Lösung 3.2:5d

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The argument of the quotient <math>i/(1+i)</math> is therefore
The argument of the quotient <math>i/(1+i)</math> is therefore
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{{Displayed math||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i)\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i)\,\textrm{.}</math>}}
We obtain the argument of <math>i</math> and <math>1+i</math> by drawing the numbers in the complex plane and using a little trigonometry.
We obtain the argument of <math>i</math> and <math>1+i</math> by drawing the numbers in the complex plane and using a little trigonometry.
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Hence, we obtain
Hence, we obtain
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{{Displayed math||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\,\textrm{.}</math>}}

Version vom 13:09, 10. Mär. 2009

When dividing two complex numbers, the numerator's magnitude is divided by the denominator's magnitude and the numerator's argument is subtracted from the numerator's argument.

The argument of the quotient \displaystyle i/(1+i) is therefore

\displaystyle \arg\frac{i}{1+i} = \arg i - \arg (1+i)\,\textrm{.}

We obtain the argument of \displaystyle i and \displaystyle 1+i by drawing the numbers in the complex plane and using a little trigonometry.

Hence, we obtain

\displaystyle \arg\frac{i}{1+i} = \arg i - \arg (1+i) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\,\textrm{.}
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.2:5d“