With the previous given information, complex numbers are defined by Laforest (2015):

A complex number is any number written in the form: z = a+ bi, where a and b are real numbers. a is known as the “real part” of z, and b as the “imaginary part”. We also define Re(z) and Im(z) as follows: Re(z) = a Im(z) = b (p. 15).

This means that complex numbers are a combination of real and imaginary numbers, we can say that R ⊂ C. In order to represent complex numbers, mathematicians use complex planes which are similar to cartesian coordinate systems but the x-axis represents the real part and the y-axis the imaginary part of the number.

What is important to remember about this topic is the imaginary unit number defined as i ² = -1, complex addition defined by:

(a+bi ) + (c+di ) = (a+c) + (b+d)i

Complex multiplication defined by:

(a+bi ) (c+di ) = ac + adi + bci + bdi ²

= (ac - bd) + (ad + bc)i

And Euler's formula:

e^{i θ} = cosθ + i sinθ

Remembering the math series expansion:

e^{i θ} = 1 + i θ + ¹⁄_2!i ²θ² + ¹⁄_3!i ³θ³ + ¹⁄_4!i ⁴θ⁴ + ...

= 1 + i θ - ¹⁄_2!θ² - ¹⁄_3!i θ³ + ¹⁄_4!θ⁴ + ...

= (1 - ¹⁄_2!θ² + ¹⁄_4!θ⁴ - ... ) + i(θ - ¹⁄_3!θ³ + ¹⁄_5!θ⁵ - ... )

= cosθ + i sinθ