Complex numbers (HL heavy)

Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0. A complex number takes the form z = a + bi, where a is the real part Re(z), b is the imaginary part Im(z), and i is the imaginary unit satisfying i² = -1.

Key Definitions:

• Complex conjugate: If z = a + bi, then z̄ = a - bi (reflects across real axis)
• Modulus: |z| = √(a² + b²) (distance from origin)
• Argument: arg(z) = θ where tan(θ) = b/a (angle from positive real axis, -π < θ ≤ π)

Forms of Complex Numbers:

1. Cartesian form: z = a + bi
2. Polar form: z = r(cos θ + i sin θ) where r = |z|
3. Euler form: z = re^(iθ) using Euler's formula: e^(iθ) = cos θ + i sin θ

Essential Operations:

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• Division: Multiply by conjugate of denominator
• De Moivre's Theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

Properties:

• |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂)
• |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg(z₁) - arg(z₂)
• z·z̄ = |z|² (always real and non-negative)

Mnemonic: CAPER for forms: Cartesian, Argand (geometric), Polar, Euler, Roots

Complex numbers aren't just mathematical abstractions—they're fundamental to modern technology. Think of i as a 90° rotation operator: multiplying by i rotates a vector counterclockwise by 90° in the complex plane.

Real-World Applications:

1. Electrical Engineering: AC circuits use complex impedance Z = R + iX where R is resistance and X is reactance. The imaginary component represents phase shift—voltage and current being "out of sync." Engineers use |Z| to find total opposition to current and arg(Z) for phase angle, critical in power transmission efficiency.

2. Signal Processing: Audio compression (MP3s) and image processing (JPEGs) use the Fast Fourier Transform, which decomposes signals into complex exponentials e^(iωt). Each frequency component has magnitude (volume/brightness) and phase (timing).

3. Quantum Mechanics: Particle wavefunctions are complex-valued: ψ(x) = A·e^(ikx). The modulus squared |ψ|² gives probability density—where you'll find the particle. Phase differences create interference patterns.

Analogy: Imagine complex numbers as GPS coordinates with direction. The real part is East-West position, imaginary part is North-South. The modulus is distance from home, and argument is compass bearing. Multiplying complex numbers is like combining movements: "walk 5 km Northeast, then 3 km at 60°" naturally adds angles and multiplies distances—exactly what arg and |z| properties describe!

Argand Diagram Visualization: Plot z = 3 + 4i as point (3,4). It's 5 units from origin (modulus) at angle arctan(4/3) ≈ 53.13° (argument). The conjugate 3 - 4i mirrors below the x-axis, like a reflection in water.

Example 1: Express z = (2 + i)/(1 - 3i) in form a + bi.

Solution: Multiply by conjugate of denominator: z = [(2 + i)(1 + 3i)] / [(1 - 3i)(1 + 3i)]

Numerator: 2(1) + 2(3i) + i(1) + i(3i) = 2 + 6i + i + 3i² = 2 + 7i - 3 = -1 + 7i Denominator: 1² - (3i)² = 1 - 9i² = 1 + 9 = 10

Therefore: z = (-1 + 7i)/10 = -1/10 + 7i/10

Examiner note: Always simplify i² = -1 immediately. Show conjugate multiplication explicitly for method marks.

Example 2: Find all solutions to z³ = -8.

Solution: Express -8 in polar form: -8 = 8(cos π + i sin π) = 8e^(iπ)

Using De Moivre's: z = 8^(1/3) · [cos((π + 2πk)/3) + i sin((π + 2πk)/3)] for k = 0, 1, 2

z = 2·[cos(π/3 + 2πk/3) + i sin(π/3 + 2πk/3)]

k = 0: z₁ = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3 k = 1: z₂ = 2(cos π + i sin π) = -2 k = 2: z₃ = 2(cos(5π/3) + i sin(5π/3)) = 2(1/2 - i√3/2) = 1 - i√3

Examiner note: Roots are equally spaced at 120° intervals. Always find ALL roots specified.

Example 3: Prove |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality).

Proof uses: |z|² = z·z̄ and (z₁ + z₂)·(z̄₁ + z̄₂) expansion, applying Cauchy-Schwarz—see full geometric interpretation in mark schemes.