Lösung 3.2:5d

Aus Online Mathematik Brückenkurs 2

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
K
Zeile 1: Zeile 1:
-
When dividing two complex numbers, the numerator's magnitude is divided by the denominator's absolute value and the numerator's argument is subtracted from the numerator's argument.
+
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
+
The argument of the quotient <math>i/(1+i)</math> is therefore
-
<math>\frac{i}{1+i}</math>
+
-
is therefore
+
 +
{{Displayed math||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i)\,\textrm{.}</math>}}
-
<math>\arg \frac{i}{1+i}=\arg i-\arg \left( 1+i \right)</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>\text{1}+i</math>
+
-
by drawing the numbers in the complex plane and using a little trigonometry:
+
-
 
+
[[Image:3_2_5_d.gif|center]]
[[Image:3_2_5_d.gif|center]]
- 
Hence, we obtain
Hence, we obtain
-
 
+
{{Displayed math||<math>\arg\frac{i}{1+i} = \arg i - \arg (1+i) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\,\textrm{.}</math>}}
-
<math>\arg \frac{i}{1+i}=\arg i-\arg \left( 1+i \right)=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}</math>
+

Version vom 12:43, 29. Okt. 2008

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“