# Number System - Complex Number - <math>\mathbb{C}</math>

### Table of Contents

## About

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. <MATH>\text{complex number} = \text{(real part)} + \text{(imaginary part)} i</MATH>

where:

- i is an imaginary number

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.

## Articles Related

## 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.

<MATH> |z| = \sqrt {a^2 + b^2} </MATH>

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

<MATH> arg z = θ </MATH>

## Law

when you multiply complex numbers:

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

## Example

- Problem: <math>(x -1)^2 = -9</math>
- Solution: <math>x = 1 + 3 i</math>