1: Complex numbers review

Complex Numbers Review

Complex numbers extend the real number system to solve equations like $x^2 + 1 = 0$. A complex number $z$ is written as $z = x + iy$, where $x, y \in \mathbb{R}$ and $i = \sqrt{-1}$ (the imaginary unit). Here, $x = \text{Re}(z)$ is the real part, and $y = \text{Im}(z)$ is the imaginary part.

Key Representations

1. Cartesian Form: $z = x + iy$.
2. Polar Form: $z = r(\cos \theta + i \sin \theta)$, where:
   • Modulus: $r = |z| = \sqrt{x^2 + y^2}$ (distance from origin in the complex plane).
   • Argument: $\theta = \arg(z)$ (angle with the positive real axis, defined modulo $2\pi$). The principal value is usually in $(-\pi, \pi]$.
3. Exponential Form: $z = re^{i\theta}$ via Euler’s formula: $$e^{i\theta} = \cos \theta + i \sin \theta.$$

Operations

• Addition/Subtraction: Combine real and imaginary parts:
  $(a + ib) \pm (c + id) = (a \pm c) + i(b \pm d)$.
• Multiplication: Use $i^2 = -1$:
  $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$.
  In polar form: $|z_1 z_2| = |z_1||z_2|$, $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$.
• Division: Multiply numerator and denominator by the complex conjugate $\bar{z} = x - iy$: $$\frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{|z_2|^2}, \quad |z_1/z_2| = |z_1|/|z_2|, \quad \arg(z_1/z_2) = \arg(z_1) - \arg(z_2).$$

Essential Properties

• Modulus: $|z| \geq 0$, $|z| = 0 \iff z = 0$, $|z_1 z_2| = |z_1||z_2|$, $|z_1 + z_2| \leq |z_1| + |z_2|$ (triangle inequality).
• Conjugate: $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$, $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$, $z \bar{z} = |z|^2$.
• de Moivre’s Theorem: For integer $n$, $$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta).$$
• Roots: Solutions to $w^n = z = re^{i\theta}$ are: $$w_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right), \quad k = 0, 1, \dots, n-1.$$ These are equally spaced on a circle of radius $r^{1/n}$ in the complex plane.