MATH1061 · Mathematics 1a
Complex Numbers
Complex numbers open the Linear Algebra stream (Weeks 1–2) and are pure procedure once you accept one new symbol: set i² = −1 and treat z = a + bi like an ordinary binomial — add, multiply, conjugate, and divide by realising the denominator. The payoff is geometric: every z is a point on the Argand plane with a modulus (length) and an argument (angle), and the polar/exponential form z = r(cosθ + i sinθ) = reiθ — via Euler's formula — turns multiplication into “multiply the moduli, add the arguments.” That makes powers (de Moivre's theorem) and n-th roots easy: the n roots sit equally spaced on a circle as the vertices of a regular n-gon.
What this chapter covers
- 01Complex arithmetic — i² = −1, add/multiply/conjugate/divide
- 02The Argand diagram — modulus and argument
- 03Polar & exponential form — Euler's formula re^{iθ}
- 04de Moivre's theorem — powers as a single angle multiplication
- 05Roots of unity and n-th roots — equally spaced on a circle
- 06The unit circle & trig connection
Worked example: complex division and a de Moivre power
- +1(a) Realise the denominator: multiply top and bottom by the conjugate 1 + i.
- +1(a) Expand: numerator (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 1 + 5i; denominator (1 − i)(1 + i) = 1 − i² = 2.
- +1(a) Divide: (1 + 5i)/2 = 1/2 + (5/2)i.
- +1(b) Polar form of 1 + i: modulus √2, argument π/4, so 1 + i = √2 (cos π/4 + i sin π/4).
- +1(b) Apply de Moivre: (1 + i)⁸ = (√2)⁸ (cos 8·π/4 + i sin 8·π/4) = 2⁴ (cos 2π + i sin 2π).
- +1(b) Evaluate: 2⁴ = 16 and cos 2π = 1, sin 2π = 0, so (1 + i)⁸ = 16.
Key terms
- Imaginary unit i
- The symbol defined by i² = −1. With it, every quadratic has roots and z = a + bi behaves like an ordinary binomial under addition and multiplication.
- Conjugate
- The conjugate of z = a + bi is z̄ = a − bi (flip the sign of the imaginary part). It satisfies z·z̄ = a² + b² = |z|², which is why multiplying by the conjugate realises (clears the i from) a denominator.
- Modulus and argument
- For z = a + bi the modulus |z| = √(a² + b²) is its distance from the origin on the Argand plane, and the argument arg z is the angle it makes with the positive real axis. Together they give the polar form.
- Euler's / exponential form
- z = r(cos θ + i sin θ) = r e^{iθ}, where r = |z| and θ = arg z. Euler's formula collapses the trig pair into an exponential, so multiplication becomes 'multiply moduli, add arguments'.
- de Moivre's theorem
- (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. In polar form, raising z to the power n raises the modulus to the nth power and multiplies the argument by n — the engine for powers and n-th roots.
Complex Numbers FAQ
How do I divide two complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. The denominator becomes the real number a² + b² (because (a + bi)(a − bi) = a² + b²), leaving a real divisor; then split into real and imaginary parts. This 'realising the denominator' is the standard first move for any z₁/z₂.
When should I switch to polar/exponential form?
Whenever you need powers, roots, or repeated multiplication. In polar form, multiplication multiplies the moduli and adds the arguments, so (re^{iθ})ⁿ = rⁿ e^{inθ} is immediate — far easier than expanding (a + bi)ⁿ by hand. For addition and subtraction, stay in Cartesian a + bi form instead.
How do I find all n-th roots of a complex number?
Write the number in polar form r e^{iθ}, then the n roots are r^(1/n) e^{i(θ + 2πk)/n} for k = 0, 1, …, n − 1. They all share the modulus r^(1/n) and are spaced 2π/n apart in angle, so they sit on a circle as the vertices of a regular n-gon. Don't forget the +2πk — leaving it out gives only one of the n roots.
Why does Euler's formula matter?
It links the exponential and trigonometric worlds: e^{iθ} = cos θ + i sin θ. That single identity is why polar multiplication adds angles, why de Moivre's theorem holds, and why complex exponentials are the clean way to handle rotations and roots. It also gives the famous e^{iπ} + 1 = 0 as the θ = π case.
Exam move
Keep two forms in play and switch deliberately: Cartesian a + bi for adding, subtracting and dividing (always realise the denominator with the conjugate), and polar/exponential reiθ for powers, roots and repeated multiplication. Memorise the polar coordinates of the common numbers (1, i, 1 + i, √3 + i) so you can convert instantly. For roots, write r1/n ei(θ+2πk)/n and list k = 0 … n − 1 — the n-gon picture is the check. Because Quiz A is no-calculator, keep exact values: surds, π/4-type angles, and exact moduli, never decimals.
