Complex Numbers — Topic Review | Year 12

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For nonzero $z = x + i y$ , $∣ z ∣ = \sqrt{x^{2} + y^{2}}$ and $z = ∣ z ∣ cis θ$ . Our principal-argument convention is $- π < Arg z \leq π$ .

Add in Cartesian form; multiply or divide conveniently in polar form. Multiply moduli and add arguments, or divide moduli and subtract arguments.

Multiplication by $ρ cis ϕ$ scales distances from the origin by $ρ$ and rotates anticlockwise through $ϕ$ .

De Moivre applies to integer powers. The nth roots of a nonzero number have equal modulus and angles separated by $2 π / n$ .

An nth root of unity is $cis (2 k π / n)$ . Their regular polygon links exact sums, products and geometry.

$∣ z - a ∣$ is the distance from a. A fixed distance gives a circle, equal distances give a perpendicular bisector, and a fixed argument gives a ray with its starting point excluded.

Complete factorisation over $ℂ$ uses linear factors and counts multiplicities. Real coefficients force non-real roots to occur in conjugate pairs.

Practice tests

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Complex Numbers — Paper 1
10 questions · 51 marks · 70 minutes · Scientific calculator
A balanced first paper on exact forms, arithmetic, powers, roots and simple geometry.
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Complex Numbers — Paper 2
10 questions · 55 marks · 70 minutes · Scientific calculator
A second balanced paper with signals, conjugates, roots, loci and reconstruction of polynomials.
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Complex Numbers — Paper 3
10 questions · 56 marks · 70 minutes · By hand
Reasoning and proof: inspect claims, derive identities and connect algebra with geometry.
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Complex Numbers — Paper 4
10 questions · 59 marks · 70 minutes · Scientific calculator
Roots and factorisations, from recognising a square to exact root geometry and real coefficient constraints.
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Complex Numbers — Paper 5
10 questions · 56 marks · 70 minutes · Scientific calculator
Applications and geometry: transformations, distance constraints, phase relations and coefficient models.
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Complex Numbers — Paper 6
10 questions · 68 marks · 75 minutes · Scientific calculator
A challenging synthesis paper with exact roots, geometric restrictions, proof and polynomial reconstruction.
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Back to Complex Numbers

Connect the ideas

Practice tests

Complex Numbers — Paper 1

Complex Numbers — Paper 2

Complex Numbers — Paper 3

Complex Numbers — Paper 4

Complex Numbers — Paper 5

Complex Numbers — Paper 6