Complex numbers.
Extend real-number algebra to complex numbers, represent them geometrically, and solve and factorise quadratics with non-real roots.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Why the imaginary unit is needed
See why real numbers are not enough for every quadratic, and introduce the imaginary unit without changing ordinary algebra.
- 02
Real and imaginary parts
Read and compare the two real components of a complex number in Cartesian form.
- 03
Complex-number arithmetic
Add, subtract and multiply complex numbers using familiar algebra, then reduce powers of i.
- 04
Conjugates and division
Use conjugates to turn complex division into a calculation with a real denominator.
- 05
Quadratic equations with non-real roots
Solve quadratics that have no real roots and check the resulting complex solutions.
- 06
Conjugate roots of real quadratics
Explain why a real quadratic's non-real roots occur in conjugate pairs, and reconstruct real quadratics from their roots.
- 07
Argand diagrams
Represent complex numbers as points on an Argand diagram and connect component calculations to geometry.
- 08
Modulus
Calculate the modulus as a geometric distance and use its relationship with conjugation, products and quotients.
- 09
Vector interpretation of addition and conjugation
Use complex addition, subtraction and conjugation as precise geometric operations.
- 10
Linear factorisation over the complex numbers
Factor real quadratics into complex linear factors and verify the factors by multiplication.
Topic review
Revisit the key ideas, then choose a practice paper.