Complex Number Division

$\frac{z_1}{z_2}=\frac{a+bi}{c+di}$

$=\frac{(a+bi)(c-di)}{c^2+d^2}$ $\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}=(c+di)(c-di)=c^2+d^2$

Theorem

$z_1\in C$ $z_2\in C$ $\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_2}{|z_2{|}^2}$

Modulus and Conjugate Distributive

$|\frac{z}{w}|=\frac{|z|}{|w|}$ $\overline{\frac{z}{w}}=\frac{\overline{z}}{\overline{w}}$