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 z_{1} ⋅ z_{2}.

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

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

Product

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 multiplication is done by multiplying out the brackets considering the relation ** i^{2}= -1**.

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

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

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

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

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

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

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