Complex Numbers

A complex number $z$ is the combination of a real number $x$ and imaginary number $iy$.

$$ z = x + iy $$

• Magnitude

• Conjugate

Complex numbers share the following properties:

1. $|a| = \sqrt{a a^{*}}$
2. $(ab)^* = a^* b^*$
3. $a + a^{*} = 2 \cdot Re\{a\}$
4. $a - a^* = 2i \cdot Im\{a\}$

Polar Coordinates in $\mathbb{C}^2$

Any complex number in the Cartesian form $z = x + iy$ can be written into the polar form:

$$ z = re^{i\theta} \text{ where } \begin{cases} r = |z| = \sqrt{x^2 + y^2} \\ \theta = \arg(z)=\tan^{-1}(\frac{y}{x})\end{cases} $$

• Derivation

Roots of Unity

A complex number $z$ is a root of unity if it is any one of the solutions to the equation:

$$ z^n = 1 $$