Complex Division/Examples/(3 - 2i) (-1 + i)^-1/Proof 2
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Example of Complex Division
- $\dfrac {3 - 2 i} {-1 + i} = \dfrac {-5 - i} 2$
Proof
\(\ds \dfrac {3 - 2 i} {-1 + i}\) | \(=\) | \(\ds a + b i\) | where $a, b \in \R$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 - 2 i\) | \(=\) | \(\ds \paren {-1 + i} \paren {a + b i}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds -a + b i + a i + b i^2\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds -a - b + \paren {a - b} i\) | simplifying |
Then:
\(\ds -a - b\) | \(=\) | \(\ds 3\) | ||||||||||||
\(\ds a - b\) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds -\frac 5 2\) | |||||||||||
\(\ds b\) | \(=\) | \(\ds -\frac 1 2\) |
Hence the result.
$\blacksquare$
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: $1 \ \text{(k)}$