# Number System - Complex Number - $\mathbb{C}$

A real number plus an imaginary number is a complex number.

Complex numbers are a number system.

A complex number has a real part and an imaginary part. $$\text{complex number} = \text{(real part)} + \text{(imaginary part)} i$$

where:

Complex Numbers are the intellectual ancestors of vectors.

Complex numbers are convenient to apply geometric transformation (such as rotation, scaling and translation) in two dimensions.

## Visualization

Since any complex number is specified by two real numbers one can visualize them by plotting a point with coordinates (a,b) in the plane for a complex number a + bi. The plane in which one plot these complex numbers is called the Complex plane, or Argand plane.

• Real numbers lie on the horizontal axis
• Imaginary numbers lie on the vertical axis

## Properties

• Length: |z| is the distance from the origin to the point z in the complex plane.

$$|z| = \sqrt {a^2 + b^2}$$

• Angle: The angle θ is called the argument of the complex number z. Notation:

$$arg z = θ$$

## Law

when you multiply complex numbers:

• their lengths get multiplied
• and their arguments get added.

## Example

• Problem: $(x -1)^2 = -9$
• Solution: $x = 1 + 3 i$