1 is called the _leading_ 1 of its row
+
+[.inline]
+* In any row, the leading 1 is always to the right of and below any leading 1 in
+  a row above.
+
+[.inline]
+* If a column contains a leading 1, then all the entries above and below that
+  leading 1 are 0s.
+
+=== Row Echelon Form
+
+The same as roe Echelon form except it doesn't require 0s above and below each
+leading 1.
+
+== Augmented matrix
+
+An augmented matrix is a compact way of representing a system of linear
+equations by combining the coefficients of the variables and the constant terms
+into a single matrix.
+
+[stem]
+++++
+\begin{cases}
+\begin{alignat*}{6}
+-x & {}-{} &  y & {}-{} & z  & =  3 \\
+4x & {}+{} & 5y &       &    & = -6 \\
+   &       &  y & {}-{} & 3z & = 5
+\end{alignat*}
+\end{cases}
+
+\qquad %
+
+\left[
+\begin{array}{ccc : c}
+-1 & -1 &  -1 &  3 \\
+ 4 &  5 &   0 & -6 \\
+ 0 &  1 &  -3 &  5
+\end{array}
+\right]
+++++
diff --git a/Linear-Algrebra/Matrix-Types.html b/Linear-Algrebra/Matrix-Types.html
new file mode 100644 (file)
index 0000000..5f3a6b5
--- /dev/null
+++ b/Linear-Algrebra/Matrix-Types.html
@@ -0,0 +1,1287 @@
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+    window.parent.postMessage({
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+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Matrix Types</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_square_matrices">Square Matrices</a></li>
+<li><a href="#_addative_identity_matrix">Addative Identity Matrix</a></li>
+<li><a href="#_multiplicative_identity_matrix">Multiplicative Identity Matrix</a></li>
+<li><a href="#_reduced_row_echelon_form_and_rown_echelon_form">Reduced Row Echelon Form and Rown Echelon Form</a>
+<ul class="sectlevel2">
+<li><a href="#_reduced_row_echelon_form">Reduced Row Echelon Form</a></li>
+<li><a href="#_row_echelon_form">Row Echelon Form</a></li>
+</ul>
+</li>
+<li><a href="#_augmented_matrix">Augmented matrix</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_square_matrices">Square Matrices</h2>
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 66.6667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Matrix Type</th>
+<th class="tableblock halign-left valign-top">Definition &amp; Example</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Upper Triangular</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>All the elements below the main diagonal are 0s.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{bmatrix}
+4 &amp; 3 &amp; 5 \\
+0 &amp; 1 &amp; -4 \\
+0 &amp; 0 &amp; 2
+\end{bmatrix}\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Lower Triangular</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>All the elements above the main diagonal are 0s.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{bmatrix}
+7 &amp; 0 &amp; 0 &amp; 0 \\
+7 &amp; 8 &amp; 0 &amp; 0 \\
+5 &amp; 2 &amp; 3 &amp; 0 \\
+5 &amp; -4 &amp; -9 &amp; 1
+\end{bmatrix}\]
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Diagonal</strong></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>All entries above and below the main diagonal are 0s.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{bmatrix}
+4 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 3 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 1 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 2
+\end{bmatrix}\]
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_addative_identity_matrix">Addative Identity Matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The identity matix needs to be the same dimensions as the original.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{cc}
+4 &amp; 3 &amp;  5 \\
+2 &amp; 1 &amp; -4 \\
+1 &amp; 2 &amp;  2 \\
+\end{array}\right] \to
+
+\left[\begin{array}{cc}
+0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_multiplicative_identity_matrix">Multiplicative Identity Matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>One&#8217;s are on the diagonal from top left to bottom right. Every other element is
+zero</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{a}
+1 &amp; 0 &amp; 0 \\
+0 &amp; 1 &amp; 0 \\
+0 &amp; 0 &amp; 1 \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_reduced_row_echelon_form_and_rown_echelon_form">Reduced Row Echelon Form and Rown Echelon Form</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{cc}
+    \text{Reduced Row Echelon Form} &amp; \text{Row Echelon Form} \\
+    \begin{bmatrix}
+    1 &amp; 0 &amp; 0 &amp;  3 &amp; 4 \\
+    0 &amp; 1 &amp; 0 &amp;  0 &amp; 2 \\
+    0 &amp; 0 &amp; 1 &amp; -1 &amp; 1 \\
+    0 &amp; 0 &amp; 0 &amp; 0  &amp; 0
+    \end{bmatrix} &amp;
+    \begin{bmatrix}
+    1 &amp; 3 &amp; 0 &amp;  2 &amp; -3 \\
+    0 &amp; 1 &amp; 4 &amp;  0 &amp;  3  \\
+    0 &amp; 0 &amp; 1 &amp;  1 &amp;  4  \\
+    0 &amp; 0 &amp; 0 &amp; 0  &amp; 0
+    \end{bmatrix}
+\end{array}\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_reduced_row_echelon_form">Reduced Row Echelon Form</h3>
+<div class="paragraph">
+<p>When a matrix is in the <em>reduced row echelon form</em>, it has the following
+characteristics:</p>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>If the matrix has any rows whose elements are all 0x, those rows are at the
+bottom of the matrix</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>Reading from left to right, the first element of a row that is <em>not</em> a 0 is
+always a 1. This first 1 is called the <em>leading</em> 1 of its row</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>In any row, the leading 1 is always to the right of and below any leading 1 in
+a row above.</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>If a column contains a leading 1, then all the entries above and below that
+leading 1 are 0s.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_row_echelon_form">Row Echelon Form</h3>
+<div class="paragraph">
+<p>The same as roe Echelon form except it doesn&#8217;t require 0s above and below each
+leading 1.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_augmented_matrix">Augmented matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>An augmented matrix is a compact way of representing a system of linear
+equations by combining the coefficients of the variables and the constant terms
+into a single matrix.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{6}
+-x &amp; {}-{} &amp;  y &amp; {}-{} &amp; z  &amp; =  3 \\
+4x &amp; {}+{} &amp; 5y &amp;       &amp;    &amp; = -6 \\
+   &amp;       &amp;  y &amp; {}-{} &amp; 3z &amp; = 5
+\end{alignat*}
+\end{cases}
+
+\qquad %
+
+\left[
+\begin{array}{ccc : c}
+-1 &amp; -1 &amp;  -1 &amp;  3 \\
+ 4 &amp;  5 &amp;   0 &amp; -6 \\
+ 0 &amp;  1 &amp;  -3 &amp;  5
+\end{array}
+\right]\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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\ No newline at end of file
diff --git a/Linear-Algrebra/Properties.adoc b/Linear-Algrebra/Properties.adoc
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+= Linear Algrebra | Properties
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Properties
+
+[cols="^.^1, ^.^1, ^.^1, <.^2", frame="none", grid="rows", options="header"]
+|===
+^| *Property* ^| *Parts* ^| *Definition* ^| *Details*
+
+| *Conditionally Commutative* | matrices
+| stem:[A B \neq B A]
+| Matrix multiplication is generally *not* commutative. Exceptions include the
+  identity matrix stem:[I], the inverse stem:[A^{-1}], or a zero matrix.
+
+| *Associative* | matrices
+| stem:[(AB)C = A(BC)]
+| Matrix multiplication is associative. As long as the order stays the same, you
+can group them however you like.
+
+| *Commutative* | scalar and matrices
+| stem:[kA = Ak]
+| Scalar multiplication is commutative—you can place the scalar stem:[k] in
+front or behind the matrix.
+
+| *Mixed Associative* | Scalar and Matrices
+| stem:[(kA)B = k(AB)]
+| You can multiply a matrix by a scalar first, or multiply the matrices first
+  and then apply the scalar.
+
+| *Mixed Associative* | scalars
+| stem:[(kl)A = k(lA)]
+| You can multiply the two scalars together first, or apply them to the matrix
+  one at a time.
+
+| *Commutative* | matrices
+| stem:[A + B = B + A]
+| Matrix addition is fully commutative. The order of operands does not change
+  the resulting sum.
+
+| *Associative* | matrices
+| stem:[(A + B) + C = A + (B + C)]
+| When adding three or more matrices, the grouping does not matter.
+
+| *Distributive* | matrices
+| stem:[A(B + C) = AB + AC]
+| Matrix multiplication distributes over matrix addition. Note: stem:[A] must
+  remain on the left side for both products.
+
+| *Distributive* | scalar and matrix
+| stem:[k(A + B) = kA + kB]
+| A scalar multiplied by the sum of two matrices is equal to the sum of the
+  scalar multiplied by each matrix individually.
+
+| *Distributive* | scalar and matrix
+| stem:[(k + l)A = kA + lA]
+| A matrix multiplied by the sum of two scalars is equal to the sum of the
+  matrix multiplied by each scalar individually.
+
+| *Distributive* | scalar and matrix
+| stem:[(A + B)k = Ak + Bk]
+| Because scalar multiplication is commutative stem:[Ak = kA], distributing a
+  scalar from the right side works exactly the same as distributing from the
+  left.
+
+| *Multiplicative Property* | determinants
+| stem:[\det(M_1 M_2) = \det(M_1)\det(M_2)]
+| The determinant of a product of matrices is the product of their determinants.
+  This holds even though matrix multiplication itself is not commutative.
+|===
+
+
+=== Multiplicative Identity Matrix 
+
+Identity matrices are only communative if it is a square matrix.
+
+When you do not have a square matrix the demensions of the identity matrix is
+different depending on the order.
+
+[stem]
+++++
+D = \left[\begin{array}{cc}
+3  & 4  \\
+2  & -1 \\
+-5 & 6 \\
+\end{array}\right]
+\\
+D * I = \left[\begin{array}{cc}
+3  &  4  \\
+2  & -1  \\
+-5 &  6  \\
+\end{array}\right] *
+\left[\begin{array}{cc}
+1  & 0  \\
+0  & 1  \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+3  &  4  \\
+2  & -1  \\
+-5 &  6  \\
+\end{array}\right]
+\\
+I * D = 
+\left[\begin{array}{cc}
+1 & 0 & 0 \\
+0 & 1 & 0 \\
+0 & 0 & 1 \\
+\end{array}\right] *
+\left[\begin{array}{cc}
+3  &  4  \\
+2  & -1  \\
+-5 &  6  \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+3  &  4  \\
+2  & -1  \\
+-5 &  6  \\
+\end{array}\right]
+++++
+
+== Invertable Matrices
+
+* A matrix with a row of 0s or a column of 0s isn't invertible -- it has no
+  inverse.
+
+== Singular Matrix
+
+A matrix that that doesn't have a inverse
+
+== Non-Singular Matrix
+
+A matrix that that has a inverse
diff --git a/Linear-Algrebra/Properties.html b/Linear-Algrebra/Properties.html
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+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Properties</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_properties">Properties</a>
+<ul class="sectlevel2">
+<li><a href="#_multiplicative_identity_matrix">Multiplicative Identity Matrix</a></li>
+</ul>
+</li>
+<li><a href="#_invertable_matrices">Invertable Matrices</a></li>
+<li><a href="#_singular_matrix">Singular Matrix</a></li>
+<li><a href="#_non_singular_matrix">Non-Singular Matrix</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_properties">Properties</h2>
+<div class="sectionbody">
+<table class="tableblock frame-none grid-rows stretch">
+<colgroup>
+<col style="width: 20%;">
+<col style="width: 20%;">
+<col style="width: 20%;">
+<col style="width: 40%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-center valign-middle"><strong>Property</strong></th>
+<th class="tableblock halign-center valign-middle"><strong>Parts</strong></th>
+<th class="tableblock halign-center valign-middle"><strong>Definition</strong></th>
+<th class="tableblock halign-center valign-middle"><strong>Details</strong></th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Conditionally Commutative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(A B \neq B A\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Matrix multiplication is generally <strong>not</strong> commutative. Exceptions include the
+  identity matrix \(I\), the inverse \(A^{-1}\), or a zero matrix.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Associative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((AB)C = A(BC)\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Matrix multiplication is associative. As long as the order stays the same, you
+can group them however you like.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Commutative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">scalar and matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(kA = Ak\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Scalar multiplication is commutative—you can place the scalar \(k\) in
+front or behind the matrix.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Mixed Associative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">Scalar and Matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((kA)B = k(AB)\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">You can multiply a matrix by a scalar first, or multiply the matrices first
+  and then apply the scalar.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Mixed Associative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">scalars</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((kl)A = k(lA)\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">You can multiply the two scalars together first, or apply them to the matrix
+  one at a time.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Commutative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(A + B = B + A\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Matrix addition is fully commutative. The order of operands does not change
+  the resulting sum.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Associative</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((A + B) + C = A + (B + C)\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">When adding three or more matrices, the grouping does not matter.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Distributive</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">matrices</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(A(B + C) = AB + AC\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Matrix multiplication distributes over matrix addition. Note: \(A\) must
+  remain on the left side for both products.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Distributive</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">scalar and matrix</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(k(A + B) = kA + kB\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">A scalar multiplied by the sum of two matrices is equal to the sum of the
+  scalar multiplied by each matrix individually.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Distributive</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">scalar and matrix</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((k + l)A = kA + lA\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">A matrix multiplied by the sum of two scalars is equal to the sum of the
+  matrix multiplied by each scalar individually.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Distributive</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">scalar and matrix</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\((A + B)k = Ak + Bk\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">Because scalar multiplication is commutative \(Ak = kA\), distributing a
+  scalar from the right side works exactly the same as distributing from the
+  left.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-center valign-middle"><p class="tableblock"><strong>Multiplicative Property</strong></p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">determinants</p></td>
+<td class="tableblock halign-center valign-middle"><p class="tableblock">\(\det(M_1 M_2) = \det(M_1)\det(M_2)\)</p></td>
+<td class="tableblock halign-left valign-middle"><p class="tableblock">The determinant of a product of matrices is the product of their determinants.
+  This holds even though matrix multiplication itself is not commutative.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="sect2">
+<h3 id="_multiplicative_identity_matrix">Multiplicative Identity Matrix</h3>
+<div class="paragraph">
+<p>Identity matrices are only communative if it is a square matrix.</p>
+</div>
+<div class="paragraph">
+<p>When you do not have a square matrix the demensions of the identity matrix is
+different depending on the order.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[D = \left[\begin{array}{cc}
+3  &amp; 4  \\
+2  &amp; -1 \\
+-5 &amp; 6 \\
+\end{array}\right]
+\\
+D * I = \left[\begin{array}{cc}
+3  &amp;  4  \\
+2  &amp; -1  \\
+-5 &amp;  6  \\
+\end{array}\right] *
+\left[\begin{array}{cc}
+1  &amp; 0  \\
+0  &amp; 1  \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+3  &amp;  4  \\
+2  &amp; -1  \\
+-5 &amp;  6  \\
+\end{array}\right]
+\\
+I * D =
+\left[\begin{array}{cc}
+1 &amp; 0 &amp; 0 \\
+0 &amp; 1 &amp; 0 \\
+0 &amp; 0 &amp; 1 \\
+\end{array}\right] *
+\left[\begin{array}{cc}
+3  &amp;  4  \\
+2  &amp; -1  \\
+-5 &amp;  6  \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+3  &amp;  4  \\
+2  &amp; -1  \\
+-5 &amp;  6  \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_invertable_matrices">Invertable Matrices</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>A matrix with a row of 0s or a column of 0s isn&#8217;t invertible&#8201;&#8212;&#8201;it has no
+inverse.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_singular_matrix">Singular Matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>A matrix that that doesn&#8217;t have a inverse</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_non_singular_matrix">Non-Singular Matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>A matrix that that has a inverse</p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+    ignoreClass: "nostem|nolatexmath"
+  },
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+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
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+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
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+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
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diff --git a/Linear-Algrebra/SCRAP b/Linear-Algrebra/SCRAP
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+3Blue 1Brown linear algrebra series:
+https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
+
+problems
+https://www.math.purdue.edu/~zhan4306/spring_2022_la/sec_1_9.pdf
+https://math.libretexts.org/Courses/University_of_California_Irvine/Linear_Algebra_(Math_3A_UCI)/02%3A_Linear_Transformations_and_Matrix_Algebra/2.07%3A_Determinants_and_Volumes/2.7.01%3A_Determinants_for_Area_and_Volume
+
+3d graphing calculator
+https://c3d.libretexts.org/CalcPlot3D/index.html
+
+
diff --git a/Linear-Algrebra/System-of-Equations.adoc b/Linear-Algrebra/System-of-Equations.adoc
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+= Linear Algrebra | System of Equations
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Solving
+
+=== Two Equations
+
+The tools or methods for solving systems of equations algebraically are
+
+* *Elimination*: You eliminate one of the variables in teh system to make it a
+  simpler system, while preserving the original relationship between variables
+
+* *Back Substitution*: You use back substitution after you've found a value for
+  one of the variables and tehn solve for the values of the other variables.
+
+.back substitution
+[example%collapsible]
+====
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{4}
+2x & {}+{} & 3y & = 23 \\
+4x & {}-{} & 3y & = 1
+\end{alignat*}
+\end{cases}
+++++
+
+[stem]
+++++
+\begin{array}{r}
+\begin{alignat*}{4}
+2x & {}+{} & 3y & = 23 \\
+2x & {}+{} & 3y & = 1 \\
+\hline
+6x &       &    &   24 \\
+x  & {}={} & \frac{24}{4} & = 4
+\end{alignat*}
+\end{array}
+++++
+
+[stem]
+++++
+\begin{gather*}
+2(4) + 3y = 23 \\
+8 + 3y = 23 \\
+3y = 15 \\
+y = 5
+\end{gather*}
+++++
+====
+
+.Elimination
+[example%collapsible]
+====
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{4}
+3x & {}+{} & 5y & = 12 \\
+4x & {}-{} & 3y & = -13
+\end{alignat*}
+\end{cases}
+++++
+
+[stem]
+++++
+\begin{alignat*}{8}
+(3x & {}+{} & 5y & = 12)3\ & =\  & 9x {}+{} & 15y & =  36 \\
+(4x & {}-{} & 3y & = -13)5\ & =\ & 2x {}-{} & 15y & = -65 \\
+\end{alignat*}
+++++
+
+[stem]
+++++
+\begin{array}{r}
+\begin{alignat*}{4}
+3x & {}+{} & 5y & = 12 \\
+4x & {}-{} & 3y & = -13 \\
+\hline
+29x &      &    & = -29 \\
+x & = & \frac{-29}{29} & = -1
+\end{alignat*}
+\end{array}
+++++
+====
+
+=== Multiple Equations
+
+The following operations may be performed (preserving the original relationship
+between the variables), giving you the solution to the original equation:
+
+* Interchange two equations (change their order)
+
+* Multiply or divide each term in an equation by a constant number
+  (stem:[\neq 0])
+
+* Add two equations together, combining like terms.
+
+* Add the multiple of one equation to another equation
+
+NOTE: There is an alternative method involving matrices
+
+.example
+[example%collapsible]
+====
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{5}
+2x & {}+{} & 3y & {}-{} & 4z & {}+{} &  w & = 15 \\
+ x & {}-{} & 2y & {}+{} & 3z & {}-{} & 2w & = -3 \\
+3x & {}+{} & 5y & {}+{} &  z & {}-{} &  w & = 20 \\
+4x & {}+{} &  y & {}-{} &  z & {}+{} &  w & = 5 \\
+\end{alignat*}
+\end{cases}
+++++
+
+remove the w
+
+[stem]
+++++
+ \begin{alignat*}{4}
+2R_1 + R_2 &\to 5x &&+ 4y &&- 5z &&= 27 \\
+R_1 + R_3  &\to 5x &&+ 8y &&- 3z &&= 35 \\
+R_4 + R_3  &\to 7x &&+ 6y &&     &&= 25
+\end{alignat*}
+++++
+
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{5}
+5x & {}+{} & 4y & {}-{} & 5z & = 15 \\
+5x & {}+{} & 8y & {}-{} & 3z & = 35 \\
+7x & {}+{} & 6y &       &    & = 25 \\
+\end{alignat*}
+\end{cases}
+++++
+
+remove the last equations
+
+[stem]
+++++
+\begin{array}{lllrcrcrcl}
+3R_1 & \to & -15x & + & 12y & - & 15z & = & 81 \\
+-5R_2 & \to & -25x & - & 40y & + & 15z & = & -175 \\
+\hline
+3R_1 + (-5R_2) & \to & -40x & - & 28y & & & = & -94
+\end{array}
+++++
+
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{4}
+  7x & {}+{} &  6y & =  25 \\
+-10x & {}-{} & 28y & = -94
+\end{alignat*}
+\end{cases}
+++++
+
+Final steps
+
+[stem]
+++++
+\begin{array}{llrcrcrcl}
+14R_1 & \to & 98x & + & 84y & = & 350 \\
+3R_2 & \to & -30x & - & 84y & = & -282 \\
+\hline
+14R_1 + 3R_2 & \to & 68x & & & = & 68
+\end{array}
+++++
+
+getting x
+
+[stem]
+++++
+\begin{align*}
+68x &= 68 \\
+x &= 1
+\end{align*}
+++++
+
+Now, substitute stem:[x = 1] into stem:[7x + 6y = 25] to get y
+
+[stem]
+++++
+\begin{align*}
+7(1) + 6y &= 25 \\
+6y &= 18 \\
+y &= 3
+\end{align*}
+++++
+
+solve for z
+
+[stem]
+++++
+\begin{align*}
+5(1) + 8(3) - 3z &= 35 \\
+5 + 24 - 3z &= 35 \\
+29 - 3z &= 35 \\
+-3z &= 6 \\
+z &= -2
+\end{align*}
+++++
+
+solve for w
+
+[stem]
+++++
+\begin{align*}
+2(1) + 3(3) - 4(-2) + w &= 15 \\
+2 + 9 + 8 + w &= 15 \\
+19 + w &= 15 \\
+w &= -4
+\end{align*}
+++++
+
+solution
+
+[stem]
+++++
+\begin{cases}
+x = 1 \\
+y = 3 \\
+z = -2 \\
+w = -4
+\end{cases}
+++++
+
+====
+
+== Linear Equations
+
+Linear equations are in the form:
+
+[stem]
+++++
+a_1x_1 + a_2x_2 + a_3x_3 + \ldots + a_nx_n
+++++
+
+== System of linear equations
+
+This is a system of linear equations. In order to solve a system of linear
+equations you have to find variables that make each line true.
+
+[stem]
+++++
+\begin{cases} 
+2x + y = 10 \\ 
+x - y = 2 
+\end{cases}
+++++
+
+answer: 
+
+[stem]
+++++
+\begin{cases}
+x = 4 \\
+y = 2
+\end{cases}
+++++
+
+=== Matrices
+
+In linear algrebra we can simplify this to be a matrix, each column is for a
+different variable. 
+
+[stem]
+++++
+\begin{cases}
+\begin{alignat*}{5}
+   x_1 & {}-{} &  2x_2 & {}+{} &  3x_3 & {}-{} &  6x_4 & = 8  \\
+   x_1 &       &       & {}-{} & 10x_3 & {}+{} &   x_4 & = 11 \\
+  4x_1 & {}+{} &   x_2 &       &       & {}-{} &  4x_4 & = 8  \\
+  3x_1 & {}+{} &  5x_2 & {}-{} &  7x_3 &       &       & = 6
+\end{alignat*}
+\end{cases}
+++++
+
+[stem]
+++++
+A = \left[\begin{array}{cc}
+1 & -2 &  3 &  6 \\
+1 & 0 & -10 &  1 \\
+4 & 1 &   0 & -4 \\
+3 & 5 &  -7 &  0 \\
+\end{array}\right]
+++++
+
+== Solutions
+
+A system of equations has a _solution_ when you have at least one set of numbers
+that replaces variabels in the equations and makes each equation read as a true
+statement. A system of equations may have more than one solution. Systems with
+more than one solution generally have an infinite number of solutions, and those
+solutions are written as an expression or rule in terms of a parameter.
+
+=== Characteristics of having just one solution
+
+1. Must contain linear equations
+
+2. The system of equations must have as many independent equations as there are
+   variables in the system.
+   * *independent equations*: This just means that none of the equations is a
+     multiple or combination of multiples of the other equations in the system.
+
+.example
+[example%collapsible]
+====
+Including the equation stem:[-4x_1 - 6x_2 = -4] would incroduce an equation that
+isn't independent. That equation is obtained by multiplying the top equation by
+-2, multiplying the bottom equation by 2, and adding the two multiples together.
+
+[stem]
+++++
+\begin{cases}
+3x_1 - 2x_2 = 16 \\
+ x_1 - 5x_2 = 14
+\end{cases}
+++++
+====
+
+=== Characteristics of having multiple solutions
+
+When systems have mutiple solutions, you'll find a pattern or rule that predicts
+what those solutions are.
+
+[stem]
+++++
+\begin{cases}
+\begin{alignat*}{4}
+   -x & {}+{} &  y & {}+{} &  z & = -1 \\
+   -x & {}+{} & 2y & {}-{} & 3z & = -4 \\
+   3x & {}-{} & 2y & {}-{} & 7z & = 0
+\end{alignat*}
+\end{cases}
+++++
+
+
+The pattern in the solutions is that x is always two less than five time z, and
+y is always three less than four times z. This has infinite solutions
+
+NOTE: linear algrebra for dummies p.67
+
+=== Graphing two lines for two equations
+
+The graph of system of equations stem:[ax + by = c] and stem:[dx + ey = f],
+where x and y area variables and a, b, c, d, e, and f are real numbers has:
+
+* Intersecting lines (exactly one solution) when the system is consistent and
+  the equations are _independent_
+
+* Parallel lines (no solutions) when the system is _inconsistent_
+
+* Coincident lines (infinit solutions) when the system in _consistent_ and the
+  equations are _dependent_
+
+=== Three equations | Exactly One Soltion (Independent)
+
+In a System of three equations, if three planes intersect at a single unique
+point then there is only one soution. (like the corner of a room where two walls
+meet the ceiling).
+
+=== Planes
+
+==== No Solutions
+Planes that have the same slant but different intersections of the axes will
+never interesect
+
+[.center]
+image::parallel-planes.png[]
+
+In mathematics, two planes are parallel if their equations have the same
+coefficients for the x, y, and z variables, but different constants.
+
+For example, look at these two planes:
+
+* *Plane 1*: stem:[3x - 2y - 7z = 0]
+
+* *Plane 2*: stem:[3x - 2y - 7z = 10]
+
+Notice that the left-hand sides are identical. In technical terms, their normal
+vectors (the direction perpendicular to the flat surface) are the same: [3, -2,
+-7]. The different constants (0 vs 10) just shift one plane "above" or
+"beside" the other without changing its angle.
+
+In linear algebra, a solution is a point where all the planes in your system
+touch or intersect. Therefore, the system is called inconsistent, and we say
+there are no solutions.
+
+==== Multiple Solutions
+
+There is only one specific case where parallel planes have multiple solutions:
+if they are actually the same plane.
+
+consider these two plane equations
+
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{4}
+x  & {}+{} & y  & {}+{} & z  & = 1 \\
+2x & {}+{} & 2y & {}+{} & 2z & = 2
+\end{alignat*}
+\end{cases}
+++++
+
+* *The "Double" Layer*: Even though there are two different mathematical
+  expressions (like stem:[x+y+z=1] and stem:[2x+2y+2z=2]), they represent the
+  same set of coordinates.
+
+* *Infinite Solutions*: Because every single point stem:[x, y, z] on the first
+  plane is also on the second, there are an infinite number of solutions.
+
+* *Linear Dependence*: In linear algebra terms, these equations are linearly
+  dependent. The second equation is simply the first equation multiplied by a
+  scalar (in this case, 2).
+
+== One Solution with Square Matrices and their Inverse
+
+To solve a system of _n_ equations with _n_ unkowns using an _n_ x _n_ matrix
+and its inverse, do the following:
+
+1. Write each of the equations in the system with the variables in the same
+   order and the constant on the other sider of the equal sign from the
+   variable.
+
+2. Construct a _coeffiecient matrix_ (a square matrix whose elements are the
+   coefficients of the variables).
+
+3. Write the _constant matrix_ (an n x 1 column matrix) using the constants in
+   the equations.
+
+4. Find the inverse of the coefficient matrix.
+
+5. Multiply the inverse of the coefficient matrix times the constant matrix.
+
+WARNING: This only works with singular matrices
+
+.example
+[%collapsible]
+====
+[stem]
+++++
+\left\{ \begin{array}{rclrclrcl}
+0  & = & 3  & + & x & + & y & + & z \\
+4x & + & 5y & + & 6 & = & 0 &   &   \\
+y  & = & 3z & + & 5 &   &   &   &   
+\end{array} \right.
+++++
+
+[stem]
+++++
+\begin{cases} 
+\begin{alignat*}{6}
+-x & {}-{} &  y & {}-{} & z   & = 3 \\
+4x & {}+{} & 5y &       &     & = 3 \\
+   &       &  y & {}-{} & 3z  & = 5
+\end{alignat*}
+\end{cases}
+++++
+
+[stem]
+++++
+A = \begin{bmatrix}
+-1 & -1 & -1 \\
+ 4 &  5 &  0 \\
+ 0 &  1 & -3 
+\end{bmatrix}
+++++
+
+[stem]
+++++
+B = \begin{bmatrix}
+3  \\
+-6 \\
+5
+\end{bmatrix}
+++++
+
+[stem]
+++++
+A^{-1} = \begin{bmatrix}
+ 15 &  4 & -5 \\
+-12 & -3 & 4  \\
+-4  & -1 & 1 
+\end{bmatrix}
+++++
+
+[stem]
+++++
+A^{-1} \otimes B = \begin{bmatrix}
+ 15 &  4 & -5 \\
+-12 & -3 & 4  \\
+-4  & -1 & 1 
+\end{bmatrix} \otimes
+\begin{bmatrix}
+ 3 \\
+-6 \\
+ 5 
+\end{bmatrix} =
+\begin{bmatrix}
+-4 \\
+ 2 \\
+-1
+\end{bmatrix}
+++++
+
+====
+
+== Solving Using an augmented matrix
+
+After wrting a system of equations as an augmented matrix, you perform row
+operations until the metrix is in either row echelon form or reduced row echelon
+form. 
+
+* *Reduced Row echelon form*:  Gives you the answer directly
+
+* *Row echelon form*:  Required back-solving or algebraic manipulations to get
+  your answer.
+
+NOTE: Works for any size matrix.
+
+=== One Solution
+.Problem
+[%collapsible]
+====
+[stem]
+++++
+\begin{cases}
+\begin{alignat*}{6}
+-x & {}-{} &  y & {}-{} & z  & =  3 \\
+4x & {}+{} & 5y &       &    & = -6 \\
+   &       &  y & {}-{} & 3z & = 5
+\end{alignat*}
+\end{cases}
+++++
+
+[stem]
+++++
+\left[
+\begin{array}{ccc : c}
+-1 & -1 &  -1 &  3 \\
+ 4 &  5 &   0 & -6 \\
+ 0 &  1 &  -3 &  5
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+-1R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &  1 &   1 & -3 \\
+ 4 &  5 &   0 & -6 \\
+ 0 &  1 &  -3 &  5
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+-4R_1 + R_2 \to R_2
+\left[
+\begin{array}{ccc : c}
+ 1 &  1 &   1 & -3 \\
+ 0 &  1 &  -4 &  6 \\
+ 0 &  1 &  -3 &  5
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+-1R_2 + R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &  0 &   5 & -9 \\
+ 0 &  1 &  -4 &  6 \\
+ 0 &  1 &  -3 &  5
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+-1R_2 + R_3 \to R_3
+\left[
+\begin{array}{ccc : c}
+ 1 &  0 &   5 & -9 \\
+ 0 &  1 &  -4 &  6 \\
+0 &  0 &   1 & -1
+\end{array}
+\right]
+++++
+====
+
+.Row Echeleon Form
+[%collapsible]
+====
+
+back substitute for z
+
+[stem]
+++++
+\begin{align*}
+y - 4(-1) &= 6 \\
+y + 4 &= 6 \\
+y &= 2
+\end{align*}
+++++
+
+back substitute for for z again
+
+[stem]
+++++
+\begin{align*}
+x + 5(-1) &= -9 \\
+x - 5 &= -9 \\
+x &= -4
+\end{align*}
+++++
+
+solution
+
+[stem]
+++++
+(x, y, z) = (-4, 2, -1)
+++++
+====
+
+.Reduced Row Echeleon Form
+[%collapsible]
+====
+[stem]
+++++
+-5R_3 + R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &  0 &   0 & -4 \\
+ 0 &  1 &  -4 &  6 \\
+ 0 &  0 &   1 & -1
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+4R_3 + R_2 \to R_2
+\left[
+\begin{array}{ccc : c}
+ 1 &  0 &   0 & -4 \\
+ 0 &  1 &   0 &  2 \\
+ 0 &  0 &   1 & -1
+\end{array}
+\right]
+++++
+
+solution
+
+[stem]
+++++
+(x, y, z) = (-4, 2, -1)
+++++
+====
+
+=== Multiple Solutions
+
+The system has more than one solution when the *Row Echelon Form*, or *Reduced
+Row Echelon Form*  has one or more rows of 0s.
+
+.example
+[%collapsible]
+====
+[stem]
+++++
+\begin{cases}
+\begin{alignat*}{6}
+ x & {}-{} &  y & {}-{} &  z  & = 1 \\
+3x & {}-{} & 2y & {}-{} & 7z  & = 0 \\
+5x & {}-{} & 4y & {}-{} & 9z  & = 2
+\end{alignat*}
+\end{cases}
+
+\qquad %
+
+\left[
+\begin{array}{ccc : c}
+ 1 & -1 & -1 & 1 \\
+ 3 & -2 & -7 & 0 \\
+ 5 & -4 & -9 & 2
+\end{array}
+\right]
+++++
+
+[stem]
+++++
+\begin{aligned}
+    \begin{matrix} 
+    -3R_1 + R_2 \to R_2 \\ 
+    -5R_1 + R_3 \to R_3 
+    \end{matrix} 
+    & \left[ \begin{array}{ccc|r}
+    1 & -1 & -1 & 1 \\
+    0 & 1 & -4 & -3 \\
+    0 & 1 & -4 & -3
+    \end{array} \right] \\[15pt]
+    -1R_2 + R_3 \to R_3 
+    & \left[ \begin{array}{ccc|r}
+    1 & -1 & -1 & 1 \\
+    0 & 1 & -4 & -3 \\
+    0 & 0 & 0 & 0
+    \end{array} \right] \\[15pt]
+    R_2 + R_1 \to R_1 
+    & \left[ \begin{array}{ccc|r}
+    1 & 0 & -5 & -2 \\
+    0 & 1 & -4 & -3 \\
+    0 & 0 & 0 & 0
+    \end{array} \right]
+\end{aligned}
+++++
+
+k = z
+
+[stem]
+++++
+\begin{array}{r}
+\begin{alignat*}{4}
+ x & {}-{} &  5k & = -2 \\
+   &       & +5k & = +5k\\
+\hline
+   &       &  x  & = -2 + 5x \\
+\end{alignat*}
+\end{array}
+++++
+
+[stem]
+++++
+\begin{array}{r}
+\begin{alignat*}{4}
+ y & {}-{} &  4k & = -3 \\
+   &       & +4k & = +4k\\
+\hline
+   &       &  y  & = -3 + 4k \\
+\end{alignat*}
+\end{array}
+++++
+
+[stem]
+++++
+\begin{cases} 
+    x - 5z = -2 \\ 
+    y - 4z = -3 
+\end{cases} 
+\text{ or } 
+\begin{cases} 
+    x = 5z - 2 \\ 
+    y = 4z - 3 
+\end{cases}
+++++
+
+Both x and y are dependent on what z is. So I choose the parameter k to
+represent some real number. I let z = k, which makes x = 5k – 2 and y = 4k – 3.
+The solution is also written (5k – 2,4k – 3,k) as an ordered triple. Once a real
+number is selected for k, then a solution is created from the ordered triple.
+For example, if k = 1, then a solution for the system is (3,1,1).
+
+====
+
+
+
+== Links
+
+* https://brilliant.org/wiki/3d-coordinate-geometry-equation-of-a-plane/[Brilliant | 3D Coordinate Geometry - Equation of a Plane]
diff --git a/Linear-Algrebra/System-of-Equations.html b/Linear-Algrebra/System-of-Equations.html
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+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | System of Equations</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_solving">Solving</a>
+<ul class="sectlevel2">
+<li><a href="#_two_equations">Two Equations</a></li>
+<li><a href="#_multiple_equations">Multiple Equations</a></li>
+</ul>
+</li>
+<li><a href="#_linear_equations">Linear Equations</a></li>
+<li><a href="#_system_of_linear_equations">System of linear equations</a>
+<ul class="sectlevel2">
+<li><a href="#_matrices">Matrices</a></li>
+</ul>
+</li>
+<li><a href="#_solutions">Solutions</a>
+<ul class="sectlevel2">
+<li><a href="#_characteristics_of_having_just_one_solution">Characteristics of having just one solution</a></li>
+<li><a href="#_characteristics_of_having_multiple_solutions">Characteristics of having multiple solutions</a></li>
+<li><a href="#_graphing_two_lines_for_two_equations">Graphing two lines for two equations</a></li>
+<li><a href="#_three_equations_exactly_one_soltion_independent">Three equations | Exactly One Soltion (Independent)</a></li>
+<li><a href="#_planes">Planes</a></li>
+</ul>
+</li>
+<li><a href="#_one_solution_with_square_matrices_and_their_inverse">One Solution with Square Matrices and their Inverse</a></li>
+<li><a href="#_solving_using_an_augmented_matrix">Solving Using an augmented matrix</a>
+<ul class="sectlevel2">
+<li><a href="#_one_solution">One Solution</a></li>
+<li><a href="#_multiple_solutions_2">Multiple Solutions</a></li>
+</ul>
+</li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_solving">Solving</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_two_equations">Two Equations</h3>
+<div class="paragraph">
+<p>The tools or methods for solving systems of equations algebraically are</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Elimination</strong>: You eliminate one of the variables in teh system to make it a
+simpler system, while preserving the original relationship between variables</p>
+</li>
+<li>
+<p><strong>Back Substitution</strong>: You use back substitution after you&#8217;ve found a value for
+one of the variables and tehn solve for the values of the other variables.</p>
+</li>
+</ul>
+</div>
+<details>
+<summary class="title">back substitution</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{4}
+2x &amp; {}+{} &amp; 3y &amp; = 23 \\
+4x &amp; {}-{} &amp; 3y &amp; = 1
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{r}
+\begin{alignat*}{4}
+2x &amp; {}+{} &amp; 3y &amp; = 23 \\
+2x &amp; {}+{} &amp; 3y &amp; = 1 \\
+\hline
+6x &amp;       &amp;    &amp;   24 \\
+x  &amp; {}={} &amp; \frac{24}{4} &amp; = 4
+\end{alignat*}
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{gather*}
+2(4) + 3y = 23 \\
+8 + 3y = 23 \\
+3y = 15 \\
+y = 5
+\end{gather*}\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Elimination</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{4}
+3x &amp; {}+{} &amp; 5y &amp; = 12 \\
+4x &amp; {}-{} &amp; 3y &amp; = -13
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{alignat*}{8}
+(3x &amp; {}+{} &amp; 5y &amp; = 12)3\ &amp; =\  &amp; 9x {}+{} &amp; 15y &amp; =  36 \\
+(4x &amp; {}-{} &amp; 3y &amp; = -13)5\ &amp; =\ &amp; 2x {}-{} &amp; 15y &amp; = -65 \\
+\end{alignat*}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{r}
+\begin{alignat*}{4}
+3x &amp; {}+{} &amp; 5y &amp; = 12 \\
+4x &amp; {}-{} &amp; 3y &amp; = -13 \\
+\hline
+29x &amp;      &amp;    &amp; = -29 \\
+x &amp; = &amp; \frac{-29}{29} &amp; = -1
+\end{alignat*}
+\end{array}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+<div class="sect2">
+<h3 id="_multiple_equations">Multiple Equations</h3>
+<div class="paragraph">
+<p>The following operations may be performed (preserving the original relationship
+between the variables), giving you the solution to the original equation:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Interchange two equations (change their order)</p>
+</li>
+<li>
+<p>Multiply or divide each term in an equation by a constant number
+(\(\neq 0\))</p>
+</li>
+<li>
+<p>Add two equations together, combining like terms.</p>
+</li>
+<li>
+<p>Add the multiple of one equation to another equation</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+There is an alternative method involving matrices
+</td>
+</tr>
+</table>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{5}
+2x &amp; {}+{} &amp; 3y &amp; {}-{} &amp; 4z &amp; {}+{} &amp;  w &amp; = 15 \\
+ x &amp; {}-{} &amp; 2y &amp; {}+{} &amp; 3z &amp; {}-{} &amp; 2w &amp; = -3 \\
+3x &amp; {}+{} &amp; 5y &amp; {}+{} &amp;  z &amp; {}-{} &amp;  w &amp; = 20 \\
+4x &amp; {}+{} &amp;  y &amp; {}-{} &amp;  z &amp; {}+{} &amp;  w &amp; = 5 \\
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>remove the w</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[ \begin{alignat*}{4}
+2R_1 + R_2 &amp;\to 5x &amp;&amp;+ 4y &amp;&amp;- 5z &amp;&amp;= 27 \\
+R_1 + R_3  &amp;\to 5x &amp;&amp;+ 8y &amp;&amp;- 3z &amp;&amp;= 35 \\
+R_4 + R_3  &amp;\to 7x &amp;&amp;+ 6y &amp;&amp;     &amp;&amp;= 25
+\end{alignat*}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{5}
+5x &amp; {}+{} &amp; 4y &amp; {}-{} &amp; 5z &amp; = 15 \\
+5x &amp; {}+{} &amp; 8y &amp; {}-{} &amp; 3z &amp; = 35 \\
+7x &amp; {}+{} &amp; 6y &amp;       &amp;    &amp; = 25 \\
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>remove the last equations</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{lllrcrcrcl}
+3R_1 &amp; \to &amp; -15x &amp; + &amp; 12y &amp; - &amp; 15z &amp; = &amp; 81 \\
+-5R_2 &amp; \to &amp; -25x &amp; - &amp; 40y &amp; + &amp; 15z &amp; = &amp; -175 \\
+\hline
+3R_1 + (-5R_2) &amp; \to &amp; -40x &amp; - &amp; 28y &amp; &amp; &amp; = &amp; -94
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{4}
+  7x &amp; {}+{} &amp;  6y &amp; =  25 \\
+-10x &amp; {}-{} &amp; 28y &amp; = -94
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Final steps</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{llrcrcrcl}
+14R_1 &amp; \to &amp; 98x &amp; + &amp; 84y &amp; = &amp; 350 \\
+3R_2 &amp; \to &amp; -30x &amp; - &amp; 84y &amp; = &amp; -282 \\
+\hline
+14R_1 + 3R_2 &amp; \to &amp; 68x &amp; &amp; &amp; = &amp; 68
+\end{array}\]
+</div>
+</div>
+<div class="paragraph">
+<p>getting x</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+68x &amp;= 68 \\
+x &amp;= 1
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Now, substitute \(x = 1\) into \(7x + 6y = 25\) to get y</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+7(1) + 6y &amp;= 25 \\
+6y &amp;= 18 \\
+y &amp;= 3
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>solve for z</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+5(1) + 8(3) - 3z &amp;= 35 \\
+5 + 24 - 3z &amp;= 35 \\
+29 - 3z &amp;= 35 \\
+-3z &amp;= 6 \\
+z &amp;= -2
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>solve for w</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+2(1) + 3(3) - 4(-2) + w &amp;= 15 \\
+2 + 9 + 8 + w &amp;= 15 \\
+19 + w &amp;= 15 \\
+w &amp;= -4
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>solution</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+x = 1 \\
+y = 3 \\
+z = -2 \\
+w = -4
+\end{cases}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_linear_equations">Linear Equations</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Linear equations are in the form:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[a_1x_1 + a_2x_2 + a_3x_3 + \ldots + a_nx_n\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_system_of_linear_equations">System of linear equations</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This is a system of linear equations. In order to solve a system of linear
+equations you have to find variables that make each line true.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+2x + y = 10 \\
+x - y = 2
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>answer:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+x = 4 \\
+y = 2
+\end{cases}\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_matrices">Matrices</h3>
+<div class="paragraph">
+<p>In linear algrebra we can simplify this to be a matrix, each column is for a
+different variable.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{5}
+   x_1 &amp; {}-{} &amp;  2x_2 &amp; {}+{} &amp;  3x_3 &amp; {}-{} &amp;  6x_4 &amp; = 8  \\
+   x_1 &amp;       &amp;       &amp; {}-{} &amp; 10x_3 &amp; {}+{} &amp;   x_4 &amp; = 11 \\
+  4x_1 &amp; {}+{} &amp;   x_2 &amp;       &amp;       &amp; {}-{} &amp;  4x_4 &amp; = 8  \\
+  3x_1 &amp; {}+{} &amp;  5x_2 &amp; {}-{} &amp;  7x_3 &amp;       &amp;       &amp; = 6
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[A = \left[\begin{array}{cc}
+1 &amp; -2 &amp;  3 &amp;  6 \\
+1 &amp; 0 &amp; -10 &amp;  1 \\
+4 &amp; 1 &amp;   0 &amp; -4 \\
+3 &amp; 5 &amp;  -7 &amp;  0 \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_solutions">Solutions</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>A system of equations has a <em>solution</em> when you have at least one set of numbers
+that replaces variabels in the equations and makes each equation read as a true
+statement. A system of equations may have more than one solution. Systems with
+more than one solution generally have an infinite number of solutions, and those
+solutions are written as an expression or rule in terms of a parameter.</p>
+</div>
+<div class="sect2">
+<h3 id="_characteristics_of_having_just_one_solution">Characteristics of having just one solution</h3>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Must contain linear equations</p>
+</li>
+<li>
+<p>The system of equations must have as many independent equations as there are
+variables in the system.</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>independent equations</strong>: This just means that none of the equations is a
+multiple or combination of multiples of the other equations in the system.</p>
+</li>
+</ul>
+</div>
+</li>
+</ol>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="paragraph">
+<p>Including the equation \(-4x_1 - 6x_2 = -4\) would incroduce an equation that
+isn&#8217;t independent. That equation is obtained by multiplying the top equation by
+-2, multiplying the bottom equation by 2, and adding the two multiples together.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+3x_1 - 2x_2 = 16 \\
+ x_1 - 5x_2 = 14
+\end{cases}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+<div class="sect2">
+<h3 id="_characteristics_of_having_multiple_solutions">Characteristics of having multiple solutions</h3>
+<div class="paragraph">
+<p>When systems have mutiple solutions, you&#8217;ll find a pattern or rule that predicts
+what those solutions are.</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{4}
+   -x &amp; {}+{} &amp;  y &amp; {}+{} &amp;  z &amp; = -1 \\
+   -x &amp; {}+{} &amp; 2y &amp; {}-{} &amp; 3z &amp; = -4 \\
+   3x &amp; {}-{} &amp; 2y &amp; {}-{} &amp; 7z &amp; = 0
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>The pattern in the solutions is that x is always two less than five time z, and
+y is always three less than four times z. This has infinite solutions</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+linear algrebra for dummies p.67
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_graphing_two_lines_for_two_equations">Graphing two lines for two equations</h3>
+<div class="paragraph">
+<p>The graph of system of equations \(ax + by = c\) and \(dx + ey = f\),
+where x and y area variables and a, b, c, d, e, and f are real numbers has:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Intersecting lines (exactly one solution) when the system is consistent and
+the equations are <em>independent</em></p>
+</li>
+<li>
+<p>Parallel lines (no solutions) when the system is <em>inconsistent</em></p>
+</li>
+<li>
+<p>Coincident lines (infinit solutions) when the system in <em>consistent</em> and the
+equations are <em>dependent</em></p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_three_equations_exactly_one_soltion_independent">Three equations | Exactly One Soltion (Independent)</h3>
+<div class="paragraph">
+<p>In a System of three equations, if three planes intersect at a single unique
+point then there is only one soution. (like the corner of a room where two walls
+meet the ceiling).</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_planes">Planes</h3>
+<div class="sect3">
+<h4 id="_no_solutions">No Solutions</h4>
+<div class="paragraph">
+<p>Planes that have the same slant but different intersections of the axes will
+never interesect</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/parallel-planes.png" alt="parallel planes">
+</div>
+</div>
+<div class="paragraph">
+<p>In mathematics, two planes are parallel if their equations have the same
+coefficients for the x, y, and z variables, but different constants.</p>
+</div>
+<div class="paragraph">
+<p>For example, look at these two planes:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Plane 1</strong>: \(3x - 2y - 7z = 0\)</p>
+</li>
+<li>
+<p><strong>Plane 2</strong>: \(3x - 2y - 7z = 10\)</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Notice that the left-hand sides are identical. In technical terms, their normal
+vectors (the direction perpendicular to the flat surface) are the same: [3, -2,
+-7]. The different constants (0 vs 10) just shift one plane "above" or
+"beside" the other without changing its angle.</p>
+</div>
+<div class="paragraph">
+<p>In linear algebra, a solution is a point where all the planes in your system
+touch or intersect. Therefore, the system is called inconsistent, and we say
+there are no solutions.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_multiple_solutions">Multiple Solutions</h4>
+<div class="paragraph">
+<p>There is only one specific case where parallel planes have multiple solutions:
+if they are actually the same plane.</p>
+</div>
+<div class="paragraph">
+<p>consider these two plane equations</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{4}
+x  &amp; {}+{} &amp; y  &amp; {}+{} &amp; z  &amp; = 1 \\
+2x &amp; {}+{} &amp; 2y &amp; {}+{} &amp; 2z &amp; = 2
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>The "Double" Layer</strong>: Even though there are two different mathematical
+expressions (like \(x+y+z=1\) and \(2x+2y+2z=2\)), they represent the
+same set of coordinates.</p>
+</li>
+<li>
+<p><strong>Infinite Solutions</strong>: Because every single point \(x, y, z\) on the first
+plane is also on the second, there are an infinite number of solutions.</p>
+</li>
+<li>
+<p><strong>Linear Dependence</strong>: In linear algebra terms, these equations are linearly
+dependent. The second equation is simply the first equation multiplied by a
+scalar (in this case, 2).</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_one_solution_with_square_matrices_and_their_inverse">One Solution with Square Matrices and their Inverse</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>To solve a system of <em>n</em> equations with <em>n</em> unkowns using an <em>n</em> x <em>n</em> matrix
+and its inverse, do the following:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Write each of the equations in the system with the variables in the same
+order and the constant on the other sider of the equal sign from the
+variable.</p>
+</li>
+<li>
+<p>Construct a <em>coeffiecient matrix</em> (a square matrix whose elements are the
+coefficients of the variables).</p>
+</li>
+<li>
+<p>Write the <em>constant matrix</em> (an n x 1 column matrix) using the constants in
+the equations.</p>
+</li>
+<li>
+<p>Find the inverse of the coefficient matrix.</p>
+</li>
+<li>
+<p>Multiply the inverse of the coefficient matrix times the constant matrix.</p>
+</li>
+</ol>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+This only works with singular matrices
+</td>
+</tr>
+</table>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\left\{ \begin{array}{rclrclrcl}
+0  &amp; = &amp; 3  &amp; + &amp; x &amp; + &amp; y &amp; + &amp; z \\
+4x &amp; + &amp; 5y &amp; + &amp; 6 &amp; = &amp; 0 &amp;   &amp;   \\
+y  &amp; = &amp; 3z &amp; + &amp; 5 &amp;   &amp;   &amp;   &amp;
+\end{array} \right.\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{6}
+-x &amp; {}-{} &amp;  y &amp; {}-{} &amp; z   &amp; = 3 \\
+4x &amp; {}+{} &amp; 5y &amp;       &amp;     &amp; = 3 \\
+   &amp;       &amp;  y &amp; {}-{} &amp; 3z  &amp; = 5
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[A = \begin{bmatrix}
+-1 &amp; -1 &amp; -1 \\
+ 4 &amp;  5 &amp;  0 \\
+ 0 &amp;  1 &amp; -3
+\end{bmatrix}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[B = \begin{bmatrix}
+3  \\
+-6 \\
+5
+\end{bmatrix}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[A^{-1} = \begin{bmatrix}
+ 15 &amp;  4 &amp; -5 \\
+-12 &amp; -3 &amp; 4  \\
+-4  &amp; -1 &amp; 1
+\end{bmatrix}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[A^{-1} \otimes B = \begin{bmatrix}
+ 15 &amp;  4 &amp; -5 \\
+-12 &amp; -3 &amp; 4  \\
+-4  &amp; -1 &amp; 1
+\end{bmatrix} \otimes
+\begin{bmatrix}
+ 3 \\
+-6 \\
+ 5
+\end{bmatrix} =
+\begin{bmatrix}
+-4 \\
+ 2 \\
+-1
+\end{bmatrix}\]
+</div>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_solving_using_an_augmented_matrix">Solving Using an augmented matrix</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>After wrting a system of equations as an augmented matrix, you perform row
+operations until the metrix is in either row echelon form or reduced row echelon
+form.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>Reduced Row echelon form</strong>:  Gives you the answer directly</p>
+</li>
+<li>
+<p><strong>Row echelon form</strong>:  Required back-solving or algebraic manipulations to get
+your answer.</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Works for any size matrix.
+</td>
+</tr>
+</table>
+</div>
+<div class="sect2">
+<h3 id="_one_solution">One Solution</h3>
+<details>
+<summary class="title">Problem</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{6}
+-x &amp; {}-{} &amp;  y &amp; {}-{} &amp; z  &amp; =  3 \\
+4x &amp; {}+{} &amp; 5y &amp;       &amp;    &amp; = -6 \\
+   &amp;       &amp;  y &amp; {}-{} &amp; 3z &amp; = 5
+\end{alignat*}
+\end{cases}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\left[
+\begin{array}{ccc : c}
+-1 &amp; -1 &amp;  -1 &amp;  3 \\
+ 4 &amp;  5 &amp;   0 &amp; -6 \\
+ 0 &amp;  1 &amp;  -3 &amp;  5
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[-1R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  1 &amp;   1 &amp; -3 \\
+ 4 &amp;  5 &amp;   0 &amp; -6 \\
+ 0 &amp;  1 &amp;  -3 &amp;  5
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[-4R_1 + R_2 \to R_2
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  1 &amp;   1 &amp; -3 \\
+ 0 &amp;  1 &amp;  -4 &amp;  6 \\
+ 0 &amp;  1 &amp;  -3 &amp;  5
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[-1R_2 + R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  0 &amp;   5 &amp; -9 \\
+ 0 &amp;  1 &amp;  -4 &amp;  6 \\
+ 0 &amp;  1 &amp;  -3 &amp;  5
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[-1R_2 + R_3 \to R_3
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  0 &amp;   5 &amp; -9 \\
+ 0 &amp;  1 &amp;  -4 &amp;  6 \\
+0 &amp;  0 &amp;   1 &amp; -1
+\end{array}
+\right]\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Row Echeleon Form</summary>
+<div class="content">
+<div class="paragraph">
+<p>back substitute for z</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+y - 4(-1) &amp;= 6 \\
+y + 4 &amp;= 6 \\
+y &amp;= 2
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>back substitute for for z again</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{align*}
+x + 5(-1) &amp;= -9 \\
+x - 5 &amp;= -9 \\
+x &amp;= -4
+\end{align*}\]
+</div>
+</div>
+<div class="paragraph">
+<p>solution</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[(x, y, z) = (-4, 2, -1)\]
+</div>
+</div>
+</div>
+</details>
+<details>
+<summary class="title">Reduced Row Echeleon Form</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[-5R_3 + R_1 \to R_1
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  0 &amp;   0 &amp; -4 \\
+ 0 &amp;  1 &amp;  -4 &amp;  6 \\
+ 0 &amp;  0 &amp;   1 &amp; -1
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[4R_3 + R_2 \to R_2
+\left[
+\begin{array}{ccc : c}
+ 1 &amp;  0 &amp;   0 &amp; -4 \\
+ 0 &amp;  1 &amp;   0 &amp;  2 \\
+ 0 &amp;  0 &amp;   1 &amp; -1
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="paragraph">
+<p>solution</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[(x, y, z) = (-4, 2, -1)\]
+</div>
+</div>
+</div>
+</details>
+</div>
+<div class="sect2">
+<h3 id="_multiple_solutions_2">Multiple Solutions</h3>
+<div class="paragraph">
+<p>The system has more than one solution when the <strong>Row Echelon Form</strong>, or <strong>Reduced
+Row Echelon Form</strong>  has one or more rows of 0s.</p>
+</div>
+<details>
+<summary class="title">example</summary>
+<div class="content">
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+\begin{alignat*}{6}
+ x &amp; {}-{} &amp;  y &amp; {}-{} &amp;  z  &amp; = 1 \\
+3x &amp; {}-{} &amp; 2y &amp; {}-{} &amp; 7z  &amp; = 0 \\
+5x &amp; {}-{} &amp; 4y &amp; {}-{} &amp; 9z  &amp; = 2
+\end{alignat*}
+\end{cases}
+
+\qquad %
+
+\left[
+\begin{array}{ccc : c}
+ 1 &amp; -1 &amp; -1 &amp; 1 \\
+ 3 &amp; -2 &amp; -7 &amp; 0 \\
+ 5 &amp; -4 &amp; -9 &amp; 2
+\end{array}
+\right]\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+    \begin{matrix}
+    -3R_1 + R_2 \to R_2 \\
+    -5R_1 + R_3 \to R_3
+    \end{matrix}
+    &amp; \left[ \begin{array}{ccc|r}
+    1 &amp; -1 &amp; -1 &amp; 1 \\
+    0 &amp; 1 &amp; -4 &amp; -3 \\
+    0 &amp; 1 &amp; -4 &amp; -3
+    \end{array} \right] \\[15pt]
+    -1R_2 + R_3 \to R_3
+    &amp; \left[ \begin{array}{ccc|r}
+    1 &amp; -1 &amp; -1 &amp; 1 \\
+    0 &amp; 1 &amp; -4 &amp; -3 \\
+    0 &amp; 0 &amp; 0 &amp; 0
+    \end{array} \right] \\[15pt]
+    R_2 + R_1 \to R_1
+    &amp; \left[ \begin{array}{ccc|r}
+    1 &amp; 0 &amp; -5 &amp; -2 \\
+    0 &amp; 1 &amp; -4 &amp; -3 \\
+    0 &amp; 0 &amp; 0 &amp; 0
+    \end{array} \right]
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>k = z</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{r}
+\begin{alignat*}{4}
+ x &amp; {}-{} &amp;  5k &amp; = -2 \\
+   &amp;       &amp; +5k &amp; = +5k\\
+\hline
+   &amp;       &amp;  x  &amp; = -2 + 5x \\
+\end{alignat*}
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{array}{r}
+\begin{alignat*}{4}
+ y &amp; {}-{} &amp;  4k &amp; = -3 \\
+   &amp;       &amp; +4k &amp; = +4k\\
+\hline
+   &amp;       &amp;  y  &amp; = -3 + 4k \\
+\end{alignat*}
+\end{array}\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{cases}
+    x - 5z = -2 \\
+    y - 4z = -3
+\end{cases}
+\text{ or }
+\begin{cases}
+    x = 5z - 2 \\
+    y = 4z - 3
+\end{cases}\]
+</div>
+</div>
+<div class="paragraph">
+<p>Both x and y are dependent on what z is. So I choose the parameter k to
+represent some real number. I let z = k, which makes x = 5k – 2 and y = 4k – 3.
+The solution is also written (5k – 2,4k – 3,k) as an ordered triple. Once a real
+number is selected for k, then a solution is created from the ordered triple.
+For example, if k = 1, then a solution for the system is (3,1,1).</p>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://brilliant.org/wiki/3d-coordinate-geometry-equation-of-a-plane/">Brilliant | 3D Coordinate Geometry - Equation of a Plane</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
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+Last updated 2026-05-08 21:58:32 -0400
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+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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\ No newline at end of file
diff --git a/Linear-Algrebra/Terminology.adoc b/Linear-Algrebra/Terminology.adoc
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+= Linear Algrebra | Terminology
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Linear Equations
+
+[stem]
+++++
+a_1x_1 + a_2x_2 + \dots + a_nx_n = b
+++++
+
+== Basis Vectors
+
+Basis vectors are the starting point of your vectors operations, They are
+typically the unit vectors
+
+== Unit Vectors
+
+* *stem:[\hat{i}]*: One unit in the x direction
+
+* *stem:[\hat{j}]*: One unit in the y direction
+
+* *stem:[\hat{k}]*: One unit in the z direction
+
+[.center]
+image::unit-vectors.png[]
+
+[.center]
+image::3D-vector-look.png[]
+
+== Linear Combination 
+
+A _linear combination_ uses vectors in a set to create a new vector that's the
+same dimension as the vectors involved in teh operations.
+
+The "Linear Combination" of stem:[\vec{v}] and stem:[\vec{w}].
+
+[stem]
+++++
+a\vec{v} + b\vec{w}
+++++
+
+If you fix one of the scalars and let the other one change it's value freely the
+tip of the resulting vector draws a straight line
+
+.scalare a changing its value feely
+[.center]
+image::linear-combination-example1.png[]
+
+== Span
+
+In linear algebra, the span of a set of vectors is the collection of all
+possible vectors you can create by scaling and adding them together (their
+linear combinations).
+
+=== Span of two vectors in 3 dimensional space
+
+[.center]
+image::span-of-two-vectors-in-3D-space.png[]
+
+== Linear Dependent
+
+When you have multiple vectors and you can remove one without reducing the
+span this means those vectors are linearly dependent. (i.e. the vectors don't
+add another demension to the span). 
+
+=== 2D Space
+
+The area of the parallellogram is zero.
+
+[.center]
+image::linear-dependent.png[]
+
+=== 3D Space
+
+The volume of the parallelepiped is zero.
+
+[.center]
+image::3d-linear-dependent.png[]
+
+== Linear Independent
+
+If each vector adds another demension to the span, they are said to be linearly
+independent.
+
+[.center]
+image::linearly-independent.png[]
+
+== Basis
+
+The basis of a vector space is a set of linearly independent vectors that span
+the full space
+
+== Shear
+In linear algebra, a shear transformation is a linear mapping that displaces
+each point in a fixed direction by an amount proportional to its distance from a
+line (in 2D) or a plane (in 3D) that remains fixed.
+
+Informally, you can think of it as "pushing" the top of an object while the base
+stays glued to the floor. Unlike scaling or rotation, a shear changes the angles
+of a shape but preserves its area (or volume).
+
+[.center]
+image::simple-shear.png[]
+
+== Composition
+
+In linear algebra, composition refers to the process of applying one linear
+transformation after another.If you have two transformations, stem:[T] and
+stem:[S], the composition (written as stem:[S \circ T]) means you first apply
+stem:[T] to a vector, and then apply $S$ to the result.
+
+[.center]
+image::linear-algrebra-composition-example1.png[]
+
+[.center]
+image::linear-combination-example2.png[]
+
+== Unit Vector Displacement
+
+If the unit vectors are displaced in any of these ways then, you can describe
+those displacement like this:
+
+[.inline]
+* *Reflection* (Orientation flip)
+** stem:[\hat{i}] now points to the left
+** stem:[\hat{j}] stays the same
+
+[.inline]
+* *Rotation* (Orientation Preserved)
+** stem:[\hat{i}] now points to the left
+** stem:[\hat{j}] now points down
+
+== Square Matrices | Singular and Non-singular Matrices
+
+A square matrix is _singular_ if it has a multiplicative inverse; a matrix is
+non-singular if it does _not_ have a multiplicative inverse.
+
+== Inverse
+
+Two numbers are multiplicative inverses if their product is 1 (the
+multiplicative identity). like 4 and 1/4, and 3/5 and 5/3. When a matrix has a
+multiplicative inverse, the product of the matrix and its inverse is equal to an
+identity matrix (multiplicative identity). And, further-more, you can multiply
+the two matrices involved in either order (communativity) and still get the
+identity.
diff --git a/Linear-Algrebra/Terminology.html b/Linear-Algrebra/Terminology.html
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+body.book #header .details span:first-child{margin-left:0!important}
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+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Terminology</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_linear_equations">Linear Equations</a></li>
+<li><a href="#_basis_vectors">Basis Vectors</a></li>
+<li><a href="#_unit_vectors">Unit Vectors</a></li>
+<li><a href="#_linear_combination">Linear Combination</a></li>
+<li><a href="#_span">Span</a>
+<ul class="sectlevel2">
+<li><a href="#_span_of_two_vectors_in_3_dimensional_space">Span of two vectors in 3 dimensional space</a></li>
+</ul>
+</li>
+<li><a href="#_linear_dependent">Linear Dependent</a>
+<ul class="sectlevel2">
+<li><a href="#_2d_space">2D Space</a></li>
+<li><a href="#_3d_space">3D Space</a></li>
+</ul>
+</li>
+<li><a href="#_linear_independent">Linear Independent</a></li>
+<li><a href="#_basis">Basis</a></li>
+<li><a href="#_shear">Shear</a></li>
+<li><a href="#_composition">Composition</a></li>
+<li><a href="#_unit_vector_displacement">Unit Vector Displacement</a></li>
+<li><a href="#_square_matrices_singular_and_non_singular_matrices">Square Matrices | Singular and Non-singular Matrices</a></li>
+<li><a href="#_inverse">Inverse</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_linear_equations">Linear Equations</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[a_1x_1 + a_2x_2 + \dots + a_nx_n = b\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_basis_vectors">Basis Vectors</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Basis vectors are the starting point of your vectors operations, They are
+typically the unit vectors</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_unit_vectors">Unit Vectors</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><strong>\(\hat{i}\)</strong>: One unit in the x direction</p>
+</li>
+<li>
+<p><strong>\(\hat{j}\)</strong>: One unit in the y direction</p>
+</li>
+<li>
+<p><strong>\(\hat{k}\)</strong>: One unit in the z direction</p>
+</li>
+</ul>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/unit-vectors.png" alt="unit vectors">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3D-vector-look.png" alt="3D vector look">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_linear_combination">Linear Combination</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>A <em>linear combination</em> uses vectors in a set to create a new vector that&#8217;s the
+same dimension as the vectors involved in teh operations.</p>
+</div>
+<div class="paragraph">
+<p>The "Linear Combination" of \(\vec{v}\) and \(\vec{w}\).</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[a\vec{v} + b\vec{w}\]
+</div>
+</div>
+<div class="paragraph">
+<p>If you fix one of the scalars and let the other one change it&#8217;s value freely the
+tip of the resulting vector draws a straight line</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linear-combination-example1.png" alt="linear combination example1">
+</div>
+<div class="title">Figure 1. scalare a changing its value feely</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_span">Span</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In linear algebra, the span of a set of vectors is the collection of all
+possible vectors you can create by scaling and adding them together (their
+linear combinations).</p>
+</div>
+<div class="sect2">
+<h3 id="_span_of_two_vectors_in_3_dimensional_space">Span of two vectors in 3 dimensional space</h3>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/span-of-two-vectors-in-3D-space.png" alt="span of two vectors in 3D space">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_linear_dependent">Linear Dependent</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When you have multiple vectors and you can remove one without reducing the
+span this means those vectors are linearly dependent. (i.e. the vectors don&#8217;t
+add another demension to the span).</p>
+</div>
+<div class="sect2">
+<h3 id="_2d_space">2D Space</h3>
+<div class="paragraph">
+<p>The area of the parallellogram is zero.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linear-dependent.png" alt="linear dependent">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_3d_space">3D Space</h3>
+<div class="paragraph">
+<p>The volume of the parallelepiped is zero.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/3d-linear-dependent.png" alt="3d linear dependent">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_linear_independent">Linear Independent</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>If each vector adds another demension to the span, they are said to be linearly
+independent.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linearly-independent.png" alt="linearly independent">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_basis">Basis</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The basis of a vector space is a set of linearly independent vectors that span
+the full space</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_shear">Shear</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In linear algebra, a shear transformation is a linear mapping that displaces
+each point in a fixed direction by an amount proportional to its distance from a
+line (in 2D) or a plane (in 3D) that remains fixed.</p>
+</div>
+<div class="paragraph">
+<p>Informally, you can think of it as "pushing" the top of an object while the base
+stays glued to the floor. Unlike scaling or rotation, a shear changes the angles
+of a shape but preserves its area (or volume).</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/simple-shear.png" alt="simple shear">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_composition">Composition</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In linear algebra, composition refers to the process of applying one linear
+transformation after another.If you have two transformations, \(T\) and
+\(S\), the composition (written as \(S \circ T\)) means you first apply
+\(T\) to a vector, and then apply $S$ to the result.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linear-algrebra-composition-example1.png" alt="linear algrebra composition example1">
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/linear-combination-example2.png" alt="linear combination example2">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_unit_vector_displacement">Unit Vector Displacement</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>If the unit vectors are displaced in any of these ways then, you can describe
+those displacement like this:</p>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>Reflection</strong> (Orientation flip)</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\(\hat{i}\) now points to the left</p>
+</li>
+<li>
+<p>\(\hat{j}\) stays the same</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>Rotation</strong> (Orientation Preserved)</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\(\hat{i}\) now points to the left</p>
+</li>
+<li>
+<p>\(\hat{j}\) now points down</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_square_matrices_singular_and_non_singular_matrices">Square Matrices | Singular and Non-singular Matrices</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>A square matrix is <em>singular</em> if it has a multiplicative inverse; a matrix is
+non-singular if it does <em>not</em> have a multiplicative inverse.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_inverse">Inverse</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Two numbers are multiplicative inverses if their product is 1 (the
+multiplicative identity). like 4 and 1/4, and 3/5 and 5/3. When a matrix has a
+multiplicative inverse, the product of the matrix and its inverse is equal to an
+identity matrix (multiplicative identity). And, further-more, you can multiply
+the two matrices involved in either order (communativity) and still get the
+identity.</p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
+  },
+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Linear-Algrebra/Vector-Math.adoc b/Linear-Algrebra/Vector-Math.adoc
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+= Linear Algrebra | Vector Math
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Normalization
+
+[stem]
+++++
+\hat{u} = \frac{\overrightarrow{v}}{|\overrightarrow{v}|}
+++++
+
+== Cross Product
+
+Only works in 3 demensions (stem:[\mathbb{R}^3]).
+
+
+[stem]
+++++
+\vec{a} = \begin{bmatrix}
+    a_1 \\
+    a_2 \\
+    a_3
+\end{bmatrix}, \quad 
+\vec{b} = \begin{bmatrix}
+    b_1 \\
+    b_2 \\
+    b_3
+\end{bmatrix} \implies
+\vec{a} \times \vec{b} = \begin{bmatrix}
+    a_2b_3 - a_3b_2 \\
+    a_3b_1 - a_1b_3 \\
+    a_1b_2 - a_2b_1
+\end{bmatrix}
+++++
+
+=== Area of Parallelogram
+
+[stem]
+++++
+Area = || a x b ||
+++++
+
+=== Area of Triangle
+
+[stem]
+++++
+Area = \frac{1}{2}|| a x b ||
+++++
+
+=== Perpendicular
+
+[.center]
+image::cross-product-perdendicular.png[]
+
+But which way?
+
+Use the right hand rule. This will tell us the direction of the vector produced
+from the cross product.
+
+[.center]
+image::cross-product-right-hand-rule.png[]
+
+=== Order Matters
+
+If v is on the right of w then the cross positive is positive
+
+[.center]
+image::cross-product-positive.png[]
+
+If v is on the left of w then the cross positive is negative
+
+[.center]
+image::cross-product-negative.png[]
+
+think of oriantation
+
+[.center]
+image::cross-product-oriantation.png[]
+
+
+== Dot Product (inner-product)
+
+Works in any demension
+
+[stem]
+++++
+\overrightarrow{a} \cdot \overrightarrow{b} \\
+\left[ \begin{array}{cc}
+a_{1} \\
+a_{2} \\
+\vdots \\
+a_{n}
+\end{array} \right]
+
+\cdot
+
+\left[ \begin{array}{cc}
+b_{1} \\
+b_{2} \\
+\vdots \\
+b_{n}
+\end{array} \right]
+=
+a_{1} * b_{1} + a_{2} * b_{2} + * a_{n} * b_{n}
+++++
+
+=== Normal Vector and Dot procuct
+
+A normal vector has a magnitude of one and can point in any direction. The dot
+product of the normal vector and a point can tell you the distance that point is
+in that direction from the starting point of the normal vector.
+
+In this image the white arrow is the normal vector and the blue dot is the
+distance we are finding in relation to the normal vector.
+
+[.center]
+image::dot_product.png[]
+
+
+[stem]
+++++
+a = normal vector \\
+b = point \\
+\\
+a.x * b.x + a.y * b.y = distance
+
+++++
+
+== Vector Addition
+
+[stem]
+++++
+\left[\begin{array}{a} 3 \\ 1 \\ \end{array}\right] + 
+\left[\begin{array}{a} 2 \\ 3 \\ \end{array}\right] = 
+\left[\begin{array}{a} 2 + 1 \\ 2 + 3 \\ \end{array}\right] = 
+\left[\begin{array}{a} 3 \\ 5 \\ \end{array}\right]
+++++
+
+[.center]
+image::vector-addition.png[]
+
+== Finding the Magnitude of a Vector
+
+[stem]
+++++
+||\overrightarrow{a}|| = \sqrt{a_{1}^2 + a_{2}^2 + \dots a_{n}^2}
+++++
+
+== Finding the angle between two vectors
+
+[stem]
+++++
+\frac{u \cdot v}{||u|| \cdot ||v||} = cos(\theta)
+++++
+
+The result is the x value of the unit cirle, from that you can determine the y
+value of the unit circle and the angle.
+
+
+== Vector and scalar multiplication
+
+[stem]
+++++
+K \left[\begin{array}{cc}
+V_1 \\
+V_2 \\
+V_3 \\
+\vdots \\
+V_n \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+KV_1 \\
+KV_2 \\
+KV_3 \\
+\vdots \\
+Kv_n \\
+\end{array}\right]
+++++
+
+== Links
+
+* https://www.cuemath.com/vector-formulas/[Cue Math | Vector Formulas]
diff --git a/Linear-Algrebra/Vector-Math.html b/Linear-Algrebra/Vector-Math.html
new file mode 100644 (file)
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Vector Math</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_normalization">Normalization</a></li>
+<li><a href="#_cross_product">Cross Product</a>
+<ul class="sectlevel2">
+<li><a href="#_area_of_parallelogram">Area of Parallelogram</a></li>
+<li><a href="#_area_of_triangle">Area of Triangle</a></li>
+<li><a href="#_perpendicular">Perpendicular</a></li>
+<li><a href="#_order_matters">Order Matters</a></li>
+</ul>
+</li>
+<li><a href="#_dot_product_inner_product">Dot Product (inner-product)</a>
+<ul class="sectlevel2">
+<li><a href="#_normal_vector_and_dot_procuct">Normal Vector and Dot procuct</a></li>
+</ul>
+</li>
+<li><a href="#_vector_addition">Vector Addition</a></li>
+<li><a href="#_finding_the_magnitude_of_a_vector">Finding the Magnitude of a Vector</a></li>
+<li><a href="#_finding_the_angle_between_two_vectors">Finding the angle between two vectors</a></li>
+<li><a href="#_vector_and_scalar_multiplication">Vector and scalar multiplication</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_normalization">Normalization</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\hat{u} = \frac{\overrightarrow{v}}{|\overrightarrow{v}|}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_cross_product">Cross Product</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Only works in 3 demensions (\(\mathbb{R}^3\)).</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\vec{a} = \begin{bmatrix}
+    a_1 \\
+    a_2 \\
+    a_3
+\end{bmatrix}, \quad
+\vec{b} = \begin{bmatrix}
+    b_1 \\
+    b_2 \\
+    b_3
+\end{bmatrix} \implies
+\vec{a} \times \vec{b} = \begin{bmatrix}
+    a_2b_3 - a_3b_2 \\
+    a_3b_1 - a_1b_3 \\
+    a_1b_2 - a_2b_1
+\end{bmatrix}\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_area_of_parallelogram">Area of Parallelogram</h3>
+<div class="stemblock">
+<div class="content">
+\[Area = || a x b ||\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_area_of_triangle">Area of Triangle</h3>
+<div class="stemblock">
+<div class="content">
+\[Area = \frac{1}{2}|| a x b ||\]
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_perpendicular">Perpendicular</h3>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/cross-product-perdendicular.png" alt="cross product perdendicular">
+</div>
+</div>
+<div class="paragraph">
+<p>But which way?</p>
+</div>
+<div class="paragraph">
+<p>Use the right hand rule. This will tell us the direction of the vector produced
+from the cross product.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/cross-product-right-hand-rule.png" alt="cross product right hand rule">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_order_matters">Order Matters</h3>
+<div class="paragraph">
+<p>If v is on the right of w then the cross positive is positive</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/cross-product-positive.png" alt="cross product positive">
+</div>
+</div>
+<div class="paragraph">
+<p>If v is on the left of w then the cross positive is negative</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/cross-product-negative.png" alt="cross product negative">
+</div>
+</div>
+<div class="paragraph">
+<p>think of oriantation</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/cross-product-oriantation.png" alt="cross product oriantation">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_dot_product_inner_product">Dot Product (inner-product)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Works in any demension</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\overrightarrow{a} \cdot \overrightarrow{b} \\
+\left[ \begin{array}{cc}
+a_{1} \\
+a_{2} \\
+\vdots \\
+a_{n}
+\end{array} \right]
+
+\cdot
+
+\left[ \begin{array}{cc}
+b_{1} \\
+b_{2} \\
+\vdots \\
+b_{n}
+\end{array} \right]
+=
+a_{1} * b_{1} + a_{2} * b_{2} + * a_{n} * b_{n}\]
+</div>
+</div>
+<div class="sect2">
+<h3 id="_normal_vector_and_dot_procuct">Normal Vector and Dot procuct</h3>
+<div class="paragraph">
+<p>A normal vector has a magnitude of one and can point in any direction. The dot
+product of the normal vector and a point can tell you the distance that point is
+in that direction from the starting point of the normal vector.</p>
+</div>
+<div class="paragraph">
+<p>In this image the white arrow is the normal vector and the blue dot is the
+distance we are finding in relation to the normal vector.</p>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/dot_product.png" alt="dot product">
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[a = normal vector \\
+b = point \\
+\\
+a.x * b.x + a.y * b.y = distance\]
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_vector_addition">Vector Addition</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\left[\begin{array}{a} 3 \\ 1 \\ \end{array}\right] +
+\left[\begin{array}{a} 2 \\ 3 \\ \end{array}\right] =
+\left[\begin{array}{a} 2 + 1 \\ 2 + 3 \\ \end{array}\right] =
+\left[\begin{array}{a} 3 \\ 5 \\ \end{array}\right]\]
+</div>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/vector-addition.png" alt="vector addition">
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_finding_the_magnitude_of_a_vector">Finding the Magnitude of a Vector</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[||\overrightarrow{a}|| = \sqrt{a_{1}^2 + a_{2}^2 + \dots a_{n}^2}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_finding_the_angle_between_two_vectors">Finding the angle between two vectors</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[\frac{u \cdot v}{||u|| \cdot ||v||} = cos(\theta)\]
+</div>
+</div>
+<div class="paragraph">
+<p>The result is the x value of the unit cirle, from that you can determine the y
+value of the unit circle and the angle.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_vector_and_scalar_multiplication">Vector and scalar multiplication</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\[K \left[\begin{array}{cc}
+V_1 \\
+V_2 \\
+V_3 \\
+\vdots \\
+V_n \\
+\end{array}\right] =
+\left[\begin{array}{cc}
+KV_1 \\
+KV_2 \\
+KV_3 \\
+\vdots \\
+Kv_n \\
+\end{array}\right]\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.cuemath.com/vector-formulas/">Cue Math | Vector Formulas</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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\ No newline at end of file
diff --git a/Linear-Algrebra/Vector-Rules.adoc b/Linear-Algrebra/Vector-Rules.adoc
new file mode 100644 (file)
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+= Linear Algrebra | Vector Rules
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem: latexmath
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Triangle inequality
+
+The magnitude of the sum of vectors is always less than or equal to the sum of
+the magnitudes of the vectos
+
+[stem]
+++++
+|| u + v ||\ \leq\ || u || + || v ||
+++++
+
+== Vector and scalar multiplication
+
+[.inline]
+* *When multiplying by a scalar that is one*
+    ** The scalar stays the same
+
+[.inline]
+* *When multiplying by a scalar greater than one*
+    ** The dierection stays the same but the size is bigger
+
+[.inline]
+* *When multiplying by a scalar between zero and one*
+    ** The dierection stays the same but the size is smaller
+
+[.inline]
+* *When multiplying by a negative scalar*
+    ** The size of the vector changes (except with -1)
+    ** The direction of the vector is reversed
+
+== Orthogonality (right angles)
+
+If the dot product of two vectors is equal to zero then the rays must be
+perendiculator to each other
+
+== Magnitude
+
+In vector math, it is impossible for the length (magnitude) to be less than
+zero. While the individual components (x, y, z) can certainly be
+negative—pointing the vector "backward" or "down"—the length itself is
+calculated using a formula that squares those values, which always results in a
+positive number.
diff --git a/Linear-Algrebra/Vector-Rules.html b/Linear-Algrebra/Vector-Rules.html
new file mode 100644 (file)
index 0000000..fdb3c66
--- /dev/null
+++ b/Linear-Algrebra/Vector-Rules.html
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+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+window.addEventListener("message", (e) => {
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+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Linear Algrebra | Vector Rules</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_triangle_inequality">Triangle inequality</a></li>
+<li><a href="#_vector_and_scalar_multiplication">Vector and scalar multiplication</a></li>
+<li><a href="#_orthogonality_right_angles">Orthogonality (right angles)</a></li>
+<li><a href="#_magnitude">Magnitude</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_triangle_inequality">Triangle inequality</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The magnitude of the sum of vectors is always less than or equal to the sum of
+the magnitudes of the vectos</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[|| u + v ||\ \leq\ || u || + || v ||\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_vector_and_scalar_multiplication">Vector and scalar multiplication</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>When multiplying by a scalar that is one</strong></p>
+<div class="ulist">
+<ul>
+<li>
+<p>The scalar stays the same</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>When multiplying by a scalar greater than one</strong></p>
+<div class="ulist">
+<ul>
+<li>
+<p>The dierection stays the same but the size is bigger</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>When multiplying by a scalar between zero and one</strong></p>
+<div class="ulist">
+<ul>
+<li>
+<p>The dierection stays the same but the size is smaller</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p><strong>When multiplying by a negative scalar</strong></p>
+<div class="ulist">
+<ul>
+<li>
+<p>The size of the vector changes (except with -1)</p>
+</li>
+<li>
+<p>The direction of the vector is reversed</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_orthogonality_right_angles">Orthogonality (right angles)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>If the dot product of two vectors is equal to zero then the rays must be
+perendiculator to each other</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_magnitude">Magnitude</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In vector math, it is impossible for the length (magnitude) to be less than
+zero. While the individual components (x, y, z) can certainly be
+negative—pointing the vector "backward" or "down"—the length itself is
+calculated using a formula that squares those values, which always results in a
+positive number.</p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+  },
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+</body>
+</html>
\ No newline at end of file
diff --git a/Linear-Algrebra/index.adoc b/Linear-Algrebra/index.adoc
new file mode 100644 (file)
index 0000000..fbf9d80
--- /dev/null
+++ b/Linear-Algrebra/index.adoc
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+= ./Linear-Algrebra
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Vector-Math.html;hb=HEAD[Vector-Math]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Math.html;hb=HEAD[Matrix-Math]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Properties.html;hb=HEAD[Properties]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Types.html;hb=HEAD[Matrix-Types]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Vector-Rules.html;hb=HEAD[Vector-Rules]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/System-of-Equations.html;hb=HEAD[System-of-Equations]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Terminology.html;hb=HEAD[Terminology]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Graphing.html;hb=HEAD[Graphing]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Operations.html;hb=HEAD[Matrix-Operations]
diff --git a/Linear-Algrebra/index.html b/Linear-Algrebra/index.html
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+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
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+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Linear-Algrebra</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Vector-Math.html;hb=HEAD">Vector-Math</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Math.html;hb=HEAD">Matrix-Math</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Properties.html;hb=HEAD">Properties</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Types.html;hb=HEAD">Matrix-Types</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Vector-Rules.html;hb=HEAD">Vector-Rules</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/System-of-Equations.html;hb=HEAD">System-of-Equations</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Terminology.html;hb=HEAD">Terminology</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Graphing.html;hb=HEAD">Graphing</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/Matrix-Operations.html;hb=HEAD">Matrix-Operations</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Makehtml b/Makehtml
new file mode 100755 (executable)
index 0000000..cf79265
--- /dev/null
+++ b/Makehtml
@@ -0,0 +1,83 @@
+#!/usr/bin/env bash
+. docinfo.conf
+
+clean () {
+rm -rf diag*.png
+rm -rf $(find . \
+            -type f \
+            -regex '.*\.html' \
+            -not -name "README.adoc" \
+            -not -name "notes.adoc" \
+            -not -name "rendered.html" \
+            | grep -v docinfo)
+rm -rf $(find . -type f -name 'index.adoc')
+}
+
+topicSearch() {
+find . -maxdepth 1 -type f -not -name 'index.adoc' -name '*.adoc'
+}
+
+createIndices () {
+echo "creating indices..."
+dirsearch="$(find . -maxdepth 1 -type d -not -name '.*' -not -name "README.html")"
+
+rm -rf index.adoc
+rd="$PWD"
+
+echo "= Math Notes
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+" > index.adoc
+
+for dir in $dirsearch; do
+    cd $dir > /dev/null
+
+    topic=$(echo ${dir} | cut -c 3-)
+
+    echo "* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=${topic}/index.html;hb=HEAD[${topic}]" >> ../index.adoc
+
+    rm -rf index.adoc
+         echo \
+"= $dir
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+" > index.adoc
+
+     for adoc in $(topicSearch); do
+        file=$(echo "$adoc" | cut -c 3-)
+        subTopic=${file%.*}
+        echo "* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=${topic}/${subTopic}.html;hb=HEAD[${subTopic}]" >> index.adoc
+
+    done
+
+    cd - > /dev/null
+done
+
+
+}
+
+build () {
+FILENAME="$(find . -type f -name "*.adoc" -not -name "README.adoc" -not -name "notes.adoc")"
+asciidoctor \
+    -a HOSTNAME="${HOSTNAME}" \
+    -a DOCINFODIR="${DOCINFODIR}" \
+    -r asciidoctor-diagram \
+    -b html5 $FILENAME
+}
+
+make () {
+clean
+createIndices
+build
+}
+
+# use this to call one of the above functions
+$@
diff --git a/Physics/Ball-Bouncing.adoc b/Physics/Ball-Bouncing.adoc
new file mode 100644 (file)
index 0000000..432ed6f
--- /dev/null
+++ b/Physics/Ball-Bouncing.adoc
@@ -0,0 +1,37 @@
+= Physics | Bouncing Ball
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images;hb=HEAD
+
+== Formulas
+
+* velocity downward
+    - stem:[v = g t]
+
+* height 
+    - stem:[h = h_0 - \frac{1}{2}t^2]
+
+* how long any object takes to fall from a given height
+    - stem:[t = \sqrt{\frac{2h}{g}}]
+
+* restitution
+    - stem:[v_1 = e * v_0]
+    - the coefficent of resitution (e) represents energy loss.
+
+* rising
+    - stem:[ v = e * v - g t ]
+
+* falling
+    - stem:[ v = g t ]
+
+== Restitution
+Typical values:
+
+* Rubber ball: stem:[e \approx 0.8 - 0.9]
+* Basketball: stem:[e \approx 0.75]
+* Tennis ball: stem:[e \approx 0.6]
diff --git a/Physics/Ball-Bouncing.html b/Physics/Ball-Bouncing.html
new file mode 100644 (file)
index 0000000..6b595c8
--- /dev/null
+++ b/Physics/Ball-Bouncing.html
@@ -0,0 +1,1164 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Physics | Bouncing Ball</title>
+<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700">
+<style>
+/*! Asciidoctor default stylesheet | MIT License | https://asciidoctor.org */
+/* Uncomment the following line when using as a custom stylesheet */
+/* @import "https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700"; */
+html{font-family:sans-serif;-webkit-text-size-adjust:100%}
+a{background:none}
+a:focus{outline:thin dotted}
+a:active,a:hover{outline:0}
+h1{font-size:2em;margin:.67em 0}
+b,strong{font-weight:bold}
+abbr{font-size:.9em}
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+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
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+});
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+// Capture scroll position while scrolling
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+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
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+        scrollPosition: scrollPosition
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+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Physics | Bouncing Ball</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_formulas">Formulas</a></li>
+<li><a href="#_restitution">Restitution</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_formulas">Formulas</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>velocity downward</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$v = g t\$</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>height</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$h = h_0 - \frac{1}{2}t^2\$</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>how long any object takes to fall from a given height</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$t = \sqrt{\frac{2h}{g}}\$</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>restitution</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$v_1 = e * v_0\$</p>
+</li>
+<li>
+<p>the coefficent of resitution (e) represents energy loss.</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>rising</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$ v = e * v - g t \$</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>falling</p>
+<div class="ulist">
+<ul>
+<li>
+<p>\$ v = g t \$</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_restitution">Restitution</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Typical values:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Rubber ball: \$e \approx 0.8 - 0.9\$</p>
+</li>
+<li>
+<p>Basketball: \$e \approx 0.75\$</p>
+</li>
+<li>
+<p>Tennis ball: \$e \approx 0.6\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
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+</html>
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diff --git a/Physics/Ball-Dynamic-Collision.adoc b/Physics/Ball-Dynamic-Collision.adoc
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+= Physics | Dynamic Collisions
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:wrap: normal
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+NOTE: This was mainly created by chatgpt
+
+
+The following explains the physics and math inside the collision‐response loop.
+All lines are wrapped to 80 characters.
+
+== Overview
+
+[source,cpp]
+----
+for (auto c : vecCollidingPairs)
+               {
+                       sBall *b1 = c.first;
+                       sBall *b2 = c.second;
+
+                       // Distance between balls
+                       float fDistance = sqrtf((b1->px - b2->px)*(b1->px - b2->px) + (b1->py - b2->py)*(b1->py - b2->py));
+
+                       // Normal
+                       float nx = (b2->px - b1->px) / fDistance;
+                       float ny = (b2->py - b1->py) / fDistance;
+
+                       // Tangent
+                       float tx = -ny;
+                       float ty = nx;
+
+                       // Dot Product Tangent
+                       float dpTan1 = b1->vx * tx + b1->vy * ty;
+                       float dpTan2 = b2->vx * tx + b2->vy * ty;
+
+                       // Dot Product Normal
+                       float dpNorm1 = b1->vx * nx + b1->vy * ny;
+                       float dpNorm2 = b2->vx * nx + b2->vy * ny;
+
+                       // Conservation of momentum in 1D
+                       float m1 = (dpNorm1 * (b1->mass - b2->mass) + 2.0f * b2->mass * dpNorm2) / (b1->mass + b2->mass);
+                       float m2 = (dpNorm2 * (b2->mass - b1->mass) + 2.0f * b1->mass * dpNorm1) / (b1->mass + b2->mass);
+
+                       // Update ball velocities
+                       b1->vx = tx * dpTan1 + nx * m1;
+                       b1->vy = ty * dpTan1 + ny * m1;
+                       b2->vx = tx * dpTan2 + nx * m2;
+                       b2->vy = ty * dpTan2 + ny * m2;
+
+                       // Wikipedia Version - Maths is smarter but same
+                       //float kx = (b1->vx - b2->vx);
+                       //float ky = (b1->vy - b2->vy);
+                       //float p = 2.0 * (nx * kx + ny * ky) / (b1->mass + b2->mass);
+                       //b1->vx = b1->vx - p * b2->mass * nx;
+                       //b1->vy = b1->vy - p * b2->mass * ny;
+                       //b2->vx = b2->vx + p * b1->mass * nx;
+                       //b2->vy = b2->vy + p * b1->mass * ny;
+               }
+----
+
+This code performs a *realistic elastic collision* between two balls. It does
+this by:
+
+. Computing the normal and tangent vectors between the balls.
+. Projecting each ball's velocity onto these axes.
+. Swapping the normal components using 1D momentum equations.
+. Reconstructing the final 2D velocity vectors.
+
+This method is standard in 2D physics engines.
+
+== Loop and Ball Pointers
+
+[source,cpp]
+----
+for (auto c : vecCollidingPairs)
+{
+    sBall *b1 = c.first;
+    sBall *b2 = c.second;
+----
+Each entry in `vecCollidingPairs` contains pointers to two balls that were
+found to be overlapping. `b1` and `b2` are those balls.
+
+== Distance Between Ball Centers
+
+[source,cpp]
+----
+float fDistance = sqrtf((b1->px - b2->px)*(b1->px - b2->px) +
+                        (b1->py - b2->py)*(b1->py - b2->py));
+----
+This computes the center‐to‐center distance.
+
+== Normal Vector Between Balls
+
+[source,cpp]
+----
+float nx = (b2->px - b1->px) / fDistance;
+float ny = (b2->py - b1->py) / fDistance;
+----
+The *normal vector* points from b1 to b2. Dividing by distance normalizes it
+(makes its length 1). This is the axis along which the collision occurs.
+
+== Tangent Vector
+
+[source,cpp]
+----
+float tx = -ny;
+float ty = nx;
+----
+The tangent vector is perpendicular to the normal. This allows us to separate
+velocity into two independent components: normal and tangent.
+
+== Project Velocities Onto Tangent Axis
+
+[source,cpp]
+----
+float dpTan1 = b1->vx * tx + b1->vy * ty;
+float dpTan2 = b2->vx * tx + b2->vy * ty;
+----
+A dot product projects each velocity onto the tangent. Tangential components
+do *not* change in a perfectly elastic collision. This tells us the velocity in
+direction of the tangent.
+
+== Project Velocities Onto Normal Axis
+
+[source,cpp]
+----
+float dpNorm1 = b1->vx * nx + b1->vy * ny;
+float dpNorm2 = b2->vx * nx + b2->vy * ny;
+----
+These are the *normal velocity components*. These are the only parts of the
+velocity that change due to collision. This tells us the velocity in the
+direction of the normal axis.
+
+== 1D Conservation of Momentum
+
+[source,cpp]
+----
+float m1 = (dpNorm1 * (b1->mass - b2->mass) +
+            2.0f * b2->mass * dpNorm2) / (b1->mass + b2->mass);
+
+float m2 = (dpNorm2 * (b2->mass - b1->mass) +
+            2.0f * b1->mass * dpNorm1) / (b1->mass + b2->mass);
+----
+This is the standard equation for a 1D elastic collision. Since the normal
+vector defines the line of impact, we treat the collision as 1D along this
+axis.
+
+The tangent components remain unchanged.
+
+== Rebuild the New Velocity Vectors
+
+[source,cpp]
+----
+b1->vx = tx * dpTan1 + nx * m1;
+b1->vy = ty * dpTan1 + ny * m1;
+
+b2->vx = tx * dpTan2 + nx * m2;
+b2->vy = ty * dpTan2 + ny * m2;
+----
+Each final velocity is:
+
+  (tangent component) + (new normal component)
+
+This gives the correct 2D elastic collision result.
+
+== Wikipedia Version
+
+The commented section uses a compact formula for the same physics. It produces
+the same results but is harder to read.
+
+== Summary
+
+* The normal vector defines the line of impact.
+* Tangent velocity stays the same.
+* Normal velocity is exchanged using elastic‐collision formulas.
+* Final velocity is rebuilt from tangent + new normal contributions.
+
+If you'd like, I can also produce:
+
+* an ASCII diagram showing normal/tangent axes,
+* a version with simplified math,
+* or a stand‐alone C example demonstrating this collision system.
+
+= What Is a Tangent Vector?
+:wrap: normal
+
+A *tangent vector* is a vector that is **perpendicular** to the *normal
+vector* at the point of collision. In circle–circle collision physics, the
+tangent vector represents the direction that is **along the surface**, while
+the normal vector represents the direction **into or out of the surface**.
+
+== Normal vs. Tangent
+
+When two balls collide, you can imagine a line between their centers. This
+line is the **normal** direction.
+
+  normal = direction from ball 1 to ball 2  
+  tangent = direction perpendicular to the normal
+
+The tangent is found simply by rotating the normal 90°:
+
+[source,cpp]
+----
+tx = -ny;
+ty =  nx;
+----
+
+== Why Do We Need a Tangent Vector?
+
+In a perfectly elastic collision:
+
+* Velocity **along the tangent** does **not change**
+* Velocity **along the normal** is the only part that collides
+
+This matches real physics:
+
+* Sliding past each other = no impact.
+* Moving toward each other = impact.
+
+To separate these two behaviors, we split velocity into:
+
+* tangential component
+* normal component
+
+using dot products.
+
+== Visual Diagram (ASCII)
+
+Here is an ASCII visualization.
+
+----
+         (tangent direction)
+                  ↑
+                  |
+        b1 ● ---> | <--- ● b2
+                  |
+                  ↓
+   <------------------------------>
+                         normal axis
+----
+
+== Key Idea
+
+The tangent vector is a direction that represents motion *that does not push
+the balls into each other*. It is used to extract and preserve the
+"sliding" part of the velocity.
+
+== Mathematical Definition
+
+Given a normalized normal vector:
+
+  n = (nx, ny)
+
+A perpendicular tangent vector is:
+
+  t = (-ny, nx)
+
+This is guaranteed to be perpendicular because:
+
+  n · t = nx * (-ny) + ny * nx = 0
+
+Thus the tangent is orthogonal to the normal.
+
+== Summary
+
+* The *normal vector* points directly from one ball to the other.
+* The *tangent vector* is perpendicular to it.
+* Only the normal component is involved in the collision.
+* Tangent velocity stays the same, representing sliding motion.
+
+== Links
+* https://www.youtube.com/watch?v=LPzyNOHY3A4&t=165s[javidx9 | Programming Balls]
+* https://www.youtube.com/watch?v=Fp7D5D8Bqjc&t=222s[OCT | Conservation of Momentum Physics Problems]
+* https://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle[Momentum-Conservation-Principle]
diff --git a/Physics/Ball-Dynamic-Collision.html b/Physics/Ball-Dynamic-Collision.html
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Physics | Dynamic Collisions</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_overview">Overview</a></li>
+<li><a href="#_loop_and_ball_pointers">Loop and Ball Pointers</a></li>
+<li><a href="#_distance_between_ball_centers">Distance Between Ball Centers</a></li>
+<li><a href="#_normal_vector_between_balls">Normal Vector Between Balls</a></li>
+<li><a href="#_tangent_vector">Tangent Vector</a></li>
+<li><a href="#_project_velocities_onto_tangent_axis">Project Velocities Onto Tangent Axis</a></li>
+<li><a href="#_project_velocities_onto_normal_axis">Project Velocities Onto Normal Axis</a></li>
+<li><a href="#_1d_conservation_of_momentum">1D Conservation of Momentum</a></li>
+<li><a href="#_rebuild_the_new_velocity_vectors">Rebuild the New Velocity Vectors</a></li>
+<li><a href="#_wikipedia_version">Wikipedia Version</a></li>
+<li><a href="#_summary">Summary</a></li>
+<li><a href="#_what_is_a_tangent_vector">What Is a Tangent Vector?</a>
+<ul class="sectlevel1">
+<li><a href="#_normal_vs_tangent">Normal vs. Tangent</a></li>
+<li><a href="#_why_do_we_need_a_tangent_vector">Why Do We Need a Tangent Vector?</a></li>
+<li><a href="#_visual_diagram_ascii">Visual Diagram (ASCII)</a></li>
+<li><a href="#_key_idea">Key Idea</a></li>
+<li><a href="#_mathematical_definition">Mathematical Definition</a></li>
+<li><a href="#_summary_2">Summary</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+This was mainly created by chatgpt
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>The following explains the physics and math inside the collision‐response loop.
+All lines are wrapped to 80 characters.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_overview">Overview</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-k">for</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-k">auto</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-w"> </span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-n">vecCollidingPairs</span><span class="tok-p">)</span>
+<span class="tok-w">           </span><span class="tok-p">{</span>
+<span class="tok-w">                   </span><span class="tok-n">sBall</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-n">b1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-p">.</span><span class="tok-n">first</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-n">sBall</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-n">b2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-p">.</span><span class="tok-n">second</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Distance between balls</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sqrtf</span><span class="tok-p">((</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">));</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Normal</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Tangent</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-n">ny</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Dot Product Tangent</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Dot Product Normal</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpNorm1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpNorm2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Conservation of momentum in 1D</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dpNorm1</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpNorm2</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">);</span>
+<span class="tok-w">                   </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dpNorm2</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpNorm1</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">);</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Update ball velocities</span>
+<span class="tok-w">                   </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-p">;</span>
+<span class="tok-w">                   </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-p">;</span>
+
+<span class="tok-w">                   </span><span class="tok-c1">// Wikipedia Version - Maths is smarter but same</span>
+<span class="tok-w">                   </span><span class="tok-c1">//float kx = (b1-&gt;vx - b2-&gt;vx);</span>
+<span class="tok-w">                   </span><span class="tok-c1">//float ky = (b1-&gt;vy - b2-&gt;vy);</span>
+<span class="tok-w">                   </span><span class="tok-c1">//float p = 2.0 * (nx * kx + ny * ky) / (b1-&gt;mass + b2-&gt;mass);</span>
+<span class="tok-w">                   </span><span class="tok-c1">//b1-&gt;vx = b1-&gt;vx - p * b2-&gt;mass * nx;</span>
+<span class="tok-w">                   </span><span class="tok-c1">//b1-&gt;vy = b1-&gt;vy - p * b2-&gt;mass * ny;</span>
+<span class="tok-w">                   </span><span class="tok-c1">//b2-&gt;vx = b2-&gt;vx + p * b1-&gt;mass * nx;</span>
+<span class="tok-w">                   </span><span class="tok-c1">//b2-&gt;vy = b2-&gt;vy + p * b1-&gt;mass * ny;</span>
+<span class="tok-w">           </span><span class="tok-p">}</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This code performs a <strong>realistic elastic collision</strong> between two balls. It does
+this by:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Computing the normal and tangent vectors between the balls.</p>
+</li>
+<li>
+<p>Projecting each ball&#8217;s velocity onto these axes.</p>
+</li>
+<li>
+<p>Swapping the normal components using 1D momentum equations.</p>
+</li>
+<li>
+<p>Reconstructing the final 2D velocity vectors.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>This method is standard in 2D physics engines.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_loop_and_ball_pointers">Loop and Ball Pointers</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-k">for</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-k">auto</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-w"> </span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-n">vecCollidingPairs</span><span class="tok-p">)</span>
+<span class="tok-p">{</span>
+<span class="tok-w">    </span><span class="tok-n">sBall</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-n">b1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-p">.</span><span class="tok-n">first</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-n">sBall</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-n">b2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">c</span><span class="tok-p">.</span><span class="tok-n">second</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each entry in <code>vecCollidingPairs</code> contains pointers to two balls that were
+found to be overlapping. <code>b1</code> and <code>b2</code> are those balls.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_distance_between_ball_centers">Distance Between Ball Centers</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sqrtf</span><span class="tok-p">((</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span>
+<span class="tok-w">                        </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">));</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This computes the center‐to‐center distance.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_normal_vector_between_balls">Normal Vector Between Balls</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+<span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>normal vector</strong> points from b1 to b2. Dividing by distance normalizes it
+(makes its length 1). This is the axis along which the collision occurs.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_tangent_vector">Tangent Vector</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-n">ny</span><span class="tok-p">;</span>
+<span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The tangent vector is perpendicular to the normal. This allows us to separate
+velocity into two independent components: normal and tangent.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_project_velocities_onto_tangent_axis">Project Velocities Onto Tangent Axis</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-p">;</span>
+<span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A dot product projects each velocity onto the tangent. Tangential components
+do <strong>not</strong> change in a perfectly elastic collision. This tells us the velocity in
+direction of the tangent.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_project_velocities_onto_normal_axis">Project Velocities Onto Normal Axis</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpNorm1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-p">;</span>
+<span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dpNorm2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These are the <strong>normal velocity components</strong>. These are the only parts of the
+velocity that change due to collision. This tells us the velocity in the
+direction of the normal axis.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_1d_conservation_of_momentum">1D Conservation of Momentum</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dpNorm1</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span>
+<span class="tok-w">            </span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpNorm2</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">);</span>
+
+<span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dpNorm2</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span>
+<span class="tok-w">            </span><span class="tok-mf">2.0f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpNorm1</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">mass</span><span class="tok-p">);</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This is the standard equation for a 1D elastic collision. Since the normal
+vector defines the line of impact, we treat the collision as 1D along this
+axis.</p>
+</div>
+<div class="paragraph">
+<p>The tangent components remain unchanged.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_rebuild_the_new_velocity_vectors">Rebuild the New Velocity Vectors</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-p">;</span>
+<span class="tok-n">b1</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan1</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m1</span><span class="tok-p">;</span>
+
+<span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">nx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-p">;</span>
+<span class="tok-n">b2</span><span class="tok-o">-&gt;</span><span class="tok-n">vy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dpTan2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">ny</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">m2</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each final velocity is:</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>(tangent component) + (new normal component)</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This gives the correct 2D elastic collision result.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_wikipedia_version">Wikipedia Version</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The commented section uses a compact formula for the same physics. It produces
+the same results but is harder to read.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_summary">Summary</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>The normal vector defines the line of impact.</p>
+</li>
+<li>
+<p>Tangent velocity stays the same.</p>
+</li>
+<li>
+<p>Normal velocity is exchanged using elastic‐collision formulas.</p>
+</li>
+<li>
+<p>Final velocity is rebuilt from tangent + new normal contributions.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>If you&#8217;d like, I can also produce:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>an ASCII diagram showing normal/tangent axes,</p>
+</li>
+<li>
+<p>a version with simplified math,</p>
+</li>
+<li>
+<p>or a stand‐alone C example demonstrating this collision system.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<h1 id="_what_is_a_tangent_vector" class="sect0">What Is a Tangent Vector?</h1>
+<div class="paragraph">
+<p>A <strong>tangent vector</strong> is a vector that is <strong>perpendicular</strong> to the <strong>normal
+vector</strong> at the point of collision. In circle–circle collision physics, the
+tangent vector represents the direction that is <strong>along the surface</strong>, while
+the normal vector represents the direction <strong>into or out of the surface</strong>.</p>
+</div>
+<div class="sect1">
+<h2 id="_normal_vs_tangent">Normal vs. Tangent</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>When two balls collide, you can imagine a line between their centers. This
+line is the <strong>normal</strong> direction.</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>normal = direction from ball 1 to ball 2
+tangent = direction perpendicular to the normal</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The tangent is found simply by rotating the normal 90°:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-n">tx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-n">ny</span><span class="tok-p">;</span>
+<span class="tok-n">ty</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w">  </span><span class="tok-n">nx</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_why_do_we_need_a_tangent_vector">Why Do We Need a Tangent Vector?</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>In a perfectly elastic collision:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Velocity <strong>along the tangent</strong> does <strong>not change</strong></p>
+</li>
+<li>
+<p>Velocity <strong>along the normal</strong> is the only part that collides</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>This matches real physics:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Sliding past each other = no impact.</p>
+</li>
+<li>
+<p>Moving toward each other = impact.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>To separate these two behaviors, we split velocity into:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>tangential component</p>
+</li>
+<li>
+<p>normal component</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>using dot products.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_visual_diagram_ascii">Visual Diagram (ASCII)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Here is an ASCII visualization.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre>         (tangent direction)
+                  ↑
+                  |
+        b1 ● ---&gt; | &lt;--- ● b2
+                  |
+                  ↓
+   &lt;------------------------------&gt;
+                         normal axis</pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_key_idea">Key Idea</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The tangent vector is a direction that represents motion <strong>that does not push
+the balls into each other</strong>. It is used to extract and preserve the
+"sliding" part of the velocity.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_mathematical_definition">Mathematical Definition</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Given a normalized normal vector:</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>n = (nx, ny)</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A perpendicular tangent vector is:</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>t = (-ny, nx)</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This is guaranteed to be perpendicular because:</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>n · t = nx * (-ny) + ny * nx = 0</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Thus the tangent is orthogonal to the normal.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_summary_2">Summary</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p>The <strong>normal vector</strong> points directly from one ball to the other.</p>
+</li>
+<li>
+<p>The <strong>tangent vector</strong> is perpendicular to it.</p>
+</li>
+<li>
+<p>Only the normal component is involved in the collision.</p>
+</li>
+<li>
+<p>Tangent velocity stays the same, representing sliding motion.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=LPzyNOHY3A4&amp;t=165s">javidx9 | Programming Balls</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=Fp7D5D8Bqjc&amp;t=222s">OCT | Conservation of Momentum Physics Problems</a></p>
+</li>
+<li>
+<p><a href="https://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle">Momentum-Conservation-Principle</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+  },
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+    ignoreClass: "nostem|noasciimath"
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+  })
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+</html>
\ No newline at end of file
diff --git a/Physics/Ball-Static-Collision.adoc b/Physics/Ball-Static-Collision.adoc
new file mode 100644 (file)
index 0000000..f5069f5
--- /dev/null
+++ b/Physics/Ball-Static-Collision.adoc
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+= Physics | Static Collisions
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+NOTE: This was mainly created by chatgpt
+
+from https://github.com/OneLoneCoder/Javidx9/blob/master/ConsoleGameEngine/BiggerProjects/Balls/OneLoneCoder_Balls1.cpp[OneLoneCoder_Balls1.cpp].
+
+The following explains what the nested loops and the collision–resolution math
+are doing.
+
+[source,cpp]
+----
+for (auto &ball : vecBalls)
+{
+    for (auto &target : vecBalls)
+    {
+        if (ball.id != target.id)
+        {
+            if (DoCirclesOverlap(ball.px, ball.py, ball.radius,
+                                 target.px, target.py, target.radius))
+            {
+                vecCollidingPairs.push_back({ &ball, &target });
+
+                float fDistance = sqrtf((ball.px - target.px)*(ball.px - target.px)
+                                      + (ball.py - target.px)*(ball.py - target.py));
+
+                float fOverlap = 0.5f * (fDistance - ball.radius - target.radius);
+
+                ball.px -= fOverlap * (ball.px - target.px) / fDistance;
+                ball.py -= fOverlap * (ball.py - target.py) / fDistance;
+
+                target.px += fOverlap * (ball.px - target.px) / fDistance;
+                target.py += fOverlap * (ball.py - target.py) / fDistance;
+            }
+        }
+    }
+}
+----
+
+== Loop Structure
+
+The code compares every ball with every other ball:
+
+* The outer loop selects the *current* ball.
+* The inner loop compares it to a *target* ball.
+* The condition `ball.id != target.id` ensures a ball is not compared with itself.
+
+== Collision Detection
+
+The function:
+
+[source,c]
+----
+DoCirclesOverlap(x1, y1, r1, x2, y2, r2)
+----
+
+checks whether the distance between centers is less than the sum of radii.
+
+If true, a collision has happened.
+
+The pair is added to:
+
+[source,c]
+----
+vecCollidingPairs
+----
+
+so it can be processed later (for physics, rendering debug markers, etc.).
+
+== Distance Between Centers
+
+[source,c]
+----
+float fDistance = sqrtf((dx * dx) + (dy * dy));
+----
+
+Where:
+
+* dx = ball.px - target.px
+* dy = ball.py - target.py
+
+This gives the center-to-center distance. This is the Pythagorean Theorem.
+
+== Overlap Amount
+
+[source,c]
+----
+float fOverlap = 0.5f * (fDistance - ball.radius - target.radius);
+----
+
+This computes how much the two circles are penetrating.
+
+The `0.5f` means **each ball moves half of the required distance**, keeping the
+system stable and centered.
+
+== Normalized Direction Vector
+
+The expressions:
+
+[source,c]
+----
+(ball.px - target.px) / fDistance  
+(ball.py - target.py) / fDistance
+----
+
+are effectively the *normalized direction vector* between the two balls.
+
+Normalization ensures the push direction has length 1. Keeping the direction but
+maintaining a distance of zero will be important so we can keep moving by
+fOverlap in the oppropriate directions.
+
+== Resolving the Collision (Separation)
+
+Each ball is moved along the collision normal:
+
+[source,c]
+----
+ball.px -= fOverlap * normalized_dx;
+ball.py -= fOverlap * normalized_dy;
+
+target.px += fOverlap * normalized_dx;
+target.py += fOverlap * normalized_dy;
+----
+
+This pushes the balls apart until they are *exactly touching*, eliminating overlap.
+
+== Summary of Physics Steps
+
+. Detect overlap (circles intersecting).
+. Compute the vector between ball centers.
+. Compute distance and normalize that vector.
+. Compute how much they overlap.
+. Push each ball away from the other along the normalized direction.
+
+This is a standard method of performing *positional correction* in 2D physics
+simulations (similar to what Box2D does).
+
+== Links
+https://www.youtube.com/watch?v=LPzyNOHY3A4&t=165s[Programming Balls #1]
diff --git a/Physics/Ball-Static-Collision.html b/Physics/Ball-Static-Collision.html
new file mode 100644 (file)
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--- /dev/null
+++ b/Physics/Ball-Static-Collision.html
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+pre.pygments .tok-cp { color: #959077 } /* Comment.Preproc */
+pre.pygments .tok-cpf { color: #959077 } /* Comment.PreprocFile */
+pre.pygments .tok-c1 { color: #959077 } /* Comment.Single */
+pre.pygments .tok-cs { color: #959077 } /* Comment.Special */
+pre.pygments .tok-gd { color: #FF4689 } /* Generic.Deleted */
+pre.pygments .tok-ge { color: #F8F8F2; font-style: italic } /* Generic.Emph */
+pre.pygments .tok-ges { color: #F8F8F2; font-weight: bold; font-style: italic } /* Generic.EmphStrong */
+pre.pygments .tok-gr { color: #F8F8F2 } /* Generic.Error */
+pre.pygments .tok-gh { color: #F8F8F2 } /* Generic.Heading */
+pre.pygments .tok-gi { color: #A6E22E } /* Generic.Inserted */
+pre.pygments .tok-go { color: #66D9EF } /* Generic.Output */
+pre.pygments .tok-gp { color: #FF4689; font-weight: bold } /* Generic.Prompt */
+pre.pygments .tok-gs { color: #F8F8F2; font-weight: bold } /* Generic.Strong */
+pre.pygments .tok-gu { color: #959077 } /* Generic.Subheading */
+pre.pygments .tok-gt { color: #F8F8F2 } /* Generic.Traceback */
+pre.pygments .tok-kc { color: #66D9EF } /* Keyword.Constant */
+pre.pygments .tok-kd { color: #66D9EF } /* Keyword.Declaration */
+pre.pygments .tok-kn { color: #FF4689 } /* Keyword.Namespace */
+pre.pygments .tok-kp { color: #66D9EF } /* Keyword.Pseudo */
+pre.pygments .tok-kr { color: #66D9EF } /* Keyword.Reserved */
+pre.pygments .tok-kt { color: #66D9EF } /* Keyword.Type */
+pre.pygments .tok-ld { color: #E6DB74 } /* Literal.Date */
+pre.pygments .tok-m { color: #AE81FF } /* Literal.Number */
+pre.pygments .tok-s { color: #E6DB74 } /* Literal.String */
+pre.pygments .tok-na { color: #A6E22E } /* Name.Attribute */
+pre.pygments .tok-nb { color: #F8F8F2 } /* Name.Builtin */
+pre.pygments .tok-nc { color: #A6E22E } /* Name.Class */
+pre.pygments .tok-no { color: #66D9EF } /* Name.Constant */
+pre.pygments .tok-nd { color: #A6E22E } /* Name.Decorator */
+pre.pygments .tok-ni { color: #F8F8F2 } /* Name.Entity */
+pre.pygments .tok-ne { color: #A6E22E } /* Name.Exception */
+pre.pygments .tok-nf { color: #A6E22E } /* Name.Function */
+pre.pygments .tok-nl { color: #F8F8F2 } /* Name.Label */
+pre.pygments .tok-nn { color: #F8F8F2 } /* Name.Namespace */
+pre.pygments .tok-nx { color: #A6E22E } /* Name.Other */
+pre.pygments .tok-py { color: #F8F8F2 } /* Name.Property */
+pre.pygments .tok-nt { color: #FF4689 } /* Name.Tag */
+pre.pygments .tok-nv { color: #F8F8F2 } /* Name.Variable */
+pre.pygments .tok-ow { color: #FF4689 } /* Operator.Word */
+pre.pygments .tok-pm { color: #F8F8F2 } /* Punctuation.Marker */
+pre.pygments .tok-w { color: #F8F8F2 } /* Text.Whitespace */
+pre.pygments .tok-mb { color: #AE81FF } /* Literal.Number.Bin */
+pre.pygments .tok-mf { color: #AE81FF } /* Literal.Number.Float */
+pre.pygments .tok-mh { color: #AE81FF } /* Literal.Number.Hex */
+pre.pygments .tok-mi { color: #AE81FF } /* Literal.Number.Integer */
+pre.pygments .tok-mo { color: #AE81FF } /* Literal.Number.Oct */
+pre.pygments .tok-sa { color: #E6DB74 } /* Literal.String.Affix */
+pre.pygments .tok-sb { color: #E6DB74 } /* Literal.String.Backtick */
+pre.pygments .tok-sc { color: #E6DB74 } /* Literal.String.Char */
+pre.pygments .tok-dl { color: #E6DB74 } /* Literal.String.Delimiter */
+pre.pygments .tok-sd { color: #E6DB74 } /* Literal.String.Doc */
+pre.pygments .tok-s2 { color: #E6DB74 } /* Literal.String.Double */
+pre.pygments .tok-se { color: #AE81FF } /* Literal.String.Escape */
+pre.pygments .tok-sh { color: #E6DB74 } /* Literal.String.Heredoc */
+pre.pygments .tok-si { color: #E6DB74 } /* Literal.String.Interpol */
+pre.pygments .tok-sx { color: #E6DB74 } /* Literal.String.Other */
+pre.pygments .tok-sr { color: #E6DB74 } /* Literal.String.Regex */
+pre.pygments .tok-s1 { color: #E6DB74 } /* Literal.String.Single */
+pre.pygments .tok-ss { color: #E6DB74 } /* Literal.String.Symbol */
+pre.pygments .tok-bp { color: #F8F8F2 } /* Name.Builtin.Pseudo */
+pre.pygments .tok-fm { color: #A6E22E } /* Name.Function.Magic */
+pre.pygments .tok-vc { color: #F8F8F2 } /* Name.Variable.Class */
+pre.pygments .tok-vg { color: #F8F8F2 } /* Name.Variable.Global */
+pre.pygments .tok-vi { color: #F8F8F2 } /* Name.Variable.Instance */
+pre.pygments .tok-vm { color: #F8F8F2 } /* Name.Variable.Magic */
+pre.pygments .tok-il { color: #AE81FF } /* Literal.Number.Integer.Long */
+</style>
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
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+
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Physics | Static Collisions</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_loop_structure">Loop Structure</a></li>
+<li><a href="#_collision_detection">Collision Detection</a></li>
+<li><a href="#_distance_between_centers">Distance Between Centers</a></li>
+<li><a href="#_overlap_amount">Overlap Amount</a></li>
+<li><a href="#_normalized_direction_vector">Normalized Direction Vector</a></li>
+<li><a href="#_resolving_the_collision_separation">Resolving the Collision (Separation)</a></li>
+<li><a href="#_summary_of_physics_steps">Summary of Physics Steps</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+This was mainly created by chatgpt
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>from <a href="https://github.com/OneLoneCoder/Javidx9/blob/master/ConsoleGameEngine/BiggerProjects/Balls/OneLoneCoder_Balls1.cpp">OneLoneCoder_Balls1.cpp</a>.</p>
+</div>
+<div class="paragraph">
+<p>The following explains what the nested loops and the collision–resolution math
+are doing.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="cpp"><span></span><span class="tok-k">for</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-k">auto</span><span class="tok-w"> </span><span class="tok-o">&amp;</span><span class="tok-n">ball</span><span class="tok-w"> </span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-n">vecBalls</span><span class="tok-p">)</span>
+<span class="tok-p">{</span>
+<span class="tok-w">    </span><span class="tok-k">for</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-k">auto</span><span class="tok-w"> </span><span class="tok-o">&amp;</span><span class="tok-n">target</span><span class="tok-w"> </span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-n">vecBalls</span><span class="tok-p">)</span>
+<span class="tok-w">    </span><span class="tok-p">{</span>
+<span class="tok-w">        </span><span class="tok-k">if</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">id</span><span class="tok-w"> </span><span class="tok-o">!=</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">id</span><span class="tok-p">)</span>
+<span class="tok-w">        </span><span class="tok-p">{</span>
+<span class="tok-w">            </span><span class="tok-k">if</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">DoCirclesOverlap</span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-p">,</span>
+<span class="tok-w">                                 </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-p">))</span>
+<span class="tok-w">            </span><span class="tok-p">{</span>
+<span class="tok-w">                </span><span class="tok-n">vecCollidingPairs</span><span class="tok-p">.</span><span class="tok-n">push_back</span><span class="tok-p">({</span><span class="tok-w"> </span><span class="tok-o">&amp;</span><span class="tok-n">ball</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-o">&amp;</span><span class="tok-n">target</span><span class="tok-w"> </span><span class="tok-p">});</span>
+
+<span class="tok-w">                </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sqrtf</span><span class="tok-p">((</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span>
+<span class="tok-w">                                      </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-o">*</span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">));</span>
+
+<span class="tok-w">                </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-mf">0.5f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-p">);</span>
+
+<span class="tok-w">                </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+<span class="tok-w">                </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+
+<span class="tok-w">                </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">+=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+<span class="tok-w">                </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">+=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-p">;</span>
+<span class="tok-w">            </span><span class="tok-p">}</span>
+<span class="tok-w">        </span><span class="tok-p">}</span>
+<span class="tok-w">    </span><span class="tok-p">}</span>
+<span class="tok-p">}</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_loop_structure">Loop Structure</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The code compares every ball with every other ball:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The outer loop selects the <strong>current</strong> ball.</p>
+</li>
+<li>
+<p>The inner loop compares it to a <strong>target</strong> ball.</p>
+</li>
+<li>
+<p>The condition <code>ball.id != target.id</code> ensures a ball is not compared with itself.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_collision_detection">Collision Detection</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The function:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-n">DoCirclesOverlap</span><span class="tok-p">(</span><span class="tok-n">x1</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">y1</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">r1</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">x2</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">y2</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">r2</span><span class="tok-p">)</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>checks whether the distance between centers is less than the sum of radii.</p>
+</div>
+<div class="paragraph">
+<p>If true, a collision has happened.</p>
+</div>
+<div class="paragraph">
+<p>The pair is added to:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-n">vecCollidingPairs</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>so it can be processed later (for physics, rendering debug markers, etc.).</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_distance_between_centers">Distance Between Centers</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sqrtf</span><span class="tok-p">((</span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dx</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-p">));</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Where:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>dx = ball.px - target.px</p>
+</li>
+<li>
+<p>dy = ball.py - target.py</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>This gives the center-to-center distance. This is the Pythagorean Theorem.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_overlap_amount">Overlap Amount</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-mf">0.5f</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">fDistance</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">radius</span><span class="tok-p">);</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This computes how much the two circles are penetrating.</p>
+</div>
+<div class="paragraph">
+<p>The <code>0.5f</code> means <strong>each ball moves half of the required distance</strong>, keeping the
+system stable and centered.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_normalized_direction_vector">Normalized Direction Vector</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The expressions:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span>
+<span class="tok-p">(</span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-n">fDistance</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>are effectively the <strong>normalized direction vector</strong> between the two balls.</p>
+</div>
+<div class="paragraph">
+<p>Normalization ensures the push direction has length 1. Keeping the direction but
+maintaining a distance of zero will be important so we can keep moving by
+fOverlap in the oppropriate directions.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_resolving_the_collision_separation">Resolving the Collision (Separation)</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>Each ball is moved along the collision normal:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">-=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">normalized_dx</span><span class="tok-p">;</span>
+<span class="tok-n">ball</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">-=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">normalized_dy</span><span class="tok-p">;</span>
+
+<span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">px</span><span class="tok-w"> </span><span class="tok-o">+=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">normalized_dx</span><span class="tok-p">;</span>
+<span class="tok-n">target</span><span class="tok-p">.</span><span class="tok-n">py</span><span class="tok-w"> </span><span class="tok-o">+=</span><span class="tok-w"> </span><span class="tok-n">fOverlap</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">normalized_dy</span><span class="tok-p">;</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This pushes the balls apart until they are <strong>exactly touching</strong>, eliminating overlap.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_summary_of_physics_steps">Summary of Physics Steps</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Detect overlap (circles intersecting).</p>
+</li>
+<li>
+<p>Compute the vector between ball centers.</p>
+</li>
+<li>
+<p>Compute distance and normalize that vector.</p>
+</li>
+<li>
+<p>Compute how much they overlap.</p>
+</li>
+<li>
+<p>Push each ball away from the other along the normalized direction.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>This is a standard method of performing <strong>positional correction</strong> in 2D physics
+simulations (similar to what Box2D does).</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p><a href="https://www.youtube.com/watch?v=LPzyNOHY3A4&amp;t=165s">Programming Balls #1</a></p>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
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+</body>
+</html>
\ No newline at end of file
diff --git a/Physics/Basics.adoc b/Physics/Basics.adoc
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+= Physics | Basics
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+
+== Kinematic's
+
+=== Symbols
+
+[.inline]
+* stem:[V] or stem:[V_{f}] -> Final velocity
+
+[.inline]
+* stem:[a] or stem:[\vec{a}] -> Acceleration
+
+[.inline]
+* stem:[t] -> Time
+
+[.inline]
+* stem:[V_{0}] -> Initial Velocity
+
+[.inline]
+* stem:[\overline{S}] -> Average Speed
+
+[.inline]
+* stem:[X_{F}] -> Final Position
+
+[.inline]
+* stem:[\Deltax] -> Displacement of x axis
+
+[.inline]
+* stem:[\Deltay] -> Displacement of y axis
+
+[.inline]
+* stem:[\vec{V_{a}}] -> Average Velocity
+
+[.inline]
+* stem:[S_{i}] -> Instantaneos Speed
+
+[.inline]
+* stem:[V_{i}] -> Instantaneos Velocity
+
+[.inline]
+* stem:[X_{0}] -> Initial Position
+
+[.inline]
+* stem:[V_{t}] -> Terminal Velocity
+
+[.inline]
+* stem:[\vec{S}] -> Displacement
+
+[.inline]
+* stem:[\vec{V}] -> Vector
+
+[.inline]
+* stem:[S_{a}] -> Speed Average
+
+[.inline]
+* stem:[w] -> weight
+
+=== Equations
+
+==== Basic
+[.inline]
+* stem:[V = V_{0}+at]
+
+[.inline]
+* stem:[(\Deltax or \Deltay) = \frac{V + V_{0}}{2}t]
+
+[.inline]
+* stem:[(\Deltax or \Deltay) = V_{0}t + 1/2at^2]
+
+[.inline]
+* stem:[V^2 = V_{0}^2 + 2a\Deltax]
+
+==== Extra
+
+[.inline]
+* stem:[\vec{V} = \vec{S}/t] or stem:[r = d/t]
+
+[.inline]
+* stem:[\vec{V_{a}} = \frac{\Deltax}{t}]
+
+[.inline]
+* stem:[\vec{a} = \frac{\Delta\vec{V}}{t}]
+
+[.inline]
+* stem:[a = \frac{V_{f} - V_{i}}{\Deltat}] 
+
+[.inline]
+* stem:[\vec{V_{a}} = \frac{V_{f} + V_{i}}{2}] ???
+
+[.inline]
+* stem:[\vec{S} = (\vec{V_{a}})\Deltat]
+
+[.inline]
+* stem:[\vec{S} = \frac{V_{f}^2-V_{i}^2}{2\vec{a}}]
+
+[.inline]
+* stem:[2\vec{a}\vec{s} = V_{f}^2 - V_{i}^2]
+
+[.inline]
+* stem:[\DeltaV = a\Deltat]
+
+[.inline]
+* stem:[\frac{\Deltaa}{\Deltat} = jerk]
+
+[.inline]
+* stem:[V_{i} = V_{f} - a\Deltat]
+
+[.inline]
+* stem:[\Deltat = \frac{V_{f} - V_{i}}{a}]
+
+[.inline]
+* stem:[S_{a} = \frac{\text{distance}}{\Deltat}]
+
+NOTE: If the divisor is t then it is assumed to be stem:[\Deltat]
+
+WARNING: kinematic formulas are only accurate if the acceleration is constant
+         during the time interval considered
+
+==== 2D Projectile Motion
+There's no acceleration in the horizontal direction since gravity does not pull
+projectiles sideways
+
+[.inline]
+* stem:[Delta x=v_xt \quad]
+
+An object can start with a horizontal component of velocity, yet have zero
+vertical component of velocity.
+
+=== Instentaneos Rate of Change
+[stem]
+++++
+\lim_{\Delta \to 0} \frac{f(x + \Deltax) - f(c)}{\Deltax} = f'(x)\
+or \
+\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)
+++++
+
+[stem]
+++++
+f'(x) = v(t) -> \text{Instentaneos Velocity}\
+f''(x) = a(t) -> \text{acceleration}\
+|f'(x)| = s(t) -> \text{Instentaneos speed}\
+f'''(x) -> jerk
+++++
+
+
+== Force
+=== Symbols
+[.inline]
+* stem:[\vec{F}] -> Force of Gravity
+
+[.inline]
+* stem:[g] -> Acceleration of Gravity
+
+[.inline]
+* stem:[m] -> mass of object
+
+[.inline]
+* stem:[r] -> distance between objects
+
+[.inline]
+* stem:[\vec{a_{g}}] -> acceleration due to gravity
+
+
+=== Equations
+[.inline]
+* stem:[\vec{F} = m \cdot g] 
+
+[.inline]
+* stem:[\vec{F} = m \cdot a]
+
+[.inline]
+* stem:[a = \vec{F}/m]
+
+[.inline]
+* stem:[a = g]
+
+[.inline]
+* stem:[\vec{F} = G \frac{m_{1}m_{2}}{r^2}]
+
+[.inline]
+* stem:[w = \frac{m}{g}]
+
+=== Gravity
+
+[stem]
+++++
+g = G \frac{m}{r^2} = 9.81 frac{m}{s^2}
+++++
+
+* stem:[m] -> Mass of the Earth
+* stem:[r] -> Radius of the Earth
+
+
+== Temrinal Velocity
+[stem]
+++++
+V_{t} = frac{sqrt{2mg}}{pAC_{d}}
+++++
+
+* stem:[V_{t}] represents terminal velocity
+* stem:[m] is the mass of the falling object
+* stem:[g] is accleration due to gravity
+* stem:[C_{d}] is the drag coefficent
+* stem:[p] is the density through which th object is falling
+* stem:[A] is the projected area of the object
+
+== Graphing
+=== Velocity vs. Time Graphs
+* Slope is acceleration and a way to find instentaneos velocity
+* Area is displacement
+
+[.center]
+image::Displacement.png[]
+
+=== Acceleration vs. Time Graphs
+* Slope is jerk
+* Area is change in velocity
+* Typical y-axis: stem:[m/s^2]
+
+[.center]
+image::Change-In-Velocity.png[]
+
+
+== Using Quadradic Formula
+
+[stem]
+++++
+\Delta y=v_{0y} t+\dfrac{1}{2}a_yt^2 \quad \text{(Start with the third kinematic formula.)}
+++++
+
+[stem]
+++++
+12.2\text{ m}=(18.3\text{ m/s})t+\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2 \quad \text{(Plug in known values.)}
+++++
+
+[stem]
+++++
+0=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2+(18.3\text{ m/s})t -12.2\text{ m} \quad \text{(Put it into the form of the quadratic equation.)}
+++++
+
+[stem]
+++++
+a=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})
+\
+b=18.3\text{ m/s}
+\
+c=-12.2\text{ m}
+++++
+
+[stem]
+++++
+t=\dfrac{-18.3\text{ m/s}\pm\sqrt{(18.3\text{ m/s})^2-4[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})(-12.2\text{ m})]}}{2[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})]}
+++++
+
+[stem]
+++++
+t=0.869\text{s}\ \text{and}\ t=2.86\text{s}
+++++
+
+== Links
+- https://en.wikipedia.org/wiki/Terminal_velocity[Terminal Velocity]
+- https://phys.libretexts.org/Bookshelves/University_Physics
+----
+https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/11%3A__Angular_Momentum/11.02%3A_Rolling_Motion
+----
diff --git a/Physics/Basics.html b/Physics/Basics.html
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+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Physics | Basics</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_kinematics">Kinematic&#8217;s</a>
+<ul class="sectlevel2">
+<li><a href="#_symbols">Symbols</a></li>
+<li><a href="#_equations">Equations</a></li>
+<li><a href="#_instentaneos_rate_of_change">Instentaneos Rate of Change</a></li>
+</ul>
+</li>
+<li><a href="#_force">Force</a>
+<ul class="sectlevel2">
+<li><a href="#_symbols_2">Symbols</a></li>
+<li><a href="#_equations_2">Equations</a></li>
+<li><a href="#_gravity">Gravity</a></li>
+</ul>
+</li>
+<li><a href="#_temrinal_velocity">Temrinal Velocity</a></li>
+<li><a href="#_graphing">Graphing</a>
+<ul class="sectlevel2">
+<li><a href="#_velocity_vs_time_graphs">Velocity vs. Time Graphs</a></li>
+<li><a href="#_acceleration_vs_time_graphs">Acceleration vs. Time Graphs</a></li>
+</ul>
+</li>
+<li><a href="#_using_quadradic_formula">Using Quadradic Formula</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_kinematics">Kinematic&#8217;s</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_symbols">Symbols</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V\$ or \$V_{f}\$ &#8594; Final velocity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$a\$ or \$\vec{a}\$ &#8594; Acceleration</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$t\$ &#8594; Time</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V_{0}\$ &#8594; Initial Velocity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\overline{S}\$ &#8594; Average Speed</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$X_{F}\$ &#8594; Final Position</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\Deltax\$ &#8594; Displacement of x axis</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\Deltay\$ &#8594; Displacement of y axis</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{V_{a}}\$ &#8594; Average Velocity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$S_{i}\$ &#8594; Instantaneos Speed</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V_{i}\$ &#8594; Instantaneos Velocity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$X_{0}\$ &#8594; Initial Position</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V_{t}\$ &#8594; Terminal Velocity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{S}\$ &#8594; Displacement</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{V}\$ &#8594; Vector</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$S_{a}\$ &#8594; Speed Average</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$w\$ &#8594; weight</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_equations">Equations</h3>
+<div class="sect3">
+<h4 id="_basic">Basic</h4>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V = V_{0}+at\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$(\Deltax or \Deltay) = \frac{V + V_{0}}{2}t\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$(\Deltax or \Deltay) = V_{0}t + 1/2at^2\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V^2 = V_{0}^2 + 2a\Deltax\$</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_extra">Extra</h4>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{V} = \vec{S}/t\$ or \$r = d/t\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{V_{a}} = \frac{\Deltax}{t}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{a} = \frac{\Delta\vec{V}}{t}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$a = \frac{V_{f} - V_{i}}{\Deltat}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{V_{a}} = \frac{V_{f} + V_{i}}{2}\$ ???</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{S} = (\vec{V_{a}})\Deltat\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{S} = \frac{V_{f}^2-V_{i}^2}{2\vec{a}}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$2\vec{a}\vec{s} = V_{f}^2 - V_{i}^2\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\DeltaV = a\Deltat\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\frac{\Deltaa}{\Deltat} = jerk\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$V_{i} = V_{f} - a\Deltat\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\Deltat = \frac{V_{f} - V_{i}}{a}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$S_{a} = \frac{\text{distance}}{\Deltat}\$</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+If the divisor is t then it is assumed to be \$\Deltat\$
+</td>
+</tr>
+</table>
+</div>
+<div class="admonitionblock warning">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-warning" title="Warning"></i>
+</td>
+<td class="content">
+kinematic formulas are only accurate if the acceleration is constant
+         during the time interval considered
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_2d_projectile_motion">2D Projectile Motion</h4>
+<div class="paragraph">
+<p>There&#8217;s no acceleration in the horizontal direction since gravity does not pull
+projectiles sideways</p>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$Delta x=v_xt \quad\$</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>An object can start with a horizontal component of velocity, yet have zero
+vertical component of velocity.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_instentaneos_rate_of_change">Instentaneos Rate of Change</h3>
+<div class="stemblock">
+<div class="content">
+\$\lim_{\Delta \to 0} \frac{f(x + \Deltax) - f(c)}{\Deltax} = f'(x)\$
+\$or\$
+\$\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$f'(x) = v(t) -&gt; \text{Instentaneos Velocity}\$
+\$f''(x) = a(t) -&gt; \text{acceleration}\$
+\$|f'(x)| = s(t) -&gt; \text{Instentaneos speed}\$
+\$f'''(x) -&gt; jerk\$
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_force">Force</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_symbols_2">Symbols</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{F}\$ &#8594; Force of Gravity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$g\$ &#8594; Acceleration of Gravity</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$m\$ &#8594; mass of object</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$r\$ &#8594; distance between objects</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{a_{g}}\$ &#8594; acceleration due to gravity</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_equations_2">Equations</h3>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{F} = m \cdot g\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{F} = m \cdot a\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$a = \vec{F}/m\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$a = g\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\vec{F} = G \frac{m_{1}m_{2}}{r^2}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$w = \frac{m}{g}\$</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_gravity">Gravity</h3>
+<div class="stemblock">
+<div class="content">
+\$g = G \frac{m}{r^2} = 9.81 frac{m}{s^2}\$
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>\$m\$ &#8594; Mass of the Earth</p>
+</li>
+<li>
+<p>\$r\$ &#8594; Radius of the Earth</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_temrinal_velocity">Temrinal Velocity</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$V_{t} = frac{sqrt{2mg}}{pAC_{d}}\$
+</div>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>\$V_{t}\$ represents terminal velocity</p>
+</li>
+<li>
+<p>\$m\$ is the mass of the falling object</p>
+</li>
+<li>
+<p>\$g\$ is accleration due to gravity</p>
+</li>
+<li>
+<p>\$C_{d}\$ is the drag coefficent</p>
+</li>
+<li>
+<p>\$p\$ is the density through which th object is falling</p>
+</li>
+<li>
+<p>\$A\$ is the projected area of the object</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_graphing">Graphing</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="_velocity_vs_time_graphs">Velocity vs. Time Graphs</h3>
+<div class="ulist">
+<ul>
+<li>
+<p>Slope is acceleration and a way to find instentaneos velocity</p>
+</li>
+<li>
+<p>Area is displacement</p>
+</li>
+</ul>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Displacement.png" alt="Displacement">
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="_acceleration_vs_time_graphs">Acceleration vs. Time Graphs</h3>
+<div class="ulist">
+<ul>
+<li>
+<p>Slope is jerk</p>
+</li>
+<li>
+<p>Area is change in velocity</p>
+</li>
+<li>
+<p>Typical y-axis: \$m/s^2\$</p>
+</li>
+</ul>
+</div>
+<div class="imageblock center">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Change-In-Velocity.png" alt="Change In Velocity">
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_using_quadradic_formula">Using Quadradic Formula</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$\Delta y=v_{0y} t+\dfrac{1}{2}a_yt^2 \quad \text{(Start with the third kinematic formula.)}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$12.2\text{ m}=(18.3\text{ m/s})t+\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2 \quad \text{(Plug in known values.)}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$0=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2+(18.3\text{ m/s})t -12.2\text{ m} \quad \text{(Put it into the form of the quadratic equation.)}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$a=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})
+\$
+\$b=18.3\text{ m/s}
+\$
+\$c=-12.2\text{ m}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$t=\dfrac{-18.3\text{ m/s}\pm\sqrt{(18.3\text{ m/s})^2-4[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})(-12.2\text{ m})]}}{2[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})]}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$t=0.869\text{s}\ \text{and}\ t=2.86\text{s}\$
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://en.wikipedia.org/wiki/Terminal_velocity">Terminal Velocity</a></p>
+</li>
+<li>
+<p><a href="https://phys.libretexts.org/Bookshelves/University_Physics" class="bare">https://phys.libretexts.org/Bookshelves/University_Physics</a></p>
+</li>
+</ul>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre>https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/11%3A__Angular_Momentum/11.02%3A_Rolling_Motion</pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
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+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
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+= Physics | Collision
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Circle and Rect collision
+
+[source,c]
+----
+bool circleRectCollide(SDL_Point circle, float r,
+                       SDL_Rect rect)
+{
+    // does this assume that circle x and circle y are the center of the circle?
+    float closestX = clamp(circle.x, rect.x, rect.x + rect.w);
+    float closestY = clamp(circle.y, rect.y, rect.y + rect.h);
+
+    // this gets the distance form the center of the circle to a side of a
+    // rectangle. It could be negative or positive. Because we are squaring them
+    // it doesn't matter.
+    float dx = circle.x - closestX;
+    float dy = circle.y - closestY;
+
+    // use the dot product to determine collision
+    return (dx*dx + dy*dy) <= (r*r);
+}
+----
+
+== Circle and Circle collision
+[source,c]
+----
+bool circlesCollide(Circle a, Circle b) {
+    float dx = b.x - a.x;
+    float dy = b.y - a.y;
+    float distance = sqrtf(dx * dx + dy * dy);
+
+    return distance <= (a.r + b.r);
+}
+----
diff --git a/Physics/Collision.html b/Physics/Collision.html
new file mode 100644 (file)
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+    --gruvbox-green: #b8bb26;
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+    --gruvbox-bg2: #665c54;
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+}
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+}
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+blockquote {
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+}
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Physics | Collision</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_circle_and_rect_collision">Circle and Rect collision</a></li>
+<li><a href="#_circle_and_circle_collision">Circle and Circle collision</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_circle_and_rect_collision">Circle and Rect collision</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-kt">bool</span><span class="tok-w"> </span><span class="tok-nf">circleRectCollide</span><span class="tok-p">(</span><span class="tok-n">SDL_Point</span><span class="tok-w"> </span><span class="tok-n">circle</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">r</span><span class="tok-p">,</span>
+<span class="tok-w">                       </span><span class="tok-n">SDL_Rect</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">)</span>
+<span class="tok-p">{</span>
+<span class="tok-w">    </span><span class="tok-c1">// does this assume that circle x and circle y are the center of the circle?</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">closestX</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">clamp</span><span class="tok-p">(</span><span class="tok-n">circle</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">w</span><span class="tok-p">);</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">closestY</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">clamp</span><span class="tok-p">(</span><span class="tok-n">circle</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">rect</span><span class="tok-p">.</span><span class="tok-n">h</span><span class="tok-p">);</span>
+
+<span class="tok-w">    </span><span class="tok-c1">// this gets the distance form the center of the circle to a side of a</span>
+<span class="tok-w">    </span><span class="tok-c1">// rectangle. It could be negative or positive. Because we are squaring them</span>
+<span class="tok-w">    </span><span class="tok-c1">// it doesn&#39;t matter.</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">circle</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">closestX</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">circle</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">closestY</span><span class="tok-p">;</span>
+
+<span class="tok-w">    </span><span class="tok-c1">// use the dot product to determine collision</span>
+<span class="tok-w">    </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">dx</span><span class="tok-o">*</span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-o">*</span><span class="tok-n">dy</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-o">&lt;=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">r</span><span class="tok-o">*</span><span class="tok-n">r</span><span class="tok-p">);</span>
+<span class="tok-p">}</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_circle_and_circle_collision">Circle and Circle collision</h2>
+<div class="sectionbody">
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="c"><span></span><span class="tok-kt">bool</span><span class="tok-w"> </span><span class="tok-nf">circlesCollide</span><span class="tok-p">(</span><span class="tok-n">Circle</span><span class="tok-w"> </span><span class="tok-n">a</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-n">Circle</span><span class="tok-w"> </span><span class="tok-n">b</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">a</span><span class="tok-p">.</span><span class="tok-n">x</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">b</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-n">a</span><span class="tok-p">.</span><span class="tok-n">y</span><span class="tok-p">;</span>
+<span class="tok-w">    </span><span class="tok-kt">float</span><span class="tok-w"> </span><span class="tok-n">distance</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-n">sqrtf</span><span class="tok-p">(</span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dx</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-n">dy</span><span class="tok-p">);</span>
+
+<span class="tok-w">    </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-n">distance</span><span class="tok-w"> </span><span class="tok-o">&lt;=</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-n">a</span><span class="tok-p">.</span><span class="tok-n">r</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-n">b</span><span class="tok-p">.</span><span class="tok-n">r</span><span class="tok-p">);</span>
+<span class="tok-p">}</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
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+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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\ No newline at end of file
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--- /dev/null
+++ b/Physics/index.adoc
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+= ./Physics
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/Basics.html;hb=HEAD[Basics]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Static-Collision.html;hb=HEAD[Ball-Static-Collision]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Dynamic-Collision.html;hb=HEAD[Ball-Dynamic-Collision]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Bouncing.html;hb=HEAD[Ball-Bouncing]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/Collision.html;hb=HEAD[Collision]
diff --git a/Physics/index.html b/Physics/index.html
new file mode 100644 (file)
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+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
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+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
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+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
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+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
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+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
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+window.addEventListener("message", (e) => {
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+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Physics</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/Basics.html;hb=HEAD">Basics</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Static-Collision.html;hb=HEAD">Ball-Static-Collision</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Dynamic-Collision.html;hb=HEAD">Ball-Dynamic-Collision</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/Ball-Bouncing.html;hb=HEAD">Ball-Bouncing</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/Collision.html;hb=HEAD">Collision</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/Probability/Pascals-Triangle.adoc b/Probability/Pascals-Triangle.adoc
new file mode 100644 (file)
index 0000000..fba732e
--- /dev/null
+++ b/Probability/Pascals-Triangle.adoc
@@ -0,0 +1,13 @@
+= Probability | Pascal's Triangle
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Links
+- https://www.youtube.com/watch?v=O4Qnsubo2tg[Die Rolling Probability]
+- https://www.youtube.com/watch?v=D9DFobRr4pw[Probability Using Pascal's Triangle]
diff --git a/Probability/Pascals-Triangle.html b/Probability/Pascals-Triangle.html
new file mode 100644 (file)
index 0000000..9d8c95c
--- /dev/null
+++ b/Probability/Pascals-Triangle.html
@@ -0,0 +1,1085 @@
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+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Probability | Pascal&#8217;s Triangle</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=O4Qnsubo2tg">Die Rolling Probability</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=D9DFobRr4pw">Probability Using Pascal&#8217;s Triangle</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
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diff --git a/Probability/Permutations-and-Combinations.adoc b/Probability/Permutations-and-Combinations.adoc
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+++ b/Probability/Permutations-and-Combinations.adoc
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+= Probability | Permutations and Combinations.adoc
+:docinfo: shared
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+== Links
+- https://www.youtube.com/watch?v=XJnIdRXUi7A&t=315s[Permutations and Combinations]
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+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
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+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
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+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Probability | Permutations and Combinations.adoc</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=XJnIdRXUi7A&amp;t=315s">Permutations and Combinations</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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diff --git a/Probability/index.adoc b/Probability/index.adoc
new file mode 100644 (file)
index 0000000..77a46eb
--- /dev/null
+++ b/Probability/index.adoc
@@ -0,0 +1,11 @@
+= ./Probability
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Probability/Permutations-and-Combinations.html;hb=HEAD[Permutations-and-Combinations]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Probability/Pascals-Triangle.html;hb=HEAD[Pascals-Triangle]
diff --git a/Probability/index.html b/Probability/index.html
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index 0000000..8a851d0
--- /dev/null
+++ b/Probability/index.html
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+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
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+    checkbox.className = 'checkbox';
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+    */
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+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Probability</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Probability/Permutations-and-Combinations.html;hb=HEAD">Permutations-and-Combinations</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Probability/Pascals-Triangle.html;hb=HEAD">Pascals-Triangle</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/README.adoc b/README.adoc
new file mode 100644 (file)
index 0000000..9aaa0cb
--- /dev/null
+++ b/README.adoc
@@ -0,0 +1,14 @@
+= My Math Notes
+
+To view the rendered notes, you can visit this site:
+https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=index.html;hb=HEAD[my-math-notes].
+
+== Building
+To convert all the _.adoc_ files into html run this command in the root directory
+
+----
+./Makehtml make
+----
+
+It will automatically create a _index.html_ page for each subject. You do not
+need to make _index.html_ pages yourself
diff --git a/Triginometry/Basics.adoc b/Triginometry/Basics.adoc
new file mode 100644 (file)
index 0000000..01a214f
--- /dev/null
+++ b/Triginometry/Basics.adoc
@@ -0,0 +1,81 @@
+= Triginometry | Basics
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== SOH CAH TOA
+
+== Using inverse functions To Find Angle
+[stem]
+++++
+\tan\theta = \frac{20.8}{7}
+++++
+
+[stem]
+++++
+\tan^-1\theta = tan^-1(\frac{20.8}{7})
+++++
+
+[stem]
+++++
+\theta = 71.4^\circ
+++++
+
+NOTE: Make sure calculator is in degrees to get the answer in degrees
+
+NOTE: The inverse version of sine, cosine, and tangent have the "arc" prefix
+
+== Trig Visualized
+https://www.youtube.com/watch?v=dUkCgTOOpQ0[Trig Visualized]
+
+In this video _d_ represents the line that is tangent to the circle. and the
+tangent also represents that slope of the radius
+
+image::trig-visualized-tangent.png[tangent]
+
+this works because we get the tangent with TOA or
+stem:[tan(\theta) = \frac{opposite}{adjacent}].
+We know that at a 45 degree angle stem:[sin(\theta)] and stem:[cos(\theta)] are
+going to be the same value. If they are divided by each other they should equal
+the length of the tangent/radius, which is 1. Hence, they have the same ratio.
+
+[source, lua]
+----
+local radii = {1, 2, 4, 5}
+
+function toRadians(degrees)
+    return (degrees * (math.pi/180))
+end
+
+for k, v in pairs(radii) do
+    print(
+        "radius: " .. 
+         v ..
+         ", " ..
+        "tangent: " .. 
+        math.tan(toRadians(45)) 
+    )
+end
+
+----
+
+.output
+[%collapsible]
+====
+radius: 2, tangent: 1
+radius: 4, tangent: 1
+radius: 5, tangent: 1
+
+====
+
+notice how the tangent is always one when you're at a 45 degree angle. No matter
+what the radius. 
+
+== Links
+* https://www.youtube.com/watch?v=jt7p-mCC0ng[Evaluating Inverse Functions]
+* https://www.youtube.com/watch?v=dUkCgTOOpQ0[trig visualization]
diff --git a/Triginometry/Basics.html b/Triginometry/Basics.html
new file mode 100644 (file)
index 0000000..f7d98b5
--- /dev/null
+++ b/Triginometry/Basics.html
@@ -0,0 +1,1277 @@
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+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>Triginometry | Basics</title>
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+pre.pygments .tok-ges { color: #F8F8F2; font-weight: bold; font-style: italic } /* Generic.EmphStrong */
+pre.pygments .tok-gr { color: #F8F8F2 } /* Generic.Error */
+pre.pygments .tok-gh { color: #F8F8F2 } /* Generic.Heading */
+pre.pygments .tok-gi { color: #A6E22E } /* Generic.Inserted */
+pre.pygments .tok-go { color: #66D9EF } /* Generic.Output */
+pre.pygments .tok-gp { color: #FF4689; font-weight: bold } /* Generic.Prompt */
+pre.pygments .tok-gs { color: #F8F8F2; font-weight: bold } /* Generic.Strong */
+pre.pygments .tok-gu { color: #959077 } /* Generic.Subheading */
+pre.pygments .tok-gt { color: #F8F8F2 } /* Generic.Traceback */
+pre.pygments .tok-kc { color: #66D9EF } /* Keyword.Constant */
+pre.pygments .tok-kd { color: #66D9EF } /* Keyword.Declaration */
+pre.pygments .tok-kn { color: #FF4689 } /* Keyword.Namespace */
+pre.pygments .tok-kp { color: #66D9EF } /* Keyword.Pseudo */
+pre.pygments .tok-kr { color: #66D9EF } /* Keyword.Reserved */
+pre.pygments .tok-kt { color: #66D9EF } /* Keyword.Type */
+pre.pygments .tok-ld { color: #E6DB74 } /* Literal.Date */
+pre.pygments .tok-m { color: #AE81FF } /* Literal.Number */
+pre.pygments .tok-s { color: #E6DB74 } /* Literal.String */
+pre.pygments .tok-na { color: #A6E22E } /* Name.Attribute */
+pre.pygments .tok-nb { color: #F8F8F2 } /* Name.Builtin */
+pre.pygments .tok-nc { color: #A6E22E } /* Name.Class */
+pre.pygments .tok-no { color: #66D9EF } /* Name.Constant */
+pre.pygments .tok-nd { color: #A6E22E } /* Name.Decorator */
+pre.pygments .tok-ni { color: #F8F8F2 } /* Name.Entity */
+pre.pygments .tok-ne { color: #A6E22E } /* Name.Exception */
+pre.pygments .tok-nf { color: #A6E22E } /* Name.Function */
+pre.pygments .tok-nl { color: #F8F8F2 } /* Name.Label */
+pre.pygments .tok-nn { color: #F8F8F2 } /* Name.Namespace */
+pre.pygments .tok-nx { color: #A6E22E } /* Name.Other */
+pre.pygments .tok-py { color: #F8F8F2 } /* Name.Property */
+pre.pygments .tok-nt { color: #FF4689 } /* Name.Tag */
+pre.pygments .tok-nv { color: #F8F8F2 } /* Name.Variable */
+pre.pygments .tok-ow { color: #FF4689 } /* Operator.Word */
+pre.pygments .tok-pm { color: #F8F8F2 } /* Punctuation.Marker */
+pre.pygments .tok-w { color: #F8F8F2 } /* Text.Whitespace */
+pre.pygments .tok-mb { color: #AE81FF } /* Literal.Number.Bin */
+pre.pygments .tok-mf { color: #AE81FF } /* Literal.Number.Float */
+pre.pygments .tok-mh { color: #AE81FF } /* Literal.Number.Hex */
+pre.pygments .tok-mi { color: #AE81FF } /* Literal.Number.Integer */
+pre.pygments .tok-mo { color: #AE81FF } /* Literal.Number.Oct */
+pre.pygments .tok-sa { color: #E6DB74 } /* Literal.String.Affix */
+pre.pygments .tok-sb { color: #E6DB74 } /* Literal.String.Backtick */
+pre.pygments .tok-sc { color: #E6DB74 } /* Literal.String.Char */
+pre.pygments .tok-dl { color: #E6DB74 } /* Literal.String.Delimiter */
+pre.pygments .tok-sd { color: #E6DB74 } /* Literal.String.Doc */
+pre.pygments .tok-s2 { color: #E6DB74 } /* Literal.String.Double */
+pre.pygments .tok-se { color: #AE81FF } /* Literal.String.Escape */
+pre.pygments .tok-sh { color: #E6DB74 } /* Literal.String.Heredoc */
+pre.pygments .tok-si { color: #E6DB74 } /* Literal.String.Interpol */
+pre.pygments .tok-sx { color: #E6DB74 } /* Literal.String.Other */
+pre.pygments .tok-sr { color: #E6DB74 } /* Literal.String.Regex */
+pre.pygments .tok-s1 { color: #E6DB74 } /* Literal.String.Single */
+pre.pygments .tok-ss { color: #E6DB74 } /* Literal.String.Symbol */
+pre.pygments .tok-bp { color: #F8F8F2 } /* Name.Builtin.Pseudo */
+pre.pygments .tok-fm { color: #A6E22E } /* Name.Function.Magic */
+pre.pygments .tok-vc { color: #F8F8F2 } /* Name.Variable.Class */
+pre.pygments .tok-vg { color: #F8F8F2 } /* Name.Variable.Global */
+pre.pygments .tok-vi { color: #F8F8F2 } /* Name.Variable.Instance */
+pre.pygments .tok-vm { color: #F8F8F2 } /* Name.Variable.Magic */
+pre.pygments .tok-il { color: #AE81FF } /* Literal.Number.Integer.Long */
+</style>
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
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+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Triginometry | Basics</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_soh_cah_toa">SOH CAH TOA</a></li>
+<li><a href="#_using_inverse_functions_to_find_angle">Using inverse functions To Find Angle</a></li>
+<li><a href="#_trig_visualized">Trig Visualized</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_soh_cah_toa">SOH CAH TOA</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+<div class="sect1">
+<h2 id="_using_inverse_functions_to_find_angle">Using inverse functions To Find Angle</h2>
+<div class="sectionbody">
+<div class="stemblock">
+<div class="content">
+\$\tan\theta = \frac{20.8}{7}\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\tan^-1\theta = tan^-1(\frac{20.8}{7})\$
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\$\theta = 71.4^\circ\$
+</div>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+Make sure calculator is in degrees to get the answer in degrees
+</td>
+</tr>
+</table>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+The inverse version of sine, cosine, and tangent have the "arc" prefix
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_trig_visualized">Trig Visualized</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p><a href="https://www.youtube.com/watch?v=dUkCgTOOpQ0">Trig Visualized</a></p>
+</div>
+<div class="paragraph">
+<p>In this video <em>d</em> represents the line that is tangent to the circle. and the
+tangent also represents that slope of the radius</p>
+</div>
+<div class="imageblock">
+<div class="content">
+<img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/trig-visualized-tangent.png" alt="tangent">
+</div>
+</div>
+<div class="paragraph">
+<p>this works because we get the tangent with TOA or
+\$tan(\theta) = \frac{opposite}{adjacent}\$.
+We know that at a 45 degree angle \$sin(\theta)\$ and \$cos(\theta)\$ are
+going to be the same value. If they are divided by each other they should equal
+the length of the tangent/radius, which is 1. Hence, they have the same ratio.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="pygments highlight"><code data-lang="lua"><span></span><span class="tok-kd">local</span><span class="tok-w"> </span><span class="tok-nv">radii</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-mi">1</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">2</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">4</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-mi">5</span><span class="tok-p">}</span>
+
+<span class="tok-kr">function</span><span class="tok-w"> </span><span class="tok-nf">toRadians</span><span class="tok-p">(</span><span class="tok-nv">degrees</span><span class="tok-p">)</span>
+<span class="tok-w">    </span><span class="tok-kr">return</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-nv">degrees</span><span class="tok-w"> </span><span class="tok-o">*</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-nb">math.pi</span><span class="tok-o">/</span><span class="tok-mi">180</span><span class="tok-p">))</span>
+<span class="tok-kr">end</span>
+
+<span class="tok-kr">for</span><span class="tok-w"> </span><span class="tok-nv">k</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nv">v</span><span class="tok-w"> </span><span class="tok-kr">in</span><span class="tok-w"> </span><span class="tok-nb">pairs</span><span class="tok-p">(</span><span class="tok-nv">radii</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-kr">do</span>
+<span class="tok-w">    </span><span class="tok-nb">print</span><span class="tok-p">(</span>
+<span class="tok-w">        </span><span class="tok-s2">&quot;radius: &quot;</span><span class="tok-w"> </span><span class="tok-o">..</span>
+<span class="tok-w">         </span><span class="tok-nv">v</span><span class="tok-w"> </span><span class="tok-o">..</span>
+<span class="tok-w">         </span><span class="tok-s2">&quot;, &quot;</span><span class="tok-w"> </span><span class="tok-o">..</span>
+<span class="tok-w">        </span><span class="tok-s2">&quot;tangent: &quot;</span><span class="tok-w"> </span><span class="tok-o">..</span>
+<span class="tok-w">        </span><span class="tok-nb">math.tan</span><span class="tok-p">(</span><span class="tok-nf">toRadians</span><span class="tok-p">(</span><span class="tok-mi">45</span><span class="tok-p">))</span>
+<span class="tok-w">    </span><span class="tok-p">)</span>
+<span class="tok-kr">end</span></code></pre>
+</div>
+</div>
+<details>
+<summary class="title">output</summary>
+<div class="content">
+<div class="paragraph">
+<p>radius: 2, tangent: 1
+radius: 4, tangent: 1
+radius: 5, tangent: 1</p>
+</div>
+</div>
+</details>
+<div class="paragraph">
+<p>notice how the tangent is always one when you&#8217;re at a 45 degree angle. No matter
+what the radius.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.youtube.com/watch?v=jt7p-mCC0ng">Evaluating Inverse Functions</a></p>
+</li>
+<li>
+<p><a href="https://www.youtube.com/watch?v=dUkCgTOOpQ0">trig visualization</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
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+</html>
\ No newline at end of file
diff --git a/Triginometry/Centers.adoc b/Triginometry/Centers.adoc
new file mode 100644 (file)
index 0000000..ad8b4d4
--- /dev/null
+++ b/Triginometry/Centers.adoc
@@ -0,0 +1,23 @@
+= Triginometry | Centers
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Incenter
+
+[.center]
+image:Incenter.jpg[]
+
+Incenter should be the center of the inscribed circle
+
+== Centroid
+[.center]
+image:Centroid.png[]
+
+== Orthocenter
+== Circumcenter
diff --git a/Triginometry/Centers.html b/Triginometry/Centers.html
new file mode 100644 (file)
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+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
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+body.book #header .details{border:0!important;display:block;padding:0!important}
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+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
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+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
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+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
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+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
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+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
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+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
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+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
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+}
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+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
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+}
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+  margin-left: 2.5em;
+}
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+
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+
+    /*
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+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
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+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
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+}
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+}
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+}
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+.highlight {
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+}
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+img {
+    margin: auto;
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+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
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+}
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+::-webkit-scrollbar-thumb {
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+}
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+}
+
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+}
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+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
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+}
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+}
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+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
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+details > .content {
+    border: solid;
+    padding: 10px;
+}
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+</style>
+
+</head>
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+<script>
+function sendHeight() {
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+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
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+/*
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+    if (e.key === "o") {
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+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
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+
+function scrollStep() {
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+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
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+        requestAnimationFrame(scrollStep);
+    } else {
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+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
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+        scrollKeys[e.key] = true;
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+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
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+    if(e.key === "o") {
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+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
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+});
+
+
+
+window.onload = function () {
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+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
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+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
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+    const text = p.textContent.trim();
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+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
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+      content = text.slice(1).trim();
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+
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+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
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+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Triginometry | Centers</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_incenter">Incenter</a></li>
+<li><a href="#_centroid">Centroid</a></li>
+<li><a href="#_orthocenter">Orthocenter</a></li>
+<li><a href="#_circumcenter">Circumcenter</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_incenter">Incenter</h2>
+<div class="sectionbody">
+<div class="paragraph center">
+<p><span class="image"><img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Incenter.jpg" alt="Incenter"></span></p>
+</div>
+<div class="paragraph">
+<p>Incenter should be the center of the inscribed circle</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_centroid">Centroid</h2>
+<div class="sectionbody">
+<div class="paragraph center">
+<p><span class="image"><img src="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=.images/Centroid.png" alt="Centroid"></span></p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_orthocenter">Orthocenter</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+<div class="sect1">
+<h2 id="_circumcenter">Circumcenter</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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+  TeX: { equationNumbers: { autoNumber: "none" } }
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+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/Triginometry/Identities.adoc b/Triginometry/Identities.adoc
new file mode 100644 (file)
index 0000000..5eb6042
--- /dev/null
+++ b/Triginometry/Identities.adoc
@@ -0,0 +1,101 @@
+= Triginometry | Identities
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:stem:
+:toc: left
+:docinfodir: {DOCINFODIR}
+:imagesdir: https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=.images
+
+== Even vs Odd Functions
+[%header, cols="1,1"]
+|===
+| Odd | Even
+| sin | cos
+| tan | sec
+| csc |
+| cot |
+|===
+
+== SOH CAH TOA
+[.inline]
+- stem:[\sin(\theta)] [.dull]#stem:[=]# stem:[\frac{opp}{hyp}]
+
+[.inline]
+- stem:[\cos(\theta)] [.dull]#stem:[=]#  stem:[\frac{adj}{hyp}]
+
+[.inline]
+- stem:[\tan(\theta)] [.dull]#stem:[=]# stem:[\frac{opp}{adj}]
+
+[.inline]
+- stem:[\csc(\theta)] [.dull]#stem:[=]# stem:[\frac{hyp}{opp}]
+
+[.inline]
+- stem:[\sec(\theta)] [.dull]#stem:[=]# stem:[\frac{hyp}{adj}]
+
+[.inline]
+- stem:[\cot(\theta)] [.dull]#stem:[=]# stem:[\frac{adj}{opp}]
+
+== Reciprocal
+
+[.inline]
+- stem:[\sin(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\csc(\theta)}]
+
+[.inline]
+- stem:[\cos(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\sec(\theta)}]
+
+[.inline]
+- stem:[\tan(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\cot(\theta)}]
+
+[.inline]
+- stem:[\csc(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\sin(\theta)}]
+
+[.inline]
+- stem:[\sec(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\cos(\theta)}]
+
+[.inline]
+- stem:[\cot(\theta)] [.dull]#stem:[=]# stem:[\frac{1}{\tan(\theta)}]
+
+== Pythagoras Therom
+[.inline]
+- stem:[a^2 + b^2] [.dull]#stem:[=]# stem:[c^2]
+
+[.inline]
+- stem:[\sin^2(\theta) + \cos^2(\theta)] [.dull]#stem:[=]# stem:[1]
+
+[.inline]
+- stem:[1 + \cot^2(\theta)] [.dull]#stem:[=]# stem:[\csc^2(\theta)]
+
+[.inline]
+- stem:[1 + \tan^2(\theta)] [.dull]#stem:[=]# stem:[\sec^2(\theta)]
+
+== Ratio Trigonometric Identities
+[.inline]
+- stem:[\tan(\theta)] [.dull]#stem:[=]# stem:[\frac{\sin(\theta)}{\cos(\theta)}]
+
+[.inline]
+- stem:[\cot(\theta)] [.dull]#stem:[=]# stem:[\frac{\cos(\theta)}{\sin(\theta)}]
+
+
+== Double Angle
+[.inline]
+- stem:[\sin(2\theta)] [.dull]#stem:[=]# stem:[2\sin(\theta)\cos(\theta)]
+
+[.inline]
+- stem:[\cos(2\theta)] [.dull]#stem:[=]# stem:[\cos^2(\theta) - \sin^2(\theta)] [.dull]#stem:[=]# stem:[2\cos^2(\theta) - 1] [.dull]#stem:[=]# stem:[1 - 2\sin^2(\theta)]
+
+[.inline]
+- stem:[\tan(2\theta)] [.dull]#stem:[=]# stem:[\frac{2\tan(\theta)}{1 - \tan^2(\theta)}]
+
+== Circle
+[.inline]
+- stem:[\sin(\theta)] [.dull]#stem:[=]# stem:[0] [.dull]#stem:[\leftrightarrow]# stem:[\theta] [.dull]#stem:[=]# stem:[K\pi]
+
+[.inline]
+- stem:[\cos(\theta)] [.dull]#stem:[=]# stem:[0] [.dull]#stem:[\leftrightarrow]# stem:[\theta] [.dull]#stem:[=]# stem:[\pi/2 + k\pi]
+
+NOTE: You can think of it like this as well stem:[sin(K\pi) = 0] and stem:[cos(\pi/2 + k\pi) = 0]
+
+== Links
+- https://byjus.com/maths/trigonometric-identities/[Trigonometric Identities]
diff --git a/Triginometry/Identities.html b/Triginometry/Identities.html
new file mode 100644 (file)
index 0000000..fe741a4
--- /dev/null
+++ b/Triginometry/Identities.html
@@ -0,0 +1,1327 @@
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+<title>Triginometry | Identities</title>
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+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
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+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
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+.hide-on-print{display:none!important}
+.print-only{display:block!important}
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+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
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+</head>
+<body class="article toc2 toc-left">
+<div id="header">
+<h1>Triginometry | Identities</h1>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#_even_vs_odd_functions">Even vs Odd Functions</a></li>
+<li><a href="#_soh_cah_toa">SOH CAH TOA</a></li>
+<li><a href="#_reciprocal">Reciprocal</a></li>
+<li><a href="#_pythagoras_therom">Pythagoras Therom</a></li>
+<li><a href="#_ratio_trigonometric_identities">Ratio Trigonometric Identities</a></li>
+<li><a href="#_double_angle">Double Angle</a></li>
+<li><a href="#_circle">Circle</a></li>
+<li><a href="#_links">Links</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_even_vs_odd_functions">Even vs Odd Functions</h2>
+<div class="sectionbody">
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Odd</th>
+<th class="tableblock halign-left valign-top">Even</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">sin</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">cos</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tan</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">sec</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">csc</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">cot</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_soh_cah_toa">SOH CAH TOA</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sin(\theta)\$ <span class="dull">\$=\$</span> \$\frac{opp}{hyp}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cos(\theta)\$ <span class="dull">\$=\$</span>  \$\frac{adj}{hyp}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\tan(\theta)\$ <span class="dull">\$=\$</span> \$\frac{opp}{adj}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\csc(\theta)\$ <span class="dull">\$=\$</span> \$\frac{hyp}{opp}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sec(\theta)\$ <span class="dull">\$=\$</span> \$\frac{hyp}{adj}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cot(\theta)\$ <span class="dull">\$=\$</span> \$\frac{adj}{opp}\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_reciprocal">Reciprocal</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sin(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\csc(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cos(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\sec(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\tan(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\cot(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\csc(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\sin(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sec(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\cos(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cot(\theta)\$ <span class="dull">\$=\$</span> \$\frac{1}{\tan(\theta)}\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_pythagoras_therom">Pythagoras Therom</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$a^2 + b^2\$ <span class="dull">\$=\$</span> \$c^2\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sin^2(\theta) + \cos^2(\theta)\$ <span class="dull">\$=\$</span> \$1\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$1 + \cot^2(\theta)\$ <span class="dull">\$=\$</span> \$\csc^2(\theta)\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$1 + \tan^2(\theta)\$ <span class="dull">\$=\$</span> \$\sec^2(\theta)\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_ratio_trigonometric_identities">Ratio Trigonometric Identities</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\tan(\theta)\$ <span class="dull">\$=\$</span> \$\frac{\sin(\theta)}{\cos(\theta)}\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cot(\theta)\$ <span class="dull">\$=\$</span> \$\frac{\cos(\theta)}{\sin(\theta)}\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_double_angle">Double Angle</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sin(2\theta)\$ <span class="dull">\$=\$</span> \$2\sin(\theta)\cos(\theta)\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cos(2\theta)\$ <span class="dull">\$=\$</span> \$\cos^2(\theta) - \sin^2(\theta)\$ <span class="dull">\$=\$</span> \$2\cos^2(\theta) - 1\$ <span class="dull">\$=\$</span> \$1 - 2\sin^2(\theta)\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\tan(2\theta)\$ <span class="dull">\$=\$</span> \$\frac{2\tan(\theta)}{1 - \tan^2(\theta)}\$</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_circle">Circle</h2>
+<div class="sectionbody">
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\sin(\theta)\$ <span class="dull">\$=\$</span> \$0\$ <span class="dull">\$\leftrightarrow\$</span> \$\theta\$ <span class="dull">\$=\$</span> \$K\pi\$</p>
+</li>
+</ul>
+</div>
+<div class="ulist inline">
+<ul>
+<li>
+<p>\$\cos(\theta)\$ <span class="dull">\$=\$</span> \$0\$ <span class="dull">\$\leftrightarrow\$</span> \$\theta\$ <span class="dull">\$=\$</span> \$\pi/2 + k\pi\$</p>
+</li>
+</ul>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+You can think of it like this as well \$sin(K\pi) = 0\$ and \$cos(\pi/2 + k\pi) = 0\$
+</td>
+</tr>
+</table>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_links">Links</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://byjus.com/maths/trigonometric-identities/">Trigonometric Identities</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 21:58:32 -0400
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
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\ No newline at end of file
diff --git a/Triginometry/index.adoc b/Triginometry/index.adoc
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+++ b/Triginometry/index.adoc
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+= ./Triginometry
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Centers.html;hb=HEAD[Centers]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Basics.html;hb=HEAD[Basics]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Identities.html;hb=HEAD[Identities]
diff --git a/Triginometry/index.html b/Triginometry/index.html
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+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
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+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
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+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
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+.colist td:not([class]):last-child{padding:.25em 0}
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+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
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+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
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+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
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+.aqua-background{background:#00fafa}
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+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
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+.gray{color:#606060}
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+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
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+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
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+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./Triginometry</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Centers.html;hb=HEAD">Centers</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Basics.html;hb=HEAD">Basics</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Triginometry/Identities.html;hb=HEAD">Identities</a></p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
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diff --git a/diagram/index.adoc b/diagram/index.adoc
new file mode 100644 (file)
index 0000000..6e8a612
--- /dev/null
+++ b/diagram/index.adoc
@@ -0,0 +1,9 @@
+= ./diagram
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+== Topics
+
diff --git a/diagram/index.html b/diagram/index.html
new file mode 100644 (file)
index 0000000..aa1b28d
--- /dev/null
+++ b/diagram/index.html
@@ -0,0 +1,1046 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 2.0.20">
+<title>./diagram</title>
+<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700">
+<style>
+/*! Asciidoctor default stylesheet | MIT License | https://asciidoctor.org */
+/* Uncomment the following line when using as a custom stylesheet */
+/* @import "https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700"; */
+html{font-family:sans-serif;-webkit-text-size-adjust:100%}
+a{background:none}
+a:focus{outline:thin dotted}
+a:active,a:hover{outline:0}
+h1{font-size:2em;margin:.67em 0}
+b,strong{font-weight:bold}
+abbr{font-size:.9em}
+abbr[title]{cursor:help;border-bottom:1px dotted #dddddf;text-decoration:none}
+dfn{font-style:italic}
+hr{height:0}
+mark{background:#ff0;color:#000}
+code,kbd,pre,samp{font-family:monospace;font-size:1em}
+pre{white-space:pre-wrap}
+q{quotes:"\201C" "\201D" "\2018" "\2019"}
+small{font-size:80%}
+sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}
+sup{top:-.5em}
+sub{bottom:-.25em}
+img{border:0}
+svg:not(:root){overflow:hidden}
+figure{margin:0}
+audio,video{display:inline-block}
+audio:not([controls]){display:none;height:0}
+fieldset{border:1px solid silver;margin:0 2px;padding:.35em .625em .75em}
+legend{border:0;padding:0}
+button,input,select,textarea{font-family:inherit;font-size:100%;margin:0}
+button,input{line-height:normal}
+button,select{text-transform:none}
+button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}
+button[disabled],html input[disabled]{cursor:default}
+input[type=checkbox],input[type=radio]{padding:0}
+button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}
+textarea{overflow:auto;vertical-align:top}
+table{border-collapse:collapse;border-spacing:0}
+*,::before,::after{box-sizing:border-box}
+html,body{font-size:100%}
+body{background:#fff;color:rgba(0,0,0,.8);padding:0;margin:0;font-family:"Noto Serif","DejaVu Serif",serif;line-height:1;position:relative;cursor:auto;-moz-tab-size:4;-o-tab-size:4;tab-size:4;word-wrap:anywhere;-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased}
+a:hover{cursor:pointer}
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+object,embed{height:100%}
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+.text-left{text-align:left!important}
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+img,object,svg{display:inline-block;vertical-align:middle}
+textarea{height:auto;min-height:50px}
+select{width:100%}
+.subheader,.admonitionblock td.content>.title,.audioblock>.title,.exampleblock>.title,.imageblock>.title,.listingblock>.title,.literalblock>.title,.stemblock>.title,.openblock>.title,.paragraph>.title,.quoteblock>.title,table.tableblock>.title,.verseblock>.title,.videoblock>.title,.dlist>.title,.olist>.title,.ulist>.title,.qlist>.title,.hdlist>.title{line-height:1.45;color:#7a2518;font-weight:400;margin-top:0;margin-bottom:.25em}
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+a:hover,a:focus{color:#1d4b8f}
+a img{border:0}
+p{line-height:1.6;margin-bottom:1.25em;text-rendering:optimizeLegibility}
+p aside{font-size:.875em;line-height:1.35;font-style:italic}
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+ul,ol{margin-left:1.5em}
+ul li ul,ul li ol{margin-left:1.25em;margin-bottom:0}
+ul.circle{list-style-type:circle}
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+dl dd{margin-bottom:1.25em}
+blockquote{margin:0 0 1.25em;padding:.5625em 1.25em 0 1.1875em;border-left:1px solid #ddd}
+blockquote,blockquote p{line-height:1.6;color:rgba(0,0,0,.85)}
+@media screen and (min-width:768px){h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{line-height:1.2}
+h1{font-size:2.75em}
+h2{font-size:2.3125em}
+h3,#toctitle,.sidebarblock>.content>.title{font-size:1.6875em}
+h4{font-size:1.4375em}}
+table{background:#fff;margin-bottom:1.25em;border:1px solid #dedede;word-wrap:normal}
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+table tr th,table tr td{padding:.5625em .625em;font-size:inherit;color:rgba(0,0,0,.8)}
+table tr.even,table tr.alt{background:#f8f8f7}
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+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    border-bottom: solid;
+    */
+    width: fit-content;
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-green) !important;
+    opacity: 80%;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+  margin-right: 12px;
+  margin-left: 12px;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+img {
+    margin: auto;
+    width: 100%;
+}
+
+/* ================= SCROLLBARS (WebKit) ================= */
+::-webkit-scrollbar {
+    width: 8px;
+}
+
+::-webkit-scrollbar-thumb {
+    background: var(--gruvbox-bg1);
+    border-radius: 4px;
+}
+
+::-webkit-scrollbar-track {
+    background: transparent;
+}
+
+div.ulist {
+    max-width: 95%;
+}
+
+.paragraph {
+    max-width: 98%;
+}
+
+.todo-list li .checkbox {
+  display: inline-block;   /* already set */
+  width: 1.5em;            /* fixed width */
+  height: 1.5em;           /* keep height same */
+  margin-right: 0.5em;
+  vertical-align: middle; /* align with text nicely */
+  text-align: center;
+}
+
+li.started .checkbox {
+  color: var(--gruvbox-yellow);
+}
+
+li.completed .checkbox {
+  color: var(--gruvbox-green);
+}
+
+li.empty .checkbox {
+  color: var(--gruvbox-red);
+}
+
+.todo-list li {
+  font-family: "inherit";      /* inherits whatever body is using */
+  line-height: 1.4em;          /* optional: keep spacing nice */
+}
+
+ul.checklist li p {
+  font-size: 1.2em;   /* same as body text */
+}
+
+
+
+
+div#header {
+    background-color: var(--gruvbox-blue);
+    height: 100px;
+    top: 0px;
+    background-color: var(--gruvbox-black);
+    z-index: 1000;
+    border-bottom: 2px solid var(--gruvbox-white);
+    padding-bottom: 2px;
+}
+
+#content {
+    /*margin-top: 105px;*/
+}
+
+details > .content {
+    border: solid;
+    padding: 10px;
+}
+
+</style>
+
+</head>
+
+<script>
+function sendHeight() {
+  const height = document.documentElement.scrollHeight;
+  parent.postMessage({ type: "HEIGHT", height }, "*");
+}
+
+window.addEventListener("load", sendHeight);
+window.addEventListener("resize", sendHeight);
+
+/*
+document.addEventListener("keydown", (e) => {
+    if (e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+*/
+let scrollKeys = { j: false, k: false };
+let isScrolling = false;
+const SCROLL_SPEED = 10; // pixels per frame
+
+function scrollStep() {
+    let delta = 0;
+    if (scrollKeys.j) delta += SCROLL_SPEED;
+    if (scrollKeys.k) delta -= SCROLL_SPEED;
+
+    if (delta !== 0) {
+        window.scrollBy(0, delta);
+        requestAnimationFrame(scrollStep);
+    } else {
+        isScrolling = false;
+    }
+}
+
+document.addEventListener("keydown", (e) => {
+    const tag = document.activeElement.tagName;
+    if (tag === "INPUT" || tag === "TEXTAREA" || document.activeElement.isContentEditable) return;
+
+    if (e.key === "j" || e.key === "k") {
+        e.preventDefault();
+        scrollKeys[e.key] = true;
+
+        if (!isScrolling) {
+            isScrolling = true;
+            requestAnimationFrame(scrollStep);
+        }
+    }
+    if(e.key === "o") {
+        e.preventDefault();
+        parent.postMessage({ type: "CLOSE_NOTES" }, "*");
+    }
+});
+
+document.addEventListener("keyup", (e) => {
+    if (e.key === "j" || e.key === "k") {
+        scrollKeys[e.key] = false;
+    }
+});
+
+
+
+window.onload = function () {
+
+  // Select all checklist ULs
+  document.querySelectorAll('ul.checklist').forEach(function(ul) {
+  ul.classList.add('todo-list');
+
+  ul.querySelectorAll('li').forEach(function(li) {
+    const p = li.querySelector('p');
+    if (!p) return;
+
+    const text = p.textContent.trim();
+
+    let state = '';
+    if (text.startsWith('❏')) {
+      state = 'empty';
+    } else if (text.startsWith('✓')) {
+      state = 'started';
+    } else if (text.startsWith('[X]')) {
+      state = 'completed';
+    } else {
+      return;
+    }
+
+    li.classList.add(state);
+
+    // Remove the marker
+    let content = '';
+    if (state === 'completed') {
+      content = text.slice(3).trim();
+    } else {
+      content = text.slice(1).trim();
+    }
+
+    // Only rewrite the first <p>, keep nested divs/lists intact
+    p.innerHTML = '';
+
+    const checkbox = document.createElement('span');
+    checkbox.className = 'checkbox';
+
+    /*
+    if (state === 'empty') checkbox.textContent = '🌑';
+    else if (state === 'started') checkbox.textContent = '🌓';
+    else if (state === 'completed') checkbox.textContent = '🌕';
+    */
+    if (state === 'empty') checkbox.textContent = '☐';
+    else if (state === 'started') checkbox.textContent = '◪';
+    else if (state === 'completed') checkbox.textContent = '▣';
+
+    p.appendChild(checkbox);
+    p.appendChild(document.createTextNode(' ' + content));
+  });
+});
+
+};
+window.addEventListener("load", e => {
+    // Request the scroll position as soon as the iframe is loaded
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_GET"
+    }, "*");
+});
+
+// Capture scroll position while scrolling
+window.addEventListener("scroll", e => {
+    const scrollPosition = window.scrollY || document.documentElement.scrollTop;
+    window.parent.postMessage({
+        type: "SCROLL_REQUEST_SET",
+        scrollPosition: scrollPosition
+    }, "*");
+});
+
+// Receive messages to set the scroll position
+window.addEventListener("message", (e) => {
+    if (e.data.type === "SCROLL_POSITION") {
+        // Set the scroll position to the iframe
+        window.scrollTo(0, e.data.scrollPosition);
+    }
+});
+</script>
+</head>
+<body class="article">
+<div id="header">
+<h1>./diagram</h1>
+</div>
+<div id="content">
+<div class="sect1">
+<h2 id="_topics">Topics</h2>
+<div class="sectionbody">
+
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
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+DOCINFODIR=/usr/local/share/asciidoctor-styles/dark1/
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diff --git a/index.adoc b/index.adoc
new file mode 100644 (file)
index 0000000..7fac565
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+= Math Notes
+:docinfo: shared
+:source-highlighter: pygments
+:pygments-style: monokai
+:icons: font
+:docinfodir: {DOCINFODIR}
+
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphing/index.html;hb=HEAD[Graphing]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Calculus/index.html;hb=HEAD[Calculus]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Physics/index.html;hb=HEAD[Physics]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Triginometry/index.html;hb=HEAD[Triginometry]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Probability/index.html;hb=HEAD[Probability]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Graphics/index.html;hb=HEAD[Graphics]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Algrebra/index.html;hb=HEAD[Algrebra]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=diagram/index.html;hb=HEAD[diagram]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Geometry/index.html;hb=HEAD[Geometry]
+* https://{HOSTNAME}/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/index.html;hb=HEAD[Linear-Algrebra]
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+.listingblock.terminal pre .command:not([data-prompt])::before{content:"$"}
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+pre.prettyprint .linenums{line-height:1.45;margin-left:2em}
+pre.prettyprint li{background:none;list-style-type:inherit;padding-left:0}
+pre.prettyprint li code[data-lang]::before{opacity:1}
+pre.prettyprint li:not(:first-child) code[data-lang]::before{display:none}
+table.linenotable{border-collapse:separate;border:0;margin-bottom:0;background:none}
+table.linenotable td[class]{color:inherit;vertical-align:top;padding:0;line-height:inherit;white-space:normal}
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+table.linenotable td.linenos,pre.pygments .linenos{border-right:1px solid;opacity:.35;padding-right:.5em;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}
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+.quoteblock.abstract blockquote::before,.quoteblock.excerpt blockquote::before,.quoteblock .quoteblock blockquote::before{display:none}
+.quoteblock.abstract blockquote,.quoteblock.abstract p,.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{line-height:1.6;word-spacing:0}
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+.quoteblock.excerpt>blockquote,.quoteblock .quoteblock{padding:0 0 .25em 1em;border-left:.25em solid #dddddf}
+.quoteblock.excerpt,.quoteblock .quoteblock{margin-left:0}
+.quoteblock.excerpt blockquote,.quoteblock.excerpt p,.quoteblock .quoteblock blockquote,.quoteblock .quoteblock p{color:inherit;font-size:1.0625rem}
+.quoteblock.excerpt .attribution,.quoteblock .quoteblock .attribution{color:inherit;font-size:.85rem;text-align:left;margin-right:0}
+p.tableblock:last-child{margin-bottom:0}
+td.tableblock>.content{margin-bottom:1.25em;word-wrap:anywhere}
+td.tableblock>.content>:last-child{margin-bottom:-1.25em}
+table.tableblock,th.tableblock,td.tableblock{border:0 solid #dedede}
+table.grid-all>*>tr>*{border-width:1px}
+table.grid-cols>*>tr>*{border-width:0 1px}
+table.grid-rows>*>tr>*{border-width:1px 0}
+table.frame-all{border-width:1px}
+table.frame-ends{border-width:1px 0}
+table.frame-sides{border-width:0 1px}
+table.frame-none>colgroup+*>:first-child>*,table.frame-sides>colgroup+*>:first-child>*{border-top-width:0}
+table.frame-none>:last-child>:last-child>*,table.frame-sides>:last-child>:last-child>*{border-bottom-width:0}
+table.frame-none>*>tr>:first-child,table.frame-ends>*>tr>:first-child{border-left-width:0}
+table.frame-none>*>tr>:last-child,table.frame-ends>*>tr>:last-child{border-right-width:0}
+table.stripes-all>*>tr,table.stripes-odd>*>tr:nth-of-type(odd),table.stripes-even>*>tr:nth-of-type(even),table.stripes-hover>*>tr:hover{background:#f8f8f7}
+th.halign-left,td.halign-left{text-align:left}
+th.halign-right,td.halign-right{text-align:right}
+th.halign-center,td.halign-center{text-align:center}
+th.valign-top,td.valign-top{vertical-align:top}
+th.valign-bottom,td.valign-bottom{vertical-align:bottom}
+th.valign-middle,td.valign-middle{vertical-align:middle}
+table thead th,table tfoot th{font-weight:bold}
+tbody tr th{background:#f7f8f7}
+tbody tr th,tbody tr th p,tfoot tr th,tfoot tr th p{color:rgba(0,0,0,.8);font-weight:bold}
+p.tableblock>code:only-child{background:none;padding:0}
+p.tableblock{font-size:1em}
+ol{margin-left:1.75em}
+ul li ol{margin-left:1.5em}
+dl dd{margin-left:1.125em}
+dl dd:last-child,dl dd:last-child>:last-child{margin-bottom:0}
+li p,ul dd,ol dd,.olist .olist,.ulist .ulist,.ulist .olist,.olist .ulist{margin-bottom:.625em}
+ul.checklist,ul.none,ol.none,ul.no-bullet,ol.no-bullet,ol.unnumbered,ul.unstyled,ol.unstyled{list-style-type:none}
+ul.no-bullet,ol.no-bullet,ol.unnumbered{margin-left:.625em}
+ul.unstyled,ol.unstyled{margin-left:0}
+li>p:empty:only-child::before{content:"";display:inline-block}
+ul.checklist>li>p:first-child{margin-left:-1em}
+ul.checklist>li>p:first-child>.fa-square-o:first-child,ul.checklist>li>p:first-child>.fa-check-square-o:first-child{width:1.25em;font-size:.8em;position:relative;bottom:.125em}
+ul.checklist>li>p:first-child>input[type=checkbox]:first-child{margin-right:.25em}
+ul.inline{display:flex;flex-flow:row wrap;list-style:none;margin:0 0 .625em -1.25em}
+ul.inline>li{margin-left:1.25em}
+.unstyled dl dt{font-weight:400;font-style:normal}
+ol.arabic{list-style-type:decimal}
+ol.decimal{list-style-type:decimal-leading-zero}
+ol.loweralpha{list-style-type:lower-alpha}
+ol.upperalpha{list-style-type:upper-alpha}
+ol.lowerroman{list-style-type:lower-roman}
+ol.upperroman{list-style-type:upper-roman}
+ol.lowergreek{list-style-type:lower-greek}
+.hdlist>table,.colist>table{border:0;background:none}
+.hdlist>table>tbody>tr,.colist>table>tbody>tr{background:none}
+td.hdlist1,td.hdlist2{vertical-align:top;padding:0 .625em}
+td.hdlist1{font-weight:bold;padding-bottom:1.25em}
+td.hdlist2{word-wrap:anywhere}
+.literalblock+.colist,.listingblock+.colist{margin-top:-.5em}
+.colist td:not([class]):first-child{padding:.4em .75em 0;line-height:1;vertical-align:top}
+.colist td:not([class]):first-child img{max-width:none}
+.colist td:not([class]):last-child{padding:.25em 0}
+.thumb,.th{line-height:0;display:inline-block;border:4px solid #fff;box-shadow:0 0 0 1px #ddd}
+.imageblock.left{margin:.25em .625em 1.25em 0}
+.imageblock.right{margin:.25em 0 1.25em .625em}
+.imageblock>.title{margin-bottom:0}
+.imageblock.thumb,.imageblock.th{border-width:6px}
+.imageblock.thumb>.title,.imageblock.th>.title{padding:0 .125em}
+.image.left,.image.right{margin-top:.25em;margin-bottom:.25em;display:inline-block;line-height:0}
+.image.left{margin-right:.625em}
+.image.right{margin-left:.625em}
+a.image{text-decoration:none;display:inline-block}
+a.image object{pointer-events:none}
+sup.footnote,sup.footnoteref{font-size:.875em;position:static;vertical-align:super}
+sup.footnote a,sup.footnoteref a{text-decoration:none}
+sup.footnote a:active,sup.footnoteref a:active{text-decoration:underline}
+#footnotes{padding-top:.75em;padding-bottom:.75em;margin-bottom:.625em}
+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
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+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
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+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+    margin-bottom: 20px !important;
+}
+
+.hdlist1 {
+  /*font-size: 1.3em;*/
+    text-decoration: underline;
+}
+
+/* for list element definitions * *define-me*: */
+li strong {
+    /*text-decoration: underline;*/
+    border-radius: .2;
+    color: grey;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+}
+
+.checklist li ul { margin-left: 3%; }
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 {
+    padding-bottom: 0;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
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+</head>
+<body class="article">
+<div id="header">
+<h1>Math Notes</h1>
+</div>
+<div id="content">
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphing/index.html;hb=HEAD">Graphing</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Calculus/index.html;hb=HEAD">Calculus</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Physics/index.html;hb=HEAD">Physics</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Triginometry/index.html;hb=HEAD">Triginometry</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Probability/index.html;hb=HEAD">Probability</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Graphics/index.html;hb=HEAD">Graphics</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Algrebra/index.html;hb=HEAD">Algrebra</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=diagram/index.html;hb=HEAD">diagram</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Geometry/index.html;hb=HEAD">Geometry</a></p>
+</li>
+<li>
+<p><a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=Linear-Algrebra/index.html;hb=HEAD">Linear-Algrebra</a></p>
+</li>
+</ul>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Last updated 2026-05-08 23:12:56 -0400
+</div>
+</div>
+</body>
+</html>
\ No newline at end of file
diff --git a/notes.adoc b/notes.adoc
new file mode 100644 (file)
index 0000000..d129cd2
--- /dev/null
+++ b/notes.adoc
@@ -0,0 +1,28 @@
+== Issues
+By defaut the stem asciidoctor extension does not support some things. Like
+writing the index of roots.
+
+You can  using latex math you can combat this
+
+----
+:stem: latexmath
+----
+
+Then, things like
+
+----
+\sqrt[5]{25}
+----
+
+Will show  up correctly
+
+== Links
+* https://www.overleaf.com/learn/latex/Subscripts_and_superscripts
+* https://math.mit.edu/~djk/calculus_beginners/chapter04/section02.html
+* https://www.youtube.com/watch?v=PahbNFypubE&t=398s[interpolation]
+* https://en.wikipedia.org/wiki/Summation[Summation]
+* https://youtu.be/I0vDXO5FeaI?si=DpAxgKE7eYFAk2el[Math Symbols]
+* https://www.cmor-faculty.rice.edu/~heinken/latex/symbols.pdf[latex math symbols]
+* https://latexeditor.lagrida.com/[live latex editor]
+* https://wiki.inkscape.org/wiki/Installing_Inkscape[inkscape | drawing diagrams]
+* https://www.youtube.com/watch?v=31A8FYRk6Ko[inkscape | graphs]
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+#footnotes hr{width:20%;min-width:6.25em;margin:-.25em 0 .75em;border-width:1px 0 0}
+#footnotes .footnote{padding:0 .375em 0 .225em;line-height:1.3334;font-size:.875em;margin-left:1.2em;margin-bottom:.2em}
+#footnotes .footnote a:first-of-type{font-weight:bold;text-decoration:none;margin-left:-1.05em}
+#footnotes .footnote:last-of-type{margin-bottom:0}
+#content #footnotes{margin-top:-.625em;margin-bottom:0;padding:.75em 0}
+div.unbreakable{page-break-inside:avoid}
+.big{font-size:larger}
+.small{font-size:smaller}
+.underline{text-decoration:underline}
+.overline{text-decoration:overline}
+.line-through{text-decoration:line-through}
+.aqua{color:#00bfbf}
+.aqua-background{background:#00fafa}
+.black{color:#000}
+.black-background{background:#000}
+.blue{color:#0000bf}
+.blue-background{background:#0000fa}
+.fuchsia{color:#bf00bf}
+.fuchsia-background{background:#fa00fa}
+.gray{color:#606060}
+.gray-background{background:#7d7d7d}
+.green{color:#006000}
+.green-background{background:#007d00}
+.lime{color:#00bf00}
+.lime-background{background:#00fa00}
+.maroon{color:#600000}
+.maroon-background{background:#7d0000}
+.navy{color:#000060}
+.navy-background{background:#00007d}
+.olive{color:#606000}
+.olive-background{background:#7d7d00}
+.purple{color:#600060}
+.purple-background{background:#7d007d}
+.red{color:#bf0000}
+.red-background{background:#fa0000}
+.silver{color:#909090}
+.silver-background{background:#bcbcbc}
+.teal{color:#006060}
+.teal-background{background:#007d7d}
+.white{color:#bfbfbf}
+.white-background{background:#fafafa}
+.yellow{color:#bfbf00}
+.yellow-background{background:#fafa00}
+span.icon>.fa{cursor:default}
+a span.icon>.fa{cursor:inherit}
+.admonitionblock td.icon [class^="fa icon-"]{font-size:2.5em;text-shadow:1px 1px 2px rgba(0,0,0,.5);cursor:default}
+.admonitionblock td.icon .icon-note::before{content:"\f05a";color:#19407c}
+.admonitionblock td.icon .icon-tip::before{content:"\f0eb";text-shadow:1px 1px 2px rgba(155,155,0,.8);color:#111}
+.admonitionblock td.icon .icon-warning::before{content:"\f071";color:#bf6900}
+.admonitionblock td.icon .icon-caution::before{content:"\f06d";color:#bf3400}
+.admonitionblock td.icon .icon-important::before{content:"\f06a";color:#bf0000}
+.conum[data-value]{display:inline-block;color:#fff!important;background:rgba(0,0,0,.8);border-radius:50%;text-align:center;font-size:.75em;width:1.67em;height:1.67em;line-height:1.67em;font-family:"Open Sans","DejaVu Sans",sans-serif;font-style:normal;font-weight:bold}
+.conum[data-value] *{color:#fff!important}
+.conum[data-value]+b{display:none}
+.conum[data-value]::after{content:attr(data-value)}
+pre .conum[data-value]{position:relative;top:-.125em}
+b.conum *{color:inherit!important}
+.conum:not([data-value]):empty{display:none}
+dt,th.tableblock,td.content,div.footnote{text-rendering:optimizeLegibility}
+h1,h2,p,td.content,span.alt,summary{letter-spacing:-.01em}
+p strong,td.content strong,div.footnote strong{letter-spacing:-.005em}
+p,blockquote,dt,td.content,td.hdlist1,span.alt,summary{font-size:1.0625rem}
+p{margin-bottom:1.25rem}
+.sidebarblock p,.sidebarblock dt,.sidebarblock td.content,p.tableblock{font-size:1em}
+.exampleblock>.content{background:#fffef7;border-color:#e0e0dc;box-shadow:0 1px 4px #e0e0dc}
+.print-only{display:none!important}
+@page{margin:1.25cm .75cm}
+@media print{*{box-shadow:none!important;text-shadow:none!important}
+html{font-size:80%}
+a{color:inherit!important;text-decoration:underline!important}
+a.bare,a[href^="#"],a[href^="mailto:"]{text-decoration:none!important}
+a[href^="http:"]:not(.bare)::after,a[href^="https:"]:not(.bare)::after{content:"(" attr(href) ")";display:inline-block;font-size:.875em;padding-left:.25em}
+abbr[title]{border-bottom:1px dotted}
+abbr[title]::after{content:" (" attr(title) ")"}
+pre,blockquote,tr,img,object,svg{page-break-inside:avoid}
+thead{display:table-header-group}
+svg{max-width:100%}
+p,blockquote,dt,td.content{font-size:1em;orphans:3;widows:3}
+h2,h3,#toctitle,.sidebarblock>.content>.title{page-break-after:avoid}
+#header,#content,#footnotes,#footer{max-width:none}
+#toc,.sidebarblock,.exampleblock>.content{background:none!important}
+#toc{border-bottom:1px solid #dddddf!important;padding-bottom:0!important}
+body.book #header{text-align:center}
+body.book #header>h1:first-child{border:0!important;margin:2.5em 0 1em}
+body.book #header .details{border:0!important;display:block;padding:0!important}
+body.book #header .details span:first-child{margin-left:0!important}
+body.book #header .details br{display:block}
+body.book #header .details br+span::before{content:none!important}
+body.book #toc{border:0!important;text-align:left!important;padding:0!important;margin:0!important}
+body.book #toc,body.book #preamble,body.book h1.sect0,body.book .sect1>h2{page-break-before:always}
+.listingblock code[data-lang]::before{display:block}
+#footer{padding:0 .9375em}
+.hide-on-print{display:none!important}
+.print-only{display:block!important}
+.hide-for-print{display:none!important}
+.show-for-print{display:inherit!important}}
+@media amzn-kf8,print{#header>h1:first-child{margin-top:1.25rem}
+.sect1{padding:0!important}
+.sect1+.sect1{border:0}
+#footer{background:none}
+#footer-text{color:rgba(0,0,0,.6);font-size:.9em}}
+@media amzn-kf8{#header,#content,#footnotes,#footer{padding:0}}
+</style>
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css">
+<head>
+<!-- <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=DotGothic16"> -->
+<!-- <link rel="icon" type="image/x-icon" href="https://raw.githubusercontent.com/A-Larsen/math-notes/refs/heads/main/favicon.png"> -->
+<!-- <link rel="icon" type="image/png" href="N2DE_favicon.png"/> -->
+<!-- <link rel="preconnect" href="https://fonts.googleapis.com"> -->
+<!-- <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> -->
+<!-- <link href="https://fonts.googleapis.com/css2?family=Abril+Fatface&display=swap" rel="stylesheet"> -->
+<style>
+
+/*--------------------------------------------------------------*/
+/*use '!important' to override default css used by asciidoctor*/
+/*--------------------------------------------------------------*/
+
+:root {
+    --gruvbox-black: #282828;
+    --gruvbox-black2: #1d2021;
+    --gruvbox-black3: #463f39;
+    --gruvbox-red: #fc7869;
+    --gruvbox-yellow: #fabd2f;
+    --gruvbox-white: #ebdbb2;
+    --gruvbox-white2: #a7a187;
+    --gruvbox-blue: #83a598;
+    --gruvbox-green: #b8bb26;
+    --gruvbox-bg1: #7c6f64;
+    --gruvbox-bg2: #665c54;
+
+}
+
+.title {
+  color: var(--gruvbox-red) !important;
+}
+
+blockquote {
+    color: var(--gruvbox-white) !important;
+}
+
+table {
+    margin: 0 auto !important;
+}
+
+.hdlist1 {
+  font-size: 1.3em;
+}
+
+cite {
+    color: var(--gruvbox-green) !important;
+}
+
+.tableblock {
+    background-color: inherit;
+    color: var(--gruvbox-white);
+    max-width: 95%;
+}
+
+.tableblock th {
+    background-color: var(--gruvbox-black3);
+    color: var(--gruvbox-white);
+    max-width: 97%;
+}
+
+code:not([data-lang]) {
+    background-color: var(--gruvbox-bg2) !important;
+    color: var(--gruvbox-white);
+    text-shadow: none;
+}
+
+.checklist {
+    text-align: left;
+    font-size: .8em;
+}
+
+.checklist li ul { margin-left: 3%;
+}
+
+#content, h1, pre {
+    border-radius: 8px !important;
+}
+
+#header, #content, #footnotes, #footer {
+  padding-left: 0 !important;
+  padding-right: 0 !important;
+}
+
+.api-type a:hover {
+    text-decoration: underline;
+}
+
+.api-type, .api-type a {
+    color: var(--gruvbox-green);
+    text-decoration: none;
+}
+
+pre {
+    border-style: solid !important;
+    border-color: white !important;
+    border-width: 2px !important;
+}
+
+h1 {
+   /* font-family: "DotGothic16", sans-serif !important;*/
+
+
+    background-color: var(--gruvbox-white2) !important;
+    color: var(--gruvbox-black) !important;
+}
+
+.admonitionblock > table td.content {
+    color: var(--gruvbox-white) !important;
+}
+
+/*
+html {
+    margin: 0;
+    height: 100%;
+    width: 100%;
+}
+
+
+*/
+/*html {*/
+
+body {
+    /*background: black url("background-image-space.jpeg") no-repeat fixed center;*/
+    /*background-color: black;*/
+    background-color: var(--gruvbox-black);
+    background-attachment: fixed;
+    background-size: cover;
+    overflow-y: scroll;
+    /*
+    margin: 0;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    */
+}
+
+
+/*
+body {
+    overflow-y: scroll;
+    background: rgba(0,0,0,0);
+    height: 100%;
+    margin: 2%;
+}
+*/
+
+#toc {
+    background-color: var(--gruvbox-black2) !important;
+    /*opacity: .9;*/
+}
+
+#toctitle {
+    color: var(--gruvbox-yellow);
+}
+
+a {
+    color: var(--gruvbox-bg1);
+}
+
+#content {
+    color: #ebdbb2;
+    background: var(--gruvbox-black);
+    /*opacity: .9;*/
+}
+
+#header > h1:nth-last-child(2) {
+    border-bottom: none !important;
+    background-color: rgba(0, 0, 0, .1);
+    padding-left: 2%;
+}
+
+.sect1 + .sect1 {
+  border-top: none !important;
+}
+
+.sect1 h2 {
+    background-size: cover;
+    background-attachment: fixed;
+    /*
+    border-bottom: 1px solid var(--gruvbox-white);
+    border-top: 1px solid var(--gruvbox-white);
+    */
+    padding: 10px;
+    text-align: left;
+    color: var(--gruvbox-red) !important;
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-black);
+    text-shadow: 3px 3px 3px var(--gruvbox-black);
+    text-shadow: 5px 5px 5px var(--gruvbox-blue);
+    */
+}
+
+h3, h4, h5 {
+    /*
+    text-shadow: 1px 1px 1px var(--gruvbox-white);
+    */
+}
+
+h3 {
+  color: var(--gruvbox-red) !important;
+}
+
+h4 {
+  color: var(--gruvbox-yellow) !important;
+}
+
+h5 {
+  color: #83a598 !important;
+}
+
+h2 ~ *:not(h1) {
+  margin-left: .5em;
+}
+
+.pygments {
+    max-width: 97% !important;
+    white-space: pre;
+    overflow-x: scroll !important;
+}
+
+::before {
+    /*makes sure that the programming language that show up on hover is within
+      pre tag*/
+    padding-right: 3%;
+}
+
+@media screen and (max-width: 800px) {
+    .checklist li ul {
+        margin-left: 6% !important;
+    }
+    #toc {
+        padding-left: 10px;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+@media screen and (max-width: 535px) {
+    h3 ~ *:not(h2):not(h3) {
+      margin-left: 0 !important;
+    }
+
+    h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+      margin-left: 0 !important;
+    }
+    h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+      margin-left: 0 !important;
+    }
+
+    footer {
+        margin-bottom: 10px !important;
+    }
+    html {
+        margin-bottom: 10px !important;
+    }
+}
+h3 ~ *:not(h2):not(h3) {
+  margin-left: 1.5em;
+}
+
+h4 ~ *:not(h2):not(h3):not(.h3):not(h4) {
+  margin-left: 2em;
+}
+h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) {
+  margin-left: 2.5em;
+}
+
+
+.sect1 h2 {
+
+    /*
+    background: linear-gradient(
+        to bottom,
+        rgba(251, 189, 47, 0.2), /* yellow */
+        rgba(0, 0, 0, 0.4)       /* black */
+    );
+    */
+    /*opacity: .9;*/
+  }
+
+#preamble p{
+  padding-left: 10px;
+  color: var(--gruvbox-white) !important;
+}
+
+#footer {
+    max-width: 62.5em !important;
+    padding-left: 10px !important;
+    text-align: center;
+}
+
+.logo img {
+    display: block !important;
+    margin-left: auto !important;
+    margin-right: auto !important;
+}
+
+#header > h1:first-child {
+  margin-top: .3em !important;
+  padding-top: .2em !important;
+  text-align: center;
+}
+
+footer {
+    max-width: 62.5em;
+    margin-left: auto;
+    margin-right: auto;
+    display: block;
+
+}
+
+details[open] > summary::before {
+   /* display: none;*/
+    border-top: none;
+    content: "▼";
+    margin-top: -12px;
+}
+
+.example.stemblock {
+    border: solid;
+    font-size: 2vw;
+    max-width: 40%;
+    margin: auto;
+}
+
+.center {
+    max-width: 500px;
+    display: flex;
+    justify-content: center;
+}
+
+.highlight {
+    color: var(--gruvbox-yellow)
+}
+
+.border {
+    padding: 4px;
+    border: 2px solid rgba(255, 0, 0, .3);
+}
+
+.dull {
+    opacity: 40%;
+}
+
+.inline {
+    margin-right: 5%;
+    display: inline-block;
+}
+
+
+</style>
+
+</head>
+
+<script src="docinfo.js"></script>
+
+
+</head>
+<body class="article">
+<div id="header">
+<h1>My Math Notes</h1>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div class="paragraph">
+<p>To view the rendered notes, you can visit this site:
+<a href="https://nyquistdev.net/git/?p=math-notes.git;a=blob_plain;f=index.html;hb=HEAD">my-math-notes</a>.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="_building">Building</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>To convert all the <em>.adoc</em> files into html run this command in the root directory</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre>./Makehtml make</pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It will automatically create a <em>index.html</em> page for each subject. You do not
+need to make <em>index.html</em> pages yourself</p>
+</div>
+</div>
+</div>
+</div>
+<script type="text/x-mathjax-config">
+MathJax.Hub.Config({
+  messageStyle: "none",
+  tex2jax: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    ignoreClass: "nostem|nolatexmath"
+  },
+  asciimath2jax: {
+    delimiters: [["\\$", "\\$"]],
+    ignoreClass: "nostem|noasciimath"
+  },
+  TeX: { equationNumbers: { autoNumber: "none" } }
+})
+MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () {
+  MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) {
+    if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) {
+      data.math.root.display = "block"
+    }
+    return data
+  })
+})
+</script>
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
Unnamed repository; edit this file 'description' to name the repository.