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
\(\ds \) \(=\) \(\ds -i\)

$\blacksquare$


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