Number System - Complex Number -


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.


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

Complex Number


  • 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>


when you multiply complex numbers:

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


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

Documentation / Reference


Discover More
Image Vector
Linear Algebra - Vector

tuple in Linear algebra are called vector. A vector is a list of scalar (real number) used to represent a When the letters are in bold in a formula, it signifies that they're vectors, To represent...
Complex Exponential Euler
Mathematics - Complex Exponential (Euler's formula)

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions (the sinusoidal functions) as weighted sums of the exponential...
Data System Architecture
Number - Field

A field is a collection of “numbers” with the operator: +, -, , / Different fields are like different classes obeying to the same interface. Same as ?? A field is also known as abody: corps...
Number System (Classification|Type)

Numbers are classified according to how they are represented or according to the properties that they have. They have what's called a type. For systems for expressing numbers, see Numeral system. ...
Number System - Imaginary Number

An imaginary numberi is one solution to and is a part of the definition of a complex number It was invented because formulas sometimes required the manipulation of square roots of negative numbers....
Data System Architecture
Number, Numeric, Quantity

A number is a mathematical object used to: count, label, and measure. Example: When we are interested in the total number of each species of tortoise, or how many individuals there are per square...
Complex Plane
Python - Complex Numbers

complex numbers in Python Python uses 'j' for the imaginary unit, not 'i'. Type Complex Addition The real and imaginary parts of a complex number can be interpreted as x and y coordinates...
Card Puncher Data Processing
R - Complex Number

A vector containing complex number (for example: 1+2i)

Share this page:
Follow us:
Task Runner