Division

There's no direct way to divide by $c + di$, so instead you multiply top and bottom by its conjugate $c - di$ — this turns the denominator into a real number:

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

Example: $\dfrac{1+2i}{3-i}$. Multiply by $\dfrac{3+i}{3+i}$:

$\frac{(1+2i)(3+i)}{(3-i)(3+i)} = \frac{3 + i + 6i + 2i^2}{9+1} = \frac{1+7i}{10} = \tfrac{1}{10} + \tfrac{7}{10}i$