H2 Maths Notes (JC 1-2): 4.1) Complex Numbers and Argand Diagrams

Before you revise\ Keep polar and exponential forms side by side—most Argand proofs demand switching between them quickly. Draw every locus and label arguments clearly.

Core Definitions

• Complex number \( z = x + iy \) with real part \( x \) and imaginary part \( y \).
• Modulus \( \lvert z \rvert = \sqrt{x^2 + y^2} \); argument \( \arg z \) is the angle from positive real axis.
• Polar form: \( z = r(\cos \theta + i \sin \theta) \) with \( r = \lvert z \rvert \space \theta = \arg z \).
• Exponential form: \( z = r e^{i\theta} \).

Algebra of Complex Numbers

• Addition/subtraction: combine real and imaginary parts.
• Multiplication: \( (a + ib)(c + id) = (ac - bd) + i(ad + bc) \).
• Division: multiply numerator and denominator by complex conjugate.
• Conjugate \( \bar{z} = x - iy \); satisfies \( z \bar{z} = \lvert z \rvert^2 \).

Example -- Division

Compute \( \frac{3 + 4i}{1 - 2i} \). \[ \frac{3 + 4i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{1 + 4} = \frac{3 + 6i + 4i + 8i^2}{5} = \frac{-5 + 10i}{5} = -1 + 2i. \]

Polar Form and De Moivre

• Convert using \( r = \sqrt{x^2 + y^2} \), \( \theta = \operatorname{atan2}(y, x) \).
• De Moivre: \( [r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta) \).
• Roots: \( z^n = w \) has roots \( r^{1/n} \left( \cos\frac{\theta + 2k\pi}{n} + i \sin\frac{\theta + 2k\pi}{n} \right) \).

Example -- Cube roots

Solve \( z^3 = 8(\cos \pi + i \sin \pi) \).

1. \( r = 8 \), \( \theta = \pi \).
2. Roots: \( 2 \left( \cos\frac{\pi + 2k\pi}{3} + i \sin\frac{\pi + 2k\pi}{3} \right) \) for \( k = 0, 1, 2 \).
3. Gives \( 2\bigl( \cos \tfrac{\pi}{3} + i \sin \tfrac{\pi}{3} \bigr), 2\bigl( \cos \pi + i \sin \pi \bigr), 2\bigl( \cos \tfrac{5\pi}{3} + i \sin \tfrac{5\pi}{3} \bigr) \).

Loci on the Argand Plane

• \( \lvert z - a \rvert = r \) represents a circle centre \( a \) radius \( r \).
• \( \arg(z - a) = \theta \) is a ray from \( a \) making angle \( \theta \).
• Linear loci: express \( z = x + iy \), substitute into condition, separate into real/imaginary parts to get Cartesian equation of line/curve.

Example -- Perpendicular bisector

Locus of points equidistant from \( 2 + i \) and \( -1 + 3i \):

\[ \lvert z - (2 + i) \rvert = \lvert z - (-1 + 3i) \rvert. \]

Let \( z = x + iy \). Squaring both sides yields \( (x - 2)^2 + (y - 1)^2 = (x + 1)^2 + (y - 3)^2 \) which simplifies to \( 6x - 4y + 2 = 0 \).

Inequalities and Regions

• \( \lvert z - a \rvert \leq r \) is a closed disk; \( < ) gives interior.
• Combined inequalities create lens or annulus regions; sketch boundaries first, then shade interior.
• Argument inequalities produce sector regions between rays.

Calculator Workflow

• Use GC complex mode to check arithmetic; ensure calculator is in radians for argument work.
• Store complex numbers in memory to avoid transcription errors during De Moivre calculations.
• For loci, use graphing mode with implicit equations to cross-check manual sketches.

Exam Watch Points

• Always state principal value of argument in the interval requested (usually \( -\pi < \arg z \leq \pi \)).
• When sketching loci, mark intercepts, centre, radius, and boundary style (solid/dashed).
• Provide exact trigonometric forms—avoid rounding angles unless final answer requires degrees.
• For De Moivre problems, list all distinct roots; avoid duplicates caused by missing the \( 2k\pi \) term.

Quick Revision Checklist

• [ ] Convert between Cartesian, polar, and exponential forms fluently.
• [ ] Perform conjugation, modulus, and division confidently.
• [ ] Sketch loci derived from modulus and argument conditions with accurate annotations.
• [ ] Apply De Moivre to powers and roots, checking for all distinct solutions.

Next steps: Continue with Topic 5.1 to reinforce differentiation techniques used throughout Section A.