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

$\blacksquare$


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