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Lösung 3.1:2b

Aus Online Mathematik Brückenkurs 2

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Zeile 3: Zeile 3:
We multiply the top and bottom of each fraction by the complex conjugate of its denominator,
We multiply the top and bottom of each fraction by the complex conjugate of its denominator,
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\frac{3i}{4-6i}-\frac{1+i}{3+2i}
\frac{3i}{4-6i}-\frac{1+i}{3+2i}
&= \frac{3i(4+6i)}{(4-6i)(4+6i)}-\frac{(1+i)(3-2i)}{(3+2i)(3-2i)}\\[5pt]
&= \frac{3i(4+6i)}{(4-6i)(4+6i)}-\frac{(1+i)(3-2i)}{(3+2i)(3-2i)}\\[5pt]
Zeile 14: Zeile 14:
Then, we multiply the top and bottom of the last fraction by 4, so as to give make both fractions have the same denominator, and after that we subtract the numerators,
Then, we multiply the top and bottom of the last fraction by 4, so as to give make both fractions have the same denominator, and after that we subtract the numerators,
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\frac{-18+12i}{52}-\frac{(5 +i)\cdot 4}{13\cdot 4}&=\frac{-18+12i}{52}-\frac{20+4i}{52}\\[5pt]
\frac{-18+12i}{52}-\frac{(5 +i)\cdot 4}{13\cdot 4}&=\frac{-18+12i}{52}-\frac{20+4i}{52}\\[5pt]
&= \frac{-18+12i-20-4i}{52}\\[5pt]
&= \frac{-18+12i-20-4i}{52}\\[5pt]

Version vom 13:05, 10. Mär. 2009

The two terms do not have the same denominator, so it is not possible to subtract them directly. It is perhaps simplest to calculate each quotient individually and then subtract the result.

We multiply the top and bottom of each fraction by the complex conjugate of its denominator,

3i4−6i−1+i3+2i=3i(4+6i)(4−6i)(4+6i)−(1+i)(3−2i)(3+2i)(3−2i)=42−(6i)23i4+3i6i−32−(2i)213−12i+i3−i2i=16+3612i+18i2−9+43−2i+3i−2i2=52−18+12i−133+(−2+3)i+2=52−18+12i−135+i.

Then, we multiply the top and bottom of the last fraction by 4, so as to give make both fractions have the same denominator, and after that we subtract the numerators,

52−18+12i−134(5+i)4=52−18+12i−5220+4i=52−18+12i−20−4i=52−38+8i=−2619+213i.
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.1:2b“