Common mistake: Wrong quadrant for the argument

Wrong quadrant for the argument — always sketch the point on the Argand diagram before writing arg(z)

Common mistake: Forgetting that arctan(b/a) gives the reference angle, not always the final argument

Forgetting that arctan(b/a) gives the reference angle, not always the final argument

Common mistake: Using degrees instead of radians for argument when the question asks for exact values

Using degrees instead of radians for argument when the question asks for exact values

Common mistake: Forgetting to apply de Moivre's theorem to the modulus as well as the argument: (r cis θ)ⁿ = rⁿ cis(nθ)

Forgetting to apply de Moivre's theorem to the modulus as well as the argument: (r cis θ)ⁿ = rⁿ cis(nθ)

VCE Units 1–4 · Specialist Maths

VCE Specialist Maths — Complex Numbers

Complex numbers extend the real number system to include √−1 = i. They appear in Specialist Units 1–2 and are used throughout Units 3–4 for solving polynomial equations, representing rotations and applying de Moivre's theorem. Precision with the argument (angle) is where most marks are lost.

Key Concepts & Formulas

Definition: i = √−1, so i² = −1
Cartesian form: z = a + bi where a = Re(z) and b = Im(z)
Modulus: |z| = √(a² + b²) — the distance from the origin on the Argand diagram
Argument: arg(z) = arctan(b/a), adjusted for the correct quadrant; always give in (−π, π]
Polar form: z = r(cos θ + i sin θ) = r cis θ where r = |z| and θ = arg(z)
Multiplication in polar form: r₁ cis θ₁ × r₂ cis θ₂ = r₁r₂ cis(θ₁ + θ₂)
Division in polar form: (r₁ cis θ₁) / (r₂ cis θ₂) = (r₁/r₂) cis(θ₁ − θ₂)
de Moivre's theorem: (r cis θ)ⁿ = rⁿ cis(nθ)
Conjugate: z̄ = a − bi; z · z̄ = |z|²; use conjugate to rationalise complex fractions
Roots of unity: the n nth roots of 1 are evenly spaced around the unit circle at angles 2πk/n

Practice Questions

5 questions

Attempt each question before reading the hint. These are styled to match VCE exam format.

Q1.Express z = −1 + i in polar form r cis θ.

2 marks

Show hint

The point is in the second quadrant — adjust the argument accordingly.

Q2.If z = 2 cis(π/3), find z³ using de Moivre's theorem and express in Cartesian form.

3 marks

Q3.Solve z² + 2z + 5 = 0 over ℂ.

2 marks

Show hint

Use the quadratic formula; √−16 = 4i.

Q4.Find |z| and arg(z) for z = (1 + i√3)/(1 − i).

3 marks

Show hint

Multiply numerator and denominator by the conjugate of the denominator.

Q5.Find all cube roots of z = 8 and plot them on an Argand diagram.

4 marks

Common Mistakes to Avoid

These are the errors that VCE students most frequently make in Complex Numbers — and that examiners are specifically watching for.

Wrong quadrant for the argument — always sketch the point on the Argand diagram before writing arg(z)
Forgetting that arctan(b/a) gives the reference angle, not always the final argument
Using degrees instead of radians for argument when the question asks for exact values
Forgetting to apply de Moivre's theorem to the modulus as well as the argument: (r cis θ)ⁿ = rⁿ cis(nθ)

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