Addition, Subtraction, Products, and Quotients in Polar Form

To add two complex numbers $z = re^{i\theta}$, $w = se^{i\phi}$ written in polar form, it is best to convert them back to rectangular form first, as this makes it easier.

For example, consider $z = 2e^{i\frac{\pi}{4}}$ and $w = 3e^{i\frac{3\pi}{4}}$. Trying to add them together in this form doesn't really get us anywhere:

$$z + w = 2e^{i\frac{\pi}{4}} + 3e^{i\frac{3\pi}{4}}.$$

We can use Euler's Formula to convert to rectangular form and then knowledge of trigonometry to simplify this expression:

$$\begin{align*} z &= 2e^{i\frac{\pi}{4}} \\ &= 2\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right) \\ &= \left(\frac{2}{\sqrt{2}} + i\frac{2}{\sqrt{2}}\right) \\ &= \frac{2}{\sqrt{2}} + i\frac{2}{\sqrt{2}}, \end{align*}$$

and,

$$\begin{align*} w &= 3e^{i\frac{3\pi}{4}} \\ &= 3\left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right) \\ &= 3\left(-\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) \\ &= -\frac{3}{\sqrt{2}} + i\frac{3}{\sqrt{2}}. \end{align*}$$

Therefore:

\begin{align*} ...