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Mathematics-Online lexicon:

Multiplication of Complex Numbers

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The product $ z_1z_2$ of two complex numbers

$\displaystyle z_k = x_k + \mathrm{i} y_k = r_k \exp(\mathrm{i}\varphi_k) $

is

$\displaystyle (x_1x_2-y_1y_2) + (x_1y_2+x_2y_1)\mathrm{i} = r_1r_2 \exp(\mathrm{i}(\varphi_1+\varphi_2))\, . $

\includegraphics[width=.5\linewidth]{a_multiplikation_bild}

Geometrically speaking, multiplication with a complex number $ z=r e^{\mathrm{i}\varphi}$ corresponds to a dilation with scale factor $ r$ and a rotation by $ \varphi$.

(Authors: Höllig/Kopf/Abele)
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  automatically generated 6/11/2007