Lösung 3.2:4d

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For magnitudes of quotients, we have the arithmetical rule
For magnitudes of quotients, we have the arithmetical rule
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{{Displayed math||<math>\left|\frac{z}{w}\right| = \frac{|z|}{|w|}\,\textrm{.}</math>}}
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<math>\left| \frac{z}{w} \right|=\frac{\left| z \right|}{\left| w \right|}</math>
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We can therefore take the magnitude of the numerator and denominator separately and then divide the magnitudes by each other,
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{{Displayed math||<math>\begin{align}
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We can therefore take the magnitude of the numerator and denominator separately and then divide the magnitudes by each other:
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\left|\frac{3-4i}{3+2i}\right|
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&= \frac{|3-4i|}{|3+2i|}
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= \frac{\sqrt{3^2+(-4)^2}}{\sqrt{3^2+2^2}}
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<math>\begin{align}
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= \frac{\sqrt{9+16}}{\sqrt{9+4}}
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& \left| \frac{3-4i}{3+2i} \right|=\frac{\left| 3-4i \right|}{\left| 3+2i \right|}=\frac{\sqrt{3^{2}+\left( -4 \right)^{2}}}{\sqrt{3^{2}+2^{2}}}=\frac{\sqrt{9+16}}{\sqrt{9+4}} \\
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= \frac{\sqrt{25}}{\sqrt{13}}
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& =\frac{\sqrt{25}}{\sqrt{13}}=\frac{5}{\sqrt{13}} \\
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= \frac{5}{\sqrt{13}}\,\textrm{.}
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\end{align}</math>
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\end{align}</math>}}

Version vom 12:16, 29. Okt. 2008

For magnitudes of quotients, we have the arithmetical rule

\displaystyle \left|\frac{z}{w}\right| = \frac{|z|}{|w|}\,\textrm{.}

We can therefore take the magnitude of the numerator and denominator separately and then divide the magnitudes by each other,

\displaystyle \begin{align}

\left|\frac{3-4i}{3+2i}\right| &= \frac{|3-4i|}{|3+2i|} = \frac{\sqrt{3^2+(-4)^2}}{\sqrt{3^2+2^2}} = \frac{\sqrt{9+16}}{\sqrt{9+4}} = \frac{\sqrt{25}}{\sqrt{13}} = \frac{5}{\sqrt{13}}\,\textrm{.} \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.2:4d“