The power operation with the complex number and the reciprocal is graphically presented. By moving the vector endpoint the complex number can be changed.

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↹#.000

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

Square Power

Cubic Power

Reciprocal

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 power of a complex number in the n-th potency is done by using the formula of Moivre.

${z}^{n}={r}^{n}(\mathrm{cos}n\phi +i\mathrm{sin}n\phi )={r}^{n}{e}^{in\phi}$

$\text{with}\phantom{\rule{1em}{0ex}}r=\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

$\text{and}\phantom{\rule{1em}{0ex}}\phi =\mathrm{atan}\frac{y}{x}$

$\text{and}\phantom{\rule{1em}{0ex}}{i}^{0}=1\text{,}\phantom{\rule{1em}{0ex}}{i}^{1}=i\text{,}\phantom{\rule{1em}{0ex}}{i}^{2}=-1\text{,}\phantom{\rule{1em}{0ex}}{i}^{3}=-i\text{...}\phantom{\rule{1em}{0ex}}$

$\text{and}\phantom{\rule{1em}{0ex}}n\in \mathbb{N}$

or with the binomial theorem

$${z}^{n}={\left(x+iy\right)}^{n}=\sum _{k=0\text{, k even}}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){\left(-1\right)}^{\frac{k}{2}}{x}^{n-k}{y}^{k}+i\sum _{k=1\text{, k odd}}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){\left(-1\right)}^{\frac{k-1}{2}}{x}^{n-k}{y}^{k}$$

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Here is a list of of further useful calculators:

Complex numbers calculator Complex numbers graphical Addition complex numbers graphical Multiplication complex numbers graphical Division complex numbers graphical Power complex numbers graphical Complex functions