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Enhance basic math notation content by adding a historical overview of mathematical symbols, including the evolution of the equals sign and the introduction of zero. Update comparison section with clearer explanations and examples, and improve styling for visual elements in the math chapter. Adjust chapter structure to allow for flexible content ordering based on importance.

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content/math/basic-math-notation/_index.md
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@ -17,6 +17,45 @@ strand = "Number & Algebra"
arithmetic yet; you only need to <strong>read</strong> the symbols like words on a sign.
</p>
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<figure class="book-figure">
<img
class="book-figure__img"
src="/chapters/math/basic-math-notation/basic-math-notation-history.png"
alt="Cartoon timeline: counting long ago, Egyptian and Babylonian marks, different symbols around the world, then equals, plus and minus, times and divide, and brackets for grouping"
width="960"
height="640"
loading="lazy"
decoding="async"
/>
<figcaption class="book-figure__caption">
From finger counting and ancient marks to short symbols on one line: how +, −, ×, ÷, =, and ( )
became the signs you read today.
</figcaption>
</figure>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="digits-and-words" class="book-prose__heading">Digits, words, and spaces</h2>
<p>
The symbols <strong>0</strong> through <strong>9</strong> are called <strong>digits</strong>. Each
@ -218,28 +257,135 @@ strand = "Number & Algebra"
<h2 id="compare" class="book-prose__heading">Less than and greater than (&lt; &gt;)</h2>
<p>
Sometimes we compare two amounts without adding or subtracting. The open mouth of the symbol
points at the <strong>bigger</strong> number.
Sometimes we compare two amounts without adding or subtracting. Think of
<strong>&lt;</strong> and <strong>&gt;</strong> as a hungry mouth: the open side points at the
<strong>bigger</strong> number — it “wants” the larger pile.
</p>
<ul>
<li><strong>&lt;</strong> means <strong>less than</strong>. Example: <strong>3 &lt; 7</strong> — three is less than seven.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong>. Example: <strong>9 &gt; 5</strong> — nine is greater than five.</li>
<li><strong>&lt;</strong> means <strong>less than</strong> — the mouth opens toward the bigger number on the right.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong> — the mouth opens toward the bigger number on the left.</li>
</ul>
<h3 id="compare-examples" class="book-prose__heading">More examples</h3>
<p>Read each line like a short sentence. The symbol tells you which side is bigger:</p>
<ul>
<li><strong>3 &lt; 7</strong> — “three is less than seven.”</li>
<li><strong>9 &gt; 5</strong> — “nine is greater than five.”</li>
<li><strong>2 &lt; 6</strong> — “two is less than six.”</li>
<li><strong>8 &gt; 3</strong> — “eight is greater than three.”</li>
<li><strong>5 &lt; 5</strong> is <strong>not</strong> true — five is the same as five, not less.</li>
<li><strong>4 &lt; 4</strong> is also false — use <strong>=</strong> when both sides match.</li>
</ul>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount.”
The same idea works either way: <strong>7 &gt; 4</strong> and <strong>4 &lt; 7</strong> both say
that seven is the bigger amount.
</p>
<h3 id="compare-chain" class="book-prose__heading">A longer compare line</h3>
<p>
You can chain comparisons when several numbers line up from <strong>biggest to smallest</strong>
(or the other way around). Read it in order, one piece at a time:
</p>
<p>
<strong>9 &gt; 7 &gt; 4 &gt; 1</strong> — “nine is greater than seven, greater than four, greater
than one.” Each number is bigger than the one to its right.
</p>
<p>
In words: nine is greater than seven; seven is greater than four; four is greater than one. The
whole chain means <strong>nine wins</strong> and <strong>one is smallest</strong>.
</p>
<figure class="notation-compare-chain" role="group" aria-label="Nine dots, then seven, then four, then one; each group has fewer dots than the group before">
<ol class="notation-compare-chain__ladder">
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">9</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">7</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">4</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">1</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span>
</span>
</li>
</ol>
<figcaption class="notation-compare-chain__caption">
Nine dots &gt; seven dots &gt; four dots &gt; one dot — the pictures shrink the same way the
numbers do.
</figcaption>
</figure>
<p>
You can flip the story for “smallest first”: <strong>1 &lt; 4 &lt; 7 &lt; 9</strong> — “one is
less than four, less than seven, less than nine.” Same order, different mouths.
</p>
<div class="book-callout" role="note">
<p class="book-callout__label">Or the same amount</p>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount,” like <strong>5 ≤ 5</strong>.
</p>
</div>
<h2 id="comma-decimal" class="book-prose__heading">Comma and decimal point</h2>
<p>
In big numbers like <strong>1,000</strong>, a <strong>comma</strong> groups digits in threes so
the number is easier to read — one thousand, not “one zero zero zero” as separate ideas.
</p>
<p>
A <strong>decimal point</strong> <strong>.</strong> sits between a whole part and a smaller part.
<strong>3.50</strong> might mean three dollars and fifty cents: the part after the point is not
whole dollars. You will learn more about decimals later; for now, remember the dot splits the
number into two pieces.
A <strong>decimal point</strong> <strong>.</strong> is a tiny dot that acts like a
<strong>fence</strong> in a number. Everything on the <strong>left</strong> counts whole things
you can line up (1, 2, 3, …). Everything on the <strong>right</strong> tells a
<strong>piece</strong> of one more thing — not another full thing.
</p>
<p>
Take <strong>3.5</strong>. Read it as <strong>“three and a half”</strong>: three full ones, plus
half of the next one. The digit <strong>5</strong> after the point does <em>not</em> mean “five
more marbles” — it names how much of the <em>next</em> whole you have filled in.
</p>
<figure class="notation-decimal-demo" role="group" aria-label="Three point five shown as three whole dots and half of a fourth">
<p class="notation-decimal-demo__label">
<span class="notation-decimal-demo__whole">3</span><span class="notation-decimal-demo__point" aria-hidden="true">.</span><span class="notation-decimal-demo__fraction">5</span>
<span class="notation-decimal-demo__words">three and a half</span>
</p>
<div class="notation-decimal-demo__picture">
<div class="notation-decimal-demo__wholes" aria-hidden="true">
<span class="notation-dots notation-dots--decimal" role="presentation">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
<span class="notation-decimal-demo__fence" aria-hidden="true">·</span>
<span class="notation-decimal-demo__half" role="presentation">
<span class="notation-decimal-demo__half-bar">
<span class="notation-decimal-demo__half-on"></span>
<span class="notation-decimal-demo__half-off"></span>
</span>
</span>
</div>
</div>
<figcaption class="notation-decimal-demo__caption">
Left of the fence: three whole dots. Right of the fence: half of the next dot filled in — that
is what <strong>3.5</strong> means.
</figcaption>
</figure>
<p>
In a shop, <strong>3.50</strong> often means three dollars and fifty cents: dollars are whole
steps on the left, cents are small pieces on the right. You will learn more about decimals later;
for now, remember the point splits “how many wholes” from “how much of the next one.”
</p>
<h2 id="fractions" class="book-prose__heading">Fractions (½ and 1/2)</h2>
@ -296,28 +442,6 @@ strand = "Number & Algebra"
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation</h2>
<h3 id="why-symbols" class="book-prose__heading">Why symbols instead of only words?</h3>