The operation with the complex numbers is graphically presented. The resulting sum is given by the red vector. By moving the vector endpoints the complex numbers can be changed. The dotted lines are the parallel moved vectors.

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z_{1} = x_{1} + *i* y_{1}

z_{2} = x_{2} + *i* y_{2}

Sum / Difference

+ *i*

+ *i*

Amount

Polar co-ordinates

Angle

Gauss plane:

The complex numbers are two-dimensional and can be used as vectors in the Gaussian plane of numbers represent. On the horizontal axis (Re) of the real part and on the vertical axis is applied (Im) of the imaginary part of the complex number. Similar to vectors complex numbers can be expressed in Cartesian coordinates (x, y) or polar coordinates (r, φ).

The addition and subtraction of complex numbers corresponding to the addition and subtraction of the position vectors. That the real and imaginary components are added or subtracted.

With
z_{1} = x_{1} + i y_{1}
and
z_{2} = x_{2} + i y_{2}
is

z_{1} + z_{2} = x_{1} + x_{2} + i ( y_{1} + y_{2} )

z_{1} - z_{2} = x_{1} - x_{2} + i ( y_{1} - y_{2} )

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