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Complex Numbers

Half algebra, half geometry. Choose the right form, and draw a locus before expanding it into x and y.

Questions in the bank
142
q/paper in 2025–26
1.00
Numeric answer
29%
Notes pages
7

Tier: Core

When you’ll see it

i, z and its conjugate, a modulus or an argument, a locus drawn by a condition on z, or a power of ω.

How this chapter is tested

Complex Numbers splits into algebra and geometry. The algebra pages — real and imaginary parts, equations in z and z̄, polar form, roots of unity, quadratics with complex roots — reward the right form: z = x + iy to compare parts, z = r(cos θ + i sin θ) for powers and rotations, and ω to reduce a power by its remainder.

The geometry pages read each condition as a shape. |z − a| = |z − b| is a line; |z − a| = k|z − b| with k ≠ 1 is a circle; a fixed argument of (z − a)/(z − b) is an arc. The greatest or least distance from a point to a circle is the centre distance plus or minus the radius. Draw the shape before any algebra.

Most questions end quickly once the form is chosen. Time is lost when a geometric condition is expanded into x and y and the result is a messy equation that a sketch would have read off. The chapter meets Quadratic Equations through complex roots, and Straight Lines and Conic Sections through its loci.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Algebra of complex numbers

    Real and imaginary parts, equations in z and z̄ solved by comparing parts, and |z₁z₂| = |z₁||z₂|.

  • Polar form and De Moivre

    Principal argument in (−π, π]; (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ; multiplying by cos α + i sin α turns about the origin by α.

  • Roots of unity

    1 + ω + ω² = 0 and ω³ = 1, so any power of ω reduces by its remainder on division by 3.

  • Quadratics with complex roots

    Sum and product as usual; high powers of the roots by a recurrence or by polar form.

  • Lines and circles

    |z − a| = |z − b| is the perpendicular bisector of a and b; a ratio of distances other than 1 is a circle.

  • Arcs and conic loci

    A fixed argument of (z − a)/(z − b) is an arc through a and b; |z − a| + |z − b| = 2k is an ellipse when 2k > |a − b|.

  • Regions and extreme distances

    On a circle with centre c₀ and radius r, |z − c| runs from ||c − c₀| − r| to |c − c₀| + r; check the extreme point lies in the region.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • The inverse tangent is not the argument

    tan⁻¹(y/x) lies in (−π/2, π/2). For a point with x < 0, add or subtract π to reach the right quadrant.

  • Conjugate roots without real coefficients

    Roots come in conjugate pairs only when every coefficient is real; z² − (3 + i)z + (2 + 2i) = 0 has roots 2 and 1 + i.

  • One arc, not the whole circle

    arg((z − a)/(z − b)) = θ is one arc; the other arc has argument θ − π. A point on the wrong arc is a distractor.

  • A bound that is never reached

    The triangle inequality gives a bound; it is the answer only if some z allowed by the question attains it.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Complex Numbers notes

Drill every Complex Numbers question

142 questions from the bank, across 7 subtopics.

Drill the 142 questions

Drill one subtopic at a time

The 7 subtopics, in teaching order.

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