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Mathematics-Online lexicon: Annotation to

Division of Complex Numbers

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The quotient $ z_1/z_2$ of two complex numbers

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

is

$\displaystyle \frac{x_1x_2+y_1y_2}{x_2^2+y_2^2} + \frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}\,\mathrm{i} = \frac{r_1}{r_2}\exp(\mathrm{i}(\varphi_1-\varphi_2))\, . $

In particular,

$\displaystyle \frac{1}{z} = \frac{1}{r^2} \bar z = \frac{1}{r} \exp(-\mathrm{i}\varphi) = \frac{x}{r^2} - \frac{y}{r^2}\,\mathrm{i} \,. $

A complex number's reciprocal can be constructed via reflection at the unit circle $ C$, as is illustrated in the following figure.

\includegraphics[height=6cm]{a_division_bild}

The complex conjugate $ w = \bar z$ is the intersection of the diagonals of the quadrilateral formed by the tangents at $ C$ passing through $ z$ and the perpendicular radii. The number $ z$ is then obtained by reflection at the real axis.

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  automatisch erstellt am 19.  8. 2013