Imaginary Part of Complex Product

Theorem
Let $z_1$ and $z_2$ be complex numbers.

Then:
 * $\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$

Proof
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.

By definition of complex multiplication:


 * $z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$

Then