I. Complex Numbers

Definition 1.1

Complex numbers are objects taking the form $a+ib$, where $a,b\in \mathbb{R}$ and $i$ satisfies the equation $i^2=-1$.

• $a$ is called the real part of $a+ib$, written $Re(a+ib)$
• $b$ is called the imaginary part $a+ib$, written $Im(a+ib)$ The set of all complex numbers is denoted $\mathbb{C}$.

$$\begin{array}{rl}\text{Arithmetic Addition: }& (a+ib)+(c+id)=(a+c)+i(b+c)\\ \text{Multiplication: }& (a+ib)\times (c+id)=ac+iad+ibc+i^{2}bd\end{array}$$

Definition 1.2

If $z=a+ib$, then the conjugate of z, denoted $\overline{z}$ is $a-ib$.

Definition 1.3

The modulus of $z=a+ib$, denoted $|z|$ is $|z|=\sqrt{a^2+b^2}$.

Remark:

$|z|$ is the distance of $z$ from $0$ in the Argand diagram (by Pythagoras).

Lemma 1.4

$z\overline{z}=|z^2|$

Proof:

$$\begin{array}{rl}z\overline{z}=& (a+ib)(a-ib)\\ =& a^2+iab-iab-i^2{b}^2\\ =& a^2+b^2=|z{|}^2\end{array}$$ $$\begin{array}{rl}& \text{What is }\frac{z}{w}\text{, where }z,w\in \mathbb{C},w\ne 0?\\ & \text{Starting with the equation }\\ & \frac{\overline{w}}{\overline{w}}=1\\ & \text{we have}\\ & \frac{z}{w}=\frac{z}{w}\times 1=\frac{z\overline{w}}{w\overline{w}}=\frac{z\overline{w}}{|w{|}^2}\\ & \boxed{\therefore \frac{z}{w}=\frac{z\overline{w}}{|w{|}^2}}\end{array}$$