# Online power of the complex number z graphically

## Square and cubic power of a complex number z

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

Scale:
Number of digits =
z1 = x1 + i y1 = + i
Square Power
Cubic Power
Reciprocal

Axes ranges

Re-min=
Re-max=
Im-min=
Im-max=

### 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).

### Powers of complex numbers

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

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

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

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

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

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

or with the binomial theorem

$zn =(x+iy)n = ∑ k = 0, k even n ( n k ) ( -1 ) k 2 x n - k y k +i ∑ k = 1, k odd n ( n k ) ( -1 ) k-1 2 x n - k y k$

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