Complex Division/Examples/(1-i) (1+i)^-1
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Example of Complex Division
- $\dfrac {1 - i} {1 + i} = -i$
Proof
\(\ds \dfrac {1 - i} {1 + i}\) | \(=\) | \(\ds \dfrac {\paren {1 - i} \paren {1 - i} } {\paren {1 + i} \paren {1 - i} }\) | multiplying top and bottom by $1 - i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - 2 i + i^2} {1^2 + 1^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {0 - 2 i} 2\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(vi)}$