Complex Numbers Review

A complex number is $z = a + bi$ where $a, b \in \mathbb{R}$ and $i^2 = -1$. We call $a = \mathrm{Re}(z)$ the real part and $b = \mathrm{Im}(z)$ the imaginary part.

Arithmetic: $(a + bi) + (c + di) = (a + c) + (b + d)i$ and $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

Proposition 1.1 (Properties of Complex Arithmetic). For all $z, w \in \mathbb{C}$:

1. $z + w = w + z$ and $zw = wz$ (commutativity)
2. $(z + w) + u = z + (w + u)$ and $(zw)u = z(wu)$ (associativity)
3. $z(w + u) = zw + zu$ (distributivity)
4. There exist additive identity $0$ and multiplicative identity $1$.
5. Every $z \neq 0$ has a multiplicative inverse $\frac{1}{z} = \frac{\bar{z}}{|z|^2}$.

Remark. The complex field $\mathbb{C}$ cannot be ordered: there is no total ordering on $\mathbb{C}$ Compatible with the field operations. In particular, $i^2 = -1$ precludes any such ordering.

Definition. The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$.

Proposition 1.2. For all $z, w \in \mathbb{C}$:

1. $\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{zw} = \bar{z}\bar{w}$
2. $z\bar{z} = |z|^2$
3. $z + \bar{z} = 2\,\mathrm{Re}(z)$ and $z - \bar{z} = 2i\,\mathrm{Im}(z)$
4. $\bar{\bar{z}} = z$

Definition. The modulus (or absolute value) of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.

Proposition 1.3 (Modulus Properties). For all $z, w \in \mathbb{C}$:

1. $|z| \geq 0$ with equality iff $z = 0$
2. $|zw| = |z||w|$
3. $|z + w| \leq |z| + |w|$ (triangle inequality)
4. $\bigl||z| - |w|\bigr| \leq |z - w|$ (reverse triangle inequality)

Proof of (3). $|z + w|^2 = (z + w)(\bar{z} + \bar{w}) = |z|^2 + z\bar{w} + \bar{z}w + |w|^2 = |z|^2 + 2\,\mathrm{Re}(z\bar{w}) + |w|^2 \leq |z|^2 + 2|z||w| + |w|^2 = (|z| + |w|)^2$. The inequality follows from $\mathrm{Re}(z\bar{w}) \leq |z\bar{w}| = |z||w|$. $\blacksquare$

Every non-zero complex number can be written in polar form:

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

Where $r = |z| = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \arg(z)$ is the argument.

Definition. The principal argument $\mathrm{Arg}(z)$ is the unique $\theta \in (-\pi, \pi]$ Such that $z = |z|e^{i\theta}$. The argument $\arg(z)$ is multi-valued: $\arg(z) = \mathrm{Arg}(z) + 2\pi k$ for $k \in \mathbb{Z}$.

Proposition 1.4. If $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$Then $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$ and $z_1/z_2 = (r_1/r_2)\, e^{i(\theta_1 - \theta_2)}$.

Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$.

De Moivre’s theorem: $(e^{i\theta})^n = e^{in\theta}$So

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

Proposition 1.5. De Moivre’s theorem holds for all integers $n$Including negative values.

Proof. For $n \geq 0$It follows by induction from the multiplication law $e^{i\alpha}e^{i\beta} = e^{i(\alpha + \beta)}$. For $n \lt 0$Write $n = -m$ with $m \gt 0$: $(\cos\theta + i\sin\theta)^n = \frac{1}{(\cos\theta + i\sin\theta)^m} = \frac{1}{\cos(m\theta) + i\sin(m\theta)} = \cos(-m\theta) + i\sin(-m\theta) = \cos(n\theta) + i\sin(n\theta)$. $\blacksquare$

Example. Compute $(1 + i)^{20}$.

$1 + i = \sqrt{2}\,e^{i\pi/4}$So $(1 + i)^{20} = (\sqrt{2})^{20}\, e^{20\pi i/4} = 2^{10}\, e^{5\pi i} = 1024\,e^{\pi i} = -1024$.

Definition. An $n$-th root of $w \in \mathbb{C}$ is a complex number $z$ such that $z^n = w$.

Proposition 1.6. Every non-zero $w \in \mathbb{C}$ has exactly $n$ distinct $n$-th roots. If $w = \rho\, e^{i\phi}$Then

$z_k = \rho^{1/n}\, e^{i(\phi + 2\pi k)/n}, \quad k = 0, 1, \ldots, n - 1$

Where $\rho^{1/n} \gt 0$ is the positive real $n$-th root of $\rho$.

Proof. If $z^n = w$Write $z = r\,e^{i\theta}$. Then $r^n e^{in\theta} = \rho\, e^{i\phi}$ So $r = \rho^{1/n}$ and $n\theta = \phi + 2\pi k$. For $k = 0, 1, \ldots, n-1$ these give distinct Values of $\theta$; for $k \geq n$ they repeat. $\blacksquare$

Remark. The $n$-th roots of $w$ lie equally spaced on a circle of radius $\rho^{1/n}$Forming a Regular $n$-gon.

The $n$-th roots of unity are the solutions of $z^n = 1$:

$z_k = e^{2\pi i k / n}, \quad k = 0, 1, \ldots, n - 1$

They form a regular $n$-gon on the unit circle in the complex plane.

Proposition 1.7. If $\omega = e^{2\pi i/n}$ is a primitive $n$-th root of unity, then $\sum_{k=0}^{n-1} \omega^k = 0$ and $\sum_{k=0}^{n-1} \omega^{jk} = 0$ for any $j$ not divisible by $n$.

Proof. The sum $\sum_{k=0}^{n-1} \omega^k = \frac{1 - \omega^n}{1 - \omega} = \frac{1 - 1}{1 - \omega} = 0$ Provided $\omega \neq 1$. For $j$ not divisible by $n$, $\omega^j$ is a non-trivial root of unity, So the same argument applies. $\blacksquare$