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Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

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<div class="book-prose">
<p>
<strong>Complex numbers</strong> extend the real number system so equations like
\(x^2+1=0\) can be solved. In high school, you usually write a complex number in
<strong>Cartesian form</strong>:
\[
z = a + bi,
\]
where \(a,b\in\mathbb{R}\) and \(i^2=-1\).
</p>
<h2 id="imaginary-unit" class="book-prose__heading">The imaginary unit \(i\)</h2>
<div class="book-callout book-callout--formula" role="note">
<p class="book-callout__label">Definition</p>
<p class="book-formula">\[ i^2 = -1 \]</p>
</div>
<p>
Useful cycle:
\[
i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1,
\]
and then the pattern repeats every 4 powers.
</p>
<h2 id="cartesian-form" class="book-prose__heading">Real and imaginary parts</h2>
<p>
For \(z=a+bi\):
</p>
<ul>
<li>\(\Re(z)=a\) (real part)</li>
<li>\(\Im(z)=b\) (imaginary coefficient)</li>
</ul>
<p><strong>Example.</strong> If \(z=3-4i\), then \(\Re(z)=3\), \(\Im(z)=-4\).</p>
<h2 id="operations" class="book-prose__heading">Basic operations</h2>
<p>
Add/subtract by combining like terms:
</p>
<p class="book-formula">
\[
(a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i.
\]
</p>
<p>
Multiply using expansion and \(i^2=-1\):
</p>
<p class="book-formula">
\[
(a+bi)(c+di)=ac+adi+bci+bd\,i^2=(ac-bd)+(ad+bc)i.
\]
</p>
<p><strong>Example.</strong> \((2+3i)(1-4i)=2-8i+3i-12i^2=14-5i.\)</p>
<h2 id="conjugate" class="book-prose__heading">Conjugate and modulus</h2>
<p>
The conjugate of \(z=a+bi\) is
\[
\overline{z}=a-bi.
\]
Multiplying by conjugates removes imaginary parts:
\[
z\overline{z}=(a+bi)(a-bi)=a^2+b^2.
\]
</p>
<p>
The modulus (distance from origin in Argand plane) is:
\[
|z|=\sqrt{a^2+b^2}.
\]
</p>
<p><strong>Example.</strong> For \(z=3-4i\), \(\overline{z}=3+4i\), \(|z|=5\).</p>
<h2 id="division" class="book-prose__heading">Division in Cartesian form</h2>
<p>
To divide by \(c+di\), multiply top and bottom by \(c-di\):
</p>
<p class="book-formula">
\[
\frac{a+bi}{c+di}
=\frac{(a+bi)(c-di)}{(c+di)(c-di)}
=\frac{(a+bi)(c-di)}{c^2+d^2}.
\]
</p>
<p><strong>Example.</strong> \(\dfrac{1+2i}{3-i}\):</p>
<p>
\[
\frac{1+2i}{3-i}\cdot\frac{3+i}{3+i}
=\frac{(1+2i)(3+i)}{10}
=\frac{1+7i}{10}
=\frac{1}{10}+\frac{7}{10}i.
\]
</p>
<h2 id="argand-plane" class="book-prose__heading">Argand plane (graph view)</h2>
<p>
Plot \(z=a+bi\) as the point \((a,b)\): horizontal axis is real part, vertical axis is
imaginary part. This lets you interpret \(|z|\) as geometric distance.
</p>
<h2 id="solving-quadratics" class="book-prose__heading">Why this matters: solving quadratics</h2>
<p>
If \(x^2+4=0\), then \(x^2=-4\), so
\[
x=\pm 2i.
\]
Complex numbers ensure every quadratic has two roots (counting multiplicity), and later this
extends to all polynomial equations.
</p>
</div>