Introduction to Complex Numbers

Complex numbers revolve around the imaginary number $i$. It follows the following identities:

\[\begin{aligned} \sqrt{-1}&=i\\ i^2&=-1\\ (-i)^2&=(-1)^2\times i^2\\&=-1 \end{aligned}\]

In general, complex numbers are written in the form:

\[z=a+ib\]

$a$ is called the real part of $z$:
- $\text{Re}(z)$
$b$ is called the imaginary part of $z$:
- - $\text{Im}(z)$
The numbers $a+ib$ ($a$ and $b$ are both real numbers) form the set of complex numbers: denoted by $C$.

The pars of real numbers $(\text{Re}(z),\text{Im}(z))\ z\in C$ form the complex plane.

Basic Operations

For addition of complex numbers you add the corresponding components together.

The real part stays the same but the imaginary part is multiplied by $-1$.

$\overline{1+i}=1-i$

\[\vert z \vert =\sqrt{\text{Re}(z)^2+\text{Im}(z)^2}\]

We take the positive root.

This is as you would expect.

You would expand the two like multiplying brackets.

In order to divide we need to emulate $v^{-1}$ in complex numbers. This is defined as so:

\[\frac{1}{v}=\frac{\bar v}{\vert v\vert^2}\]

Additionally $v\neq 0$

We can check this by multiplying $v$ with $v^{-1}$ to get $1$.