Given two complex numbers, $a+bi$ and $c+di,$ we can ask about the quotient:

There are two ways to proceed. One is to remember that when you multiply a complex number by its complex conjugate the result is a real number. So we get this:

$\frac{a+bi}{c+di}~=~\frac{a+bi}{c+di}\times\frac{c-di}{c-di}$

The right-hand side multiples through, using the rules for multiplying rational numbers, to:

$\frac{a+bi}{c+di}\times\frac{c-di}{c-di}~=~\frac{(a+bi)(c-di)}{(c+di)(c-di)}$

$\frac{(a+bi)(c-di)}{(c+di)(c-di)}~=~\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

Alternatively, we can express the numbers using the polar representation of a complex number:

$z_1=re^{i\theta},\quad\quad~z_2=se^{i\phi}$

Then the quotient is:

$\frac{z_1}{z_2}~=~\frac{r}{s}e^{i(\theta-\phi)}$

This uses the index laws that $\frac{x^a}{x^b}~=~x^{a-b}.$

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Dividing Complex Numbers