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Mathematics-Online course: Prepcourse Mathematics - Basics - Complex Numbers

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)
The product of $ 1+\mathrm{i}= \sqrt{2}\exp(\mathrm{i}\pi/4)$ and $ \sqrt{3}+3\mathrm{i}=2\sqrt{3}\exp(\mathrm{i}\pi/3)$ can be calculated by multiplication of the standard forms via

$\displaystyle (1+\mathrm{i})(\sqrt{3}+3\mathrm{i}) = \sqrt{3}-3 + \left(\sqrt{3}+3\right)\mathrm{i} $

or in polar coordinates via

$\displaystyle \sqrt{2}\exp(\mathrm{i}\pi/4)\cdot 2\sqrt{3}\exp(\mathrm{i}\pi/3) = 2\sqrt{6} \exp(7\mathrm{i}\pi/12)\, . $

For

$\displaystyle z = 3+\sqrt{3}\mathrm{i}= 2\sqrt{3}\exp(\mathrm{i}\pi/6) $

we have

$\displaystyle z^2 = 6 + 6\sqrt{3}\mathrm{i} = 12 \exp(\mathrm{i}\pi/3) \,. $

The above examples show that -generally speaking- multiplication in polar cooordinates is easier than multiplication in standard form. In particular the computation of powers is facilitated.

(Authors: Höllig/Kopf/Abele)
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  automatically generated 10/23/2009