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

1 - 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>


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.

3 - 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

4 - 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>

5 - Law

when you multiply complex numbers:

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

6 - Example

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

7 - Documentation / Reference

Data Science
Data Analysis
Data Science
Linear Algebra Mathematics

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