# Division of complex numbers graphically

## Division of the complex numbers z1 and z2

The ratio of the complex numbers is graphically presented. By moving the vector endpoints the numbers can be changed. The red arrow shows the result of the division z1 / z2.

Scale:
Number of digits =
z1 = x1 + i y1 = + i
z2 = x2 + i y2 = + i
Quotient

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).

### Division of complex numbers

The division is carried out by the fraction is expanded with the complex conjugate of the denominator.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\phantom{\rule{1.5em}{0ex}}\text{is}$

$\frac{{z}_{1}}{{z}_{2}}=\frac{{x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}}{{x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}}$ $=\frac{{x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}}{{x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}}\frac{{x}_{2}-i\phantom{\rule{0.3em}{0ex}}{y}_{2}}{{x}_{2}-i\phantom{\rule{0.3em}{0ex}}{y}_{2}}$ $=\frac{{x}_{1}{x}_{2}+{y}_{1}{y}_{2}}{{x}_{2}^{2}+{y}_{2}^{2}}+i\phantom{\rule{0.3em}{0ex}}\frac{{x}_{2}{y}_{1}-{x}_{1}{y}_{2}}{{x}_{2}^{2}+{y}_{2}^{2}}$

The division of complex numbers can also be done in trigonometric or exponential form.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={r}_{1}\left(\mathrm{cos}{\phi }_{1}+i\mathrm{sin}{\phi }_{1}\right)={r}_{1}{e}^{i{\phi }_{1}}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={r}_{2}\left(\mathrm{cos}{\phi }_{2}+i\mathrm{sin}{\phi }_{2}\right)={r}_{2}{e}^{i{\phi }_{2}}\phantom{\rule{1.5em}{0ex}}\text{ist}\phantom{\rule{1.5em}{0ex}}$

$\frac{{z}_{1}}{{z}_{2}}$ $=\frac{{r}_{1}}{{r}_{2}}\left(\mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right)+i\mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right)\right)$ $=\frac{{r}_{1}}{{r}_{2}}{e}^{i\left({\phi }_{1}-{\phi }_{2}\right)}$

$\text{with}\phantom{\rule{1em}{0ex}}r=|z|=\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\phi =\mathrm{atan}\frac{y}{x}$

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