Complex numbers

A complex number is just a real number and an imaginary number that expressed together.

Rectangular Form
Very simple, $$z = x+yj$$, where:
 * z is the complex number
 * x is the real part
 * y is the imaginary part

Polar Form
The polar form of a complex number is stupid thing but it is. The polar form of imaginary number is represented as:

$$z = r\angle\theta $$ or $$ z = r(\cos\theta + j\sin\theta) $$

Where:
 * z is a complex number
 * r is the magnitude
 * $$\theta$$ is the angle in radians. Can also be in degrees sometimes.
 * $$\angle$$ is just a representation of the angle and doesn't stand for any number or anything.

See the graph for a more intuitive understanding, but it is basically a spinning line from the origin with an angle that ranges from 0 to 2&pi;.

https://i.imgur.com/whNrQKf.png

Euler Form
This form requires some deviation, but by using Euler's Theorem, we can express the complex number in this form. $$ z = r^{j\theta}$$. This is because in short:

$$ e^{j\theta} = \cos\theta + j \sin \theta \rightarrow re^{j\theta} = r\cos\theta + rj\sin\theta \\$$

As this includes e, it is often used for deviations. In addition there are some useful deviations on the Euler page as since this is basically just his theorem

Polar (and euler) to Rectangle
$$z = r\angle\theta \\ z = x + yj \\ x = r\cos\theta \\ yj = r\sin\theta j $$

Rectangle to Polar (and euler)
$$z = x + yj \\ z = r\angle\theta \\ r = \left\vert x+yj \right\vert = \sqrt{x^2 + y^2} \\ \theta = \tan{\frac{y}{x}} \\ $$

Addition/Subtraction
The general idea is that it is easiest in rectangular form:

$$ (x_1 + y_1j) + (x_2 + y_2j) = (x_1 + x_2) + j(y_1 + y_2) \\ (x_1 + y_1j) - (x_2 + y_2j) = (x_1 - x_2) + j(y_1 - y_2) \\ $$

Multiplication/Division
The general idea is that it is easiest in polar form:

$$ r_1 \angle \theta_1 \times r_2 \angle \theta_2 = r_1 r_2 \angle (\theta_1+\theta_2) \\ \frac{r_1\angle\theta_1}{r_2\angle \theta_2} = \frac{r_1}{r_2} \angle (\theta_1 - \theta_2) \\ $$