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      <text>10.1.2.12 Dividing one complex number by another</text>
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      <text><![CDATA[<h3>Dividing complex numbers </h3>

<p>Here we consider the operation of dividing one complex number <span class="math inline">\(z=a+i b\)</span> by another, <span class="math inline">\(w=c+i d\)</span>: <span class="math inline">\[\qquad
{z\over w}={a+i b\over {c+i d}}\]</span> We wish to simplify the right-hand side into the standard form of a complex number : <span class="math display">\[\text{(Real part)}\ +i\ \text{(Imaginary part)}\]</span> or the equivalent: <span class="math display">\[\text{(Real part)}\ +\ \text{(Imaginary part)}\ i\]</span> To do this we multiply ‘top and bottom’ by the complex conjugate of the bottom (the denominator), that is, by <span class="math inline">\(c-i d\)</span> (this is called <span><strong>rationalising</strong></span>): <span class="math inline">\[\qquad
{z\over w}={a+i b\over {c+i d}}={a+i b\over {c+i d}}\times {c-i d\over
{c-i d}}\]</span> and then carry out the multiplication, top and bottom: <span class="math display">\[\begin{aligned}
{z\over w}&amp;={(ac+bd)+i(bc-ad)\over {c^2+d^2}}\\
&amp;=\left({ac+bd\over
{c^2+d^2}}\right)+i\left({bc-ad\over
{c^2+d^2}}\right)\qquad\end{aligned}\]</span> which is now in the required form. The reason for rationalising is to get a real number in the denominator since a complex number divided by a real number is easy to evaluate.</p>]]></text>
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