RedCrab Math Tutorial

# Geometric addition and subtraction of complex numbers

## Addition in the Gaussian number plane

Complex numbers are added by adding the real parts and the imaginary parts separately. For the addition of the two complex numbers

z1 = a1 + b1i and z2 = a2 + b2i

applies

A complex number is uniquely defined by a pair of numbers (a, b), or geometrically by a point in the Gaussian plane. Each pair of numbers can be assigned a unique vector.

This vector can be represented in the Gaussian plane by a line or an arrow with the starting point 0 and the end point z.

The addition of two complex numbers z1 and z2 corresponds to the addition of the associated vectors in the Gaussian plane

Vectors are added by adding the components separately. The first component corresponds to the real part and the second to the imaginary part.

The following figure shows the complex numbers z1 = 3 + i and z2 = 1 + 2i and the visualized result of the complex addition.

## Subtraction in the Gaussian Plane

The geometric subtraction of two complex numbers z1 and z2 is similar. It is true that complex numbers are subtracted by subtracting the real parts and imaginary parts separately - as well as subtracting vectors.

The subtraction of the vectors z1 and z2 s carried out in practice such that to the vector of z1 is added the invers vector of z2 , that is the vector -z2

z1- z2 = z1+ (-z2)

The following figure shows the geometric subtraction:

The difference z1 - z2 can be represented by the vector from 0 to z1 - z2 or also by the vector from z2 to z1 dargestellt werden. Both vectors have the same length, direction and orientation. That is, both vectors are the same.

The vectors are also identical from 0 to z2 and from (z1 - z2) to z1.

Depending on the one or the other representation may be beneficial.

## Complex numbers

Basics of complex numbers
Computing with complex numbers
Multiplication of complex numbers
Conjugate complex numbers
Division of complex numbers