# Online graphical multiplication of complex numbers

## Multiplication of the complex numbers z1 and z2

The operation with the complex numbers is graphically presented. By moving the vector endpoints the complex numbers can be changed. The red arrow shows the result of the multiplication z1 ⋅ z2.

↹#.000
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z1 = x1 + i y1 = + i
z2 = x2 + i y2 = + i
Product

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).

### Multiplication of complex numbers

The multiplication is done by multiplying out the brackets considering the relation i2= -1.

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

${z}_{1}\cdot {z}_{2}=\left({x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\right)\cdot \left({x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\right)$ $={x}_{1}\cdot {x}_{2}-{y}_{1}\cdot {y}_{2}+i\phantom{\rule{0.3em}{0ex}}\left({x}_{1}\cdot {y}_{2}+{y}_{1}\cdot {x}_{2}\right)$

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

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

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

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

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