Understanding complex multiplication with Euler’s formula

The simplest way to understand multiplication, division, and powers of complex numbers

A complex number consists of two parts, a real part and an imaginary part. We often represent it as an (x, y) point on an Argand diagram.

We can also represent a complex number using polar coordinates, and it turns out this is extremely useful when we look at multiplication and powers of complex numbers.

Argand diagram

We will start by looking at the complex number z given by:

Here it is represented on an Argand diagram:

In fact, this is the number 3 + 4i, and it is represented by the point (3, 4).

Polar representation

We can also represent the same number in polar form, also known as the modulus-argument form. This graph shows a complex number in polar form, with the previous (x, y) representation shown alongside it: