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      <text>10.1.2.5 Multiplication of complex numbers</text>
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    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>In general the square of a complex number is not necessarily a positive real number; it may not even be real at all. For example if <span class="math inline">\(z=-2+i\)</span> then <span class="math inline">\[\qquad
z^2=(-2+i)^2=4-4i+i^2=4-4i-1=3-4i.\]</span> However, the product of a complex number with its conjugate is always a non-negative real number. If <span class="math inline">\(z=a+i b\)</span> then <span class="math display">\[\begin{aligned}
zz^*&amp;=(a+i b)(a-i b)\\
&amp;=a^2-a(i b)+(i b)a-i^2b^2\\
&amp;=a^2-i^2b^2\\
&amp;=a^2+b^2\qquad \text{since}\ \ i^2=-1
\qquad\end{aligned}\]</span> For example, if <span class="math inline">\(z=2+i\)</span> then <span class="math inline">\[\qquad
zz^*=(2+i)(2-i)=4+1=5.\]</span></p>]]></text>
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