Complex Division/Examples/(3 - 2i) (-1 + i)^-1/Proof 1
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Example of Complex Division
- $\dfrac {3 - 2 i} {-1 + i} = \dfrac {-5 - i} 2$
Proof
\(\ds \dfrac {3 - 2 i} {-1 + i}\) | \(=\) | \(\ds \dfrac {\paren {3 - 2 i} \paren {-1 - i} } {\paren {-1 + i} \paren {-1 - i} }\) | multiplying top and bottom by $-1 - i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-3 - i + 2 i^2} {1^2 + 1^2}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-5 - i} 2\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(k)}$