
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 
z_{1} = a_{1} + b_{1}i and z_{2} = a_{2} + b_{2}i

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 z_{1} and z_{2} 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 z_{1} = 3 + i and z_{2} = 1 + 2i and the visualized result of the complex addition. 

Subtraction in the Gaussian Plane 
The geometric subtraction of two complex numbers z1 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} s 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} 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 z_{2 }and from (z_{1 } z_{2}) to z_{1}.
Depending on the one or the other representation may be beneficial. 
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