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.
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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>
Documentation / Reference
* https://www.math.wisc.edu/~angenent/Free-Lecture-Notes/freecomplexnumbers.pdf