Complex Numbers in Polar Form

All the complex numbers you’ve dealt with this unit have been in rectangular form (a + bi). Polar form complex numbers are written in terms of r (radius length) and an angle (θ). The numbers a and b are related to r and θ by the formulas below:


In the first video below, you’ll be shown how to convert from one form to another. Here, polar form is shown as:


Although it can also be written as:


In literature, this form is sometimes referred to as trigonometric form. But does it look familiar? Notice it models a ± bi, where a is replaced with the right side of equation (3), and b is replaced with the right side of equation (4).

Multiply and Divide Polar Complex Numbers

So far you’ve been shown two ways to represent polar complex numbers. To make matters simple, two videos have been prepared to show you how to multiply/divide polar complex numbers in their alternate forms. The first video shows how to multiply polar complex numbers simplified in the r∠θ format, while the link below it shows you how to multiply, divide, and power while in trigonometric format.

Similar computations, except when in trigonometric form are shown below ( ↓ ).

The formulas used in both videos are display here:

Raising Polar Complex Numbers to a Power

So far we’ve looked at how to multiple and divide polar complex numbers, but what if you wanted to raise it to a power? This is where DeMoivre’s theorem helps.

This tells use that to raise a complex number to a power, you simply multiply it by itself the proper number of times. Demonstrations are shown in the video below.

An Alternate Form – Exponential Form

In case you’re expected to write the complex number in exponential form, follow the instructions in the first video below. Once you’re comfortable with the process, watch the second one to learn how you can multiply, divide, and power these numbers. Just keep in mind that when using exponential form, your angle must be presented in radians, not degrees.