Introduction

Real numbers, $\mathbb{R}$, are any numbers that make sense as a displacement. Ex. ($-2, \pi$). Some expressions are meaningless; $2/0$, $\sqrt{-7}$

To express them a new variable/value is introduced: $i$. The unit for an Imaginary Number can be defined as: $i = \sqrt{-1}$ and $\sqrt{-7} = 7i$

All all algebraic properties of $\mathbb{R}$ also hold true in $\mathbb{C}$ ex. (commutativity, associativity) except order ($\leq$)

Properties of the Imaginary Number

1. $i^2 = -1$
2. $i^3 = -i$
3. $i^4 = 1$
4. $i^5 = i$
5. repeat

Definition of Complex Numbers

$\mathbb{C} = \{ a + bi | a,b \in \mathbb{R}, i = \sqrt{-1}\}$, which translates to "set of things like $a+bi$ satisfying $a,b$ are in the set of $\mathbb{R}$ and $i = \sqrt{-1}$"

Some notation

$z \in \mathbb{C}, z = a + bi$

• $a = \textbf{Re}(z)$ where $Re$ is real part
• $b = \textbf{Im}(z)$ where $Im$ is imaginary part

If $\textbf{Im}(z) = 0, z = a = \mathbb{R}$, then every real number is complex

Modulus, Argument and Complex Conjugate

The modulus of a complex number can be referred to as the magintude and is illustrated as $r$ the above graph. Hence, $r = |z| = \sqrt{a^2+b^2}$

The argument is the represented by the angle in the graph above.

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. For example, if $z = a + bi$ then $z$'s complex conjugate is $z' = \overline{z} = a - bi$

Polar Form

$z \in \mathbb{C} \Rightarrow z = r\cos\theta + ir\sin\theta = r cis\theta$

• $\theta$ is the argument of $z$, $r = |z|$
• $z = a + bi \Rightarrow \cos\theta = \frac{a}{r}, \sin\theta = \frac{b}{r}$

Euler Form

$z = e^{i\theta}$

• $\overline{z} = e^{-i\theta}$
• Multiplication $\Rightarrow 2e^{i\frac{\pi}{2}}*(\frac{1}{3}e^{\frac{3\pi}{4}}) = \frac{2}{3}e^{i\frac{5\pi}{4}}$