Division via polar form

A direct consequence of the above development would be to understand division of complex numbers involves subtraction of arguments.

First of all,

\[ \begin{align*} \frac{1}{z_2} & = \frac{1}{r_2(\cos \theta_2 + i \sin \theta_2)} \\ & = \frac{1}{r_2} \cdot \frac{\cos \theta_2 - i \sin \theta_2}{\cos^2 \theta_2 + \sin^2 \theta_2} \\ & = \frac{1}{r_2}(\cos \theta_2 - i \sin \theta_2) \\ & = \frac{1}{r_2}[\cos(-\theta_2) + i \sin(-\theta_2)]. \end{align*} \]

The effect of taking the reciprocal of a complex number

• reciprocates its length; and
• reflects it about the positive real axis.

In particular, when \(r_2=1\), i.e., when \(z_2\) is a complex number of unit length, then reciprocating it is equivalent to conjugating it.

Next, we proceed to divide \(z_1\) by \(z_2\).

\[\frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2} = [r_1(\cos \theta_1 + i \sin \theta_1)][\frac{1}{r_2}(\cos(-\theta_2) + i \sin(-\theta_2))] = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)].\]