Complex Numbers

$z = a + bi$ where $a$ = real part, $b$ = imaginary part.

Operations

Addition, subtraction, multiplication (FOIL with $i^2 = -1$), division (multiply by conjugate).

Conjugate

$\bar{z} = a - bi$. $z\bar{z} = a^2 + b^2 = |z|^2$.

Modulus-Argument (Polar) Form

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

$r = |z| = \sqrt{a^2 + b^2}$, $\theta = \arg(z) = \arctan(\frac{b}{a})$.

De Moivre's Theorem

$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

$(3+2i)(1-i) = 3 - 3i + 2i - 2i^2 = 5 - i$.

$|3+4i| = 5$. $\arg(3+4i) = \arctan(\frac{4}{3}) \approx 53.1°$.