Complex Numbers - An Overview

This article is a work in progress.

Foreword

Complex numbers were the bane of my existence in the beginning - well, not necessarily the numbers themselves, but certainly DeMoivre’s formula and polar forms. I’m going to try and break down all of the following concepts in nice, easy to digest chunks. We’ll start off with the basics.

Introduction

The first thing to remember is the complex numbers are not necessarily complex - this is just a name they’ve been given! Think of it like a Chernobyl packet - sounds like it’ll kill you, but it definitely won’t. (Thanks to XKCD for the inspiration there)

Complex numbers are simply numbers with the format: $a+bi$
$a$ and $b$ are real numbers and $i$ is defined as $i^2=-1$, the Imaginary part. And yes, it’s actually called imaginary. This is basically just a symbol ($i$) that gives you -1 when you square it - don’t overthink it!

We also use the symbols Re($a+bi$) = $a$ and Im($a+bi$) = $b$ for the real and imaginary parts of a complex number, respectively.

Take the following example:
$3+4i$ is our complex number.
Re($3+4i$) = $3$, as $3$ is the real part.
Inversely, Im($3+4i$) = $4$.
Please note that we don’t include the imaginary unit $i$ as the imaginary part of a complex number is defined to just be $b$, not $bi$, as seen above.

There are other formats that complex numbers may be written in, but when it is in the form $a+bi$ then it’s in Cartesian form.

Arithmetic

Let’s start with basic addition of complex numbers.
Let’s say we have two of them: $a+bi$ and $c+di$.
To add them, we use the following formula:
$(a+bi)+(c+di)=(a+c)+(b+d)i$
So we are basically just adding their real parts and then their imaginary parts.

Addition Example

Take the complex numbers $(5+4i)+(6+9i)$ and use the formula above.
$(5+6)+(4+9)i$
$=11+13i$

See? Simple!

Subtraction is just as easy - we just change up the formula a tiny bit.
Now we’ll be using the following:

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

Subtraction Example