Polar form arithmetic

Complex numbers in polar form, especially when expressed in exponential form ${r\mathrm{e}^{\mathrm{i}\theta},}$ lends itself especially well to multiplication, division and conjugation.

$$\begin{aligned} (r_1\mathrm{e}^{\mathrm{i}\theta_1}) \cdot (r_2\mathrm{e}^{\mathrm{i}\theta_2}) &= (r_1r_2)\mathrm{e}^{\mathrm{i}(\theta_1 + \theta_2)} \\ \left| z_1 z_2 \right| &= \left| z_1 \right| \left| z_2 \right| \\ \arg(z_1 z_2) &= \arg(z_1) + \arg(z_2) \end{aligned}$$

${\begin{aligned} \frac{r_1\mathrm{e}^{\mathrm{i}\theta_1}}{r_2\mathrm{e}^{\mathrm{i}\theta_2}} &= \frac{r_1}{r_2}\mathrm{e}^{\mathrm{i}(\theta_1 - \theta_2)} \\ \left| \frac{z_1}{z_2} \right| &= \frac{\left| z_1 \right|}{\left| z_2 \right|} \\ \arg\left(\frac{z_1}{z_2}\right) &= \arg\left(z_1\right) - \arg\left(z_2\right) \end{aligned}}$

$$\begin{aligned} (r\mathrm{e}^{\mathrm{i}\theta})^* &= r\mathrm{e}^{-\mathrm{i}\theta} \\ \left| z^* \right| &= \left| z \right| \\ \arg(z^*) &= -\arg(z) \end{aligned}$$

Principal argument

The problem with angles are that there are more than one way to express the same angle (by going multiple rounds). For example, the angles $\frac{1}{4}\pi$ represents the same angle as $\frac{9}{4}\pi$ and $-\frac{3}{4}\pi$.

We resolve this issue by defining the principal argument of a complex number to be between $-\pi$ and $\pi$.

Principal argument: ${-\pi < \theta \leq \pi.}$

If any of our arguments are outside of this range, we can add or subtract $2k\pi$ from it to bring it back.

Examples