Complex Division/Examples/(3+4i) (1+2i)^-1
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Example of Complex Division
- $\dfrac {3 + 4 i} {1 + 2 i} = \dfrac {11} 5 - \dfrac 2 5 i$
Proof
\(\ds \dfrac {3 + 4 i} {1 + 2 i}\) | \(=\) | \(\ds \dfrac {\paren {3 + 4 i} \paren {1 - 2 i} } {\paren {1 + 2 i} \paren {1 - 2 i} }\) | multiplying top and bottom by $1 - 2 i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 + 4 i - 6 i - 8 i^2} {1^2 + 2^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {11 - 2 i} 5\) | simplifying |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(v)}$