Geometric addition and subtraction of complex numbers in the Gaussian number plane with examples

## 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:

$$z_1=a_1+b_1i$$   und   $$z_2=a_2+b_2i$$   applies   $$z_1 +z_2=(a_1+a_2)+(b_1+b_2)i$$

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

$$\begin{bmatrix}a_1 \cr b_1\end{bmatrix} + \begin{bmatrix}a_2 \cr b_2\end{bmatrix} = \begin{bmatrix}a_1 + a_2 \cr b_1 + b_2\end{bmatrix}$$

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 $$z_1$$ and $$z_2$$ 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 $$z_1$$ and $$z_2$$ carried out in practice such that to the vector of $$z_1$$ is added the invers vector of $$z_2$$ that is the vector $$-z_2$$.

$$z_1- z_2 = z_1+ (-z_2)$$

The following figure shows the geometric subtraction

The difference $$z_1 - z_2$$ can be represented by the vector from $$0$$ to $$z_1 - z_2$$ or also by the vector from $$z_2$$ to $$z_1$$ Both vectors have the same length, direction and orientation. That is, both vectors are the same.

The vectors are also identical from $$0$$ to $$z_2$$ and from $$z_1 - z_2$$ to $$z_1$$.

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