Division of Complex Numbers

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Theorem

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers such that $z_2 \ne 0$.


The operation of division is performed on $z_1$ by $z_2$ as follows:

$\dfrac {z_1} {z_2} = \dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \dfrac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}$


Proof

\(\displaystyle \frac {z_1} {z_2}\) \(=\) \(\displaystyle z_1 \paren {z_2}^{-1}\) Definition of Complex Division
\(\displaystyle \) \(=\) \(\displaystyle \paren {a_1 + i b_1} \dfrac {a_2 - i b_2} {a_2^2 + b_2^2}\) Inverse for Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {a_1 a_2 + b_1 b_2} + i \paren {a_2 b_1 - a_1 b_2} } {a_2^2 + b_2^2}\) Definition of Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \frac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \frac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}\)

$\blacksquare$


Sources