🌀 Complex Numbers

Complex numbers extend real numbers and are essential for AMC 12. Master algebra, polar form, and De Moivre’s theorem.

🎯 Key Ideas

Definition: Complex numbers have form $z = a + bi$ where $i = \sqrt{-1}$ and $a, b \in \mathbb{R}$.

Polar Form: $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$ where $r = |z|$ and $\theta = \arg(z)$.

De Moivre’s Theorem: $(re^{i\theta})^n = r^n e^{in\theta}$ for integer $n$.

🔢 Complex Algebra

Basic Operations

  • Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$
  • Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Division: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$

Example: Simplify $(2 + 3i)(1 - 2i)$

Solution:

  • $(2 + 3i)(1 - 2i) = 2(1) + 2(-2i) + 3i(1) + 3i(-2i)$
  • $= 2 - 4i + 3i - 6i^2$
  • $= 2 - i - 6(-1)$ (since $i^2 = -1$)
  • $= 2 - i + 6 = 8 - i$

📐 Modulus and Argument

Modulus (Magnitude)

$$|z| = |a + bi| = \sqrt{a^2 + b^2}$$

Argument (Angle)

$$\arg(z) = \arctan\left(\frac{b}{a}\right) \text{ (with quadrant consideration)}$$

Example: Find modulus and argument of $z = 1 + i$

Solution:

  • Modulus: $|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$
  • Argument: $\arg(z) = \arctan(1) = \frac{\pi}{4}$ (first quadrant)

🌀 Polar Form

Conversion to Polar

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

where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$.

Conversion from Polar

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

Example: Convert $z = 2 + 2\sqrt{3}i$ to polar form

Solution:

  • Modulus: $r = \sqrt{2^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$
  • Argument: $\theta = \arctan\left(\frac{2\sqrt{3}}{2}\right) = \arctan(\sqrt{3}) = \frac{\pi}{3}$
  • Polar form: $z = 4e^{i\pi/3} = 4(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})$

⚡ De Moivre’s Theorem

Statement

For complex number $z = re^{i\theta}$ and integer $n$: $$z^n = (re^{i\theta})^n = r^n e^{in\theta} = r^n(\cos(n\theta) + i\sin(n\theta))$$

Applications

  • Powers: Calculate $z^n$ easily
  • Roots: Find $n$-th roots of complex numbers
  • Trig identities: Derive multiple angle formulas

Example: Calculate $(1 + i)^6$

Solution:

  1. Convert to polar: $1 + i = \sqrt{2}e^{i\pi/4}$
  2. Apply De Moivre’s: $(1 + i)^6 = (\sqrt{2})^6 e^{i6\pi/4} = 8e^{i3\pi/2}$
  3. Convert back: $8e^{i3\pi/2} = 8(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}) = 8(0 + i(-1)) = -8i$

🌟 Roots of Unity

Definition

The $n$-th roots of unity are solutions to $z^n = 1$.

Formula

$$\omega_k = e^{i2\pi k/n} = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right)$$

for $k = 0, 1, 2, \ldots, n-1$.

Properties

  • Sum: $\omega_0 + \omega_1 + \cdots + \omega_{n-1} = 0$
  • Product: $\omega_0 \omega_1 \cdots \omega_{n-1} = (-1)^{n-1}$
  • Geometric: Form regular $n$-gon on unit circle

Example: Find all 4th roots of unity

Solution:

  • $\omega_k = e^{i2\pi k/4} = e^{i\pi k/2}$ for $k = 0, 1, 2, 3$
  • $\omega_0 = e^{i0} = 1$
  • $\omega_1 = e^{i\pi/2} = i$
  • $\omega_2 = e^{i\pi} = -1$
  • $\omega_3 = e^{i3\pi/2} = -i$
  • Roots: $1, i, -1, -i$

🔄 Complex Conjugate

Definition

For $z = a + bi$, the conjugate is $\overline{z} = a - bi$.

Properties

  • $z + \overline{z} = 2a$ (twice real part)
  • $z - \overline{z} = 2bi$ (twice imaginary part)
  • $z \cdot \overline{z} = |z|^2 = a^2 + b^2$
  • $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
  • $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$

🎯 AMC-Style Worked Example

Problem: Find all complex numbers $z$ such that $z^3 = -8$.

Solution:

  1. Write in polar form: $-8 = 8e^{i\pi}$
  2. Find all cube roots: $z_k = 8^{1/3} e^{i(\pi + 2\pi k)/3}$ for $k = 0, 1, 2$
  3. Calculate each root:
    • $z_0 = 2e^{i\pi/3} = 2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}) = 2(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = 1 + i\sqrt{3}$
    • $z_1 = 2e^{i\pi} = 2(\cos\pi + i\sin\pi) = 2(-1 + i \cdot 0) = -2$
    • $z_2 = 2e^{i5\pi/3} = 2(\cos\frac{5\pi}{3} + i\sin\frac{5\pi}{3}) = 2(\frac{1}{2} - i\frac{\sqrt{3}}{2}) = 1 - i\sqrt{3}$

Answer: $z = -2, 1 + i\sqrt{3}, 1 - i\sqrt{3}$

🔍 Common Traps & Fixes

Trap: Wrong quadrant in argument

Fix: Use $\arctan$ but adjust by $\pi$ for quadrants II and III.

Trap: Forgetting all roots of unity

Fix: Remember there are $n$ distinct $n$-th roots, not just one.

Trap: Confusing polar and rectangular forms

Fix: $re^{i\theta} = r(\cos\theta + i\sin\theta)$, not $r + i\theta$.

Trap: Wrong sign in complex multiplication

Fix: Remember $i^2 = -1$ when expanding $(a + bi)(c + di)$.

📋 Quick Reference

Essential Formulas

  • $|a + bi| = \sqrt{a^2 + b^2}$
  • $\arg(a + bi) = \arctan\frac{b}{a}$ (with quadrant check)
  • $re^{i\theta} = r(\cos\theta + i\sin\theta)$
  • $(re^{i\theta})^n = r^n e^{in\theta}$

Roots of Unity

  • $n$-th roots: $\omega_k = e^{i2\pi k/n}$ for $k = 0, 1, \ldots, n-1$
  • Sum: $\sum_{k=0}^{n-1} \omega_k = 0$

Complex Conjugate

  • $\overline{a + bi} = a - bi$
  • $z \cdot \overline{z} = |z|^2$

Prev: Laws of Sines & Cosines
Next: Complex Plane Geometry
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