Complex numbers
Complex numbers tests arithmetic in the form a + bi, conjugates, modulus and argument, the Argand diagram, modulus–argument form, and the roots of polynomial equations with real coefficients. De Moivre's theorem and geometric interpretations appear in the more demanding questions.
Arithmetic and the conjugate
With i² = −1, addition and subtraction combine real and imaginary parts separately, while multiplication expands as a binomial. To divide, multiply numerator and denominator by the conjugate of the denominator, a − bi, which makes the denominator real and equal to a² + b².
Modulus, argument and the Argand diagram
The modulus is |z| = √(a² + b²) and the argument is the angle of z from the positive real axis, chosen in the correct quadrant. The modulus–argument form is z = r(cos θ + i sin θ) = re^(iθ), so multiplying multiplies moduli and adds arguments, and dividing divides moduli and subtracts arguments. On the Argand diagram, |z − z₀| = r represents a circle centred at z₀.
Polynomial equations
If a polynomial has real coefficients, any non-real root occurs with its complex conjugate. Quadratic equations with negative discriminants give conjugate pairs. De Moivre's theorem, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, generates multiple roots and trigonometric identities.
Worked example
Express (1 + i)/(2 − i) in the form a + bi.
Multiply by the conjugate: (1 + i)(2 + i)/((2 − i)(2 + i)) = (2 + i + 2i + i²)/5 = (1 + 3i)/5 = 0.2 + 0.6i.
Traps that cost marks
- Writing i² = 1 instead of −1 during expansion.
- Taking the argument in the wrong quadrant when the real part is negative.
- Forgetting to multiply both numerator and denominator by the conjugate.
- Assuming a real-coefficient polynomial has only real roots.
- Losing the second, conjugate root when a question asks for all roots.