Complex Numbers

$z = a + bi$ where $a$ is the real part and $b$ is the imaginary part.

• Addition: $(a + bi) + (c + di) = (a+c) + (b+d)i$
• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Conjugate: $\overline{a + bi} = a - bi$
• Magnitude: $|a + bi| = \sqrt{a^2 + b^2}$

Polar Form and Euler's Formula

Euler's formula: $e^{i x} = \cos x + i \sin x$

$(a, b) = (r \cos \theta, r \sin \theta)$ where $r = \sqrt{a^2 + b^2}$ and $\theta = \arctan(\frac{b}{a})$.

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

• Addition: $z_1 + z_2 = r_1 e^{i \theta_1} + r_2 e^{i \theta_2}$
• Multiplication: $z_1 z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)}$
• Power: $z^n = r^n e^{i n \theta}$

N-th Roots of Unity

The $n$-th roots of unity satisfy $z^n = 1$. These roots are evenly spaced on the unit circle in the complex plane, with each root corresponding to a point at an angle of $\frac{2\pi k}{n}$ radians from the positive real axis for $k = 0, 1, \ldots, n-1$.

$$\omega_n = (1, \frac{2\pi}{n}) = e^{i \frac{2\pi}{n}} = \cos(\frac{2\pi}{n}) + i \sin(\frac{2\pi}{n}) \\ and\\ \omega_n^k = (1, \frac{2\pi k}{n}) = e^{i \frac{2\pi k}{n}} = \cos(\frac{2\pi k}{n}) + i \sin(\frac{2\pi k}{n}) \\$$

The principal $n$-th root of unity is $\omega_n = e^{i \frac{2\pi}{n}}$, and the other $n$-th roots of unity are given by $\omega_n^k$.

Properties:

• $\omega_n^n = 1$
• $\omega_n^{n/2} = -1$
• Square: $(\omega_n^{k})^2 = \omega_{n/2}^k$
• Conjugate: $\overline{\omega_n^k} = \omega_n^{-k}$
• Conjugate Symmetry: $\omega_n^k = -\omega_n^{k+\frac{n}{2}}$.
• Inverse: $\omega_n^{-k} = \omega_n^{n-k}$ and $\omega_n^k \cdot \omega_n^{-k} = 1$.
• Sum: $\sum\limits_{k=0}^{n-1} \omega_n^k = 0$ for $n > 1$.
  • $\sum\limits_{k=0}^{n-1} \omega_n^k = \frac{\omega_n^n - 1}{\omega_n - 1} = \frac{1 - 1}{\omega_n - 1} = 0$
• Product: $\prod\limits_{k=0}^{n-1} \omega_n^k = (-1)^{n-1}$.
  • $\prod\limits_{k=0}^{n-1} \omega_n^k = \omega_n^{\sum\limits_{k=0}^{n-1} k} = \omega_n^{\frac{n(n-1)}{2}} = e^{i \pi (n-1)} = (-1)^{n-1}$