Complex Division/Examples/(3 - 2i) (-1 + i)^-1
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Examples of Complex Division
- $\dfrac {3 - 2 i} {-1 + i} = \dfrac {-5 - i} 2$
Proof 1
| \(\ds \dfrac {3 - 2 i} {-1 + i}\) | \(=\) | \(\ds \dfrac {\paren {3 - 2 i} \paren {-1 - i} } {\paren {-1 + i} \paren {-1 - i} }\) | 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds -i\) |
$\blacksquare$
Proof 2
| \(\ds \dfrac {3 - 2 i} {-1 + i}\) | \(=\) | \(\ds a + b i\) | 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.
$\blacksquare$