Complex Numbers.
Use polar geometry, exact complex arithmetic and roots to solve equations, describe loci and factor polynomials with real or complex coefficients.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Cartesian/polar forms
Describe the same complex number using coordinates or a distance and direction.
- 02
Modulus and argument
Use modulus and principal argument with explicit angle conventions.
- 03
Complex arithmetic in both forms
Choose Cartesian form for adding and polar form for multiplying or dividing.
- 04
Multiplication as rotation/dilation
Interpret multiplication as a combined turn and change of scale.
- 05
De Moivre's theorem
Raise polar complex numbers to integer powers and justify de Moivre’s rule.
- 06
Powers and roots
Find every nth root by accounting for all possible full turns.
- 07
Roots of unity
Use the regular geometry and algebra of roots of unity.
- 08
Complex loci and geometry
Translate distance and angle conditions into sets of points.
- 09
Factorisation over the complex numbers
Factor polynomials completely over the complex numbers.
- 10
Conjugate-root results for real polynomials
Use conjugation to explain paired non-real roots of real polynomials.
Topic review
Revisit the key ideas, then choose a practice paper.