Examples and practice
Complex numbers and the Argand plane
Back to lessonPractice section 1 · Questions 1–6
Explain why the new quadratic roots are not real · Relate real numbers to complex numbers · +4 more
- Explain why the new quadratic roots are not real
- Relate real numbers to complex numbers
- Complete the real and imaginary parts table
- Compare real and imaginary components to find two real unknowns
- Plot six complex operations and explain their geometry
- Practise three operations for each of three complex pairs
Practice section 2 · Questions 7–12
Find paired real unknowns by comparing components · Practise comparing components with real parameters · +4 more
Practice section 3 · Questions 13–18
Recall rationalisation before complex division · Divide exactly and identify real and imaginary parts · +4 more
- Recall rationalisation before complex division
- Divide exactly and identify real and imaginary parts
- Practise exact division and combine complex fractions
- Solve simultaneous equations without a graphing calculator
- Use a known complex root to recover and factor a cubic
- Verify a quartic root and find the other roots
Practice section 4 · Questions 19–24
Recover a quartic and deduce reciprocal roots · Reuse a cube after multiplying by i · +4 more
Practice section 5 · Questions 25–30
Sketch complex sums and prove a square · Apply the nine modulus and argument rules · +4 more
Practice section 6 · Questions 31–36
Real coefficients and a known complex root · A quadratic with complex coefficients · +4 more
Practice section 7 · Questions 37–42
Real parts, imaginary parts, moduli and arguments · Use both conjugate constraints to determine z · +4 more
- Real parts, imaginary parts, moduli and arguments
- Use both conjugate constraints to determine z
- Divide exactly using the conjugate of a surd denominator
- Solve coupled equations involving a conjugate
- Use conjugate roots and a reciprocal substitution
- Recover a quartic and reuse its real roots for intersections
Practice section 8 · Questions 43–48
Reuse a known square after multiplying by i · Reuse a square root before undoing a scale and translation · +4 more
- Reuse a known square after multiplying by i
- Reuse a square root before undoing a scale and translation
- Read cubic roots and rotate roots of a complex quadratic
- Distinguish a modulus square from a complex square
- Evaluate a complex quadratic discriminant exactly
- Solve coupled equations by three algebraic methods
Practice section 9 · Questions 49–54
Use a purely imaginary root to factor and transform a quartic · Use rotation to find area and real quadratic factors · +4 more
- Use a purely imaginary root to factor and transform a quartic
- Use rotation to find area and real quadratic factors
- Plot polar points and interpret a complex multiplier
- Find an exact quotient modulus without a calculator
- Solve a conjugate equation without assuming conjugate roots
- Factor a cubic and choose exact principal arguments