Complex Numbers
Complex numbers are examined through arithmetic, the Argand diagram, modulus–argument form, De Moivre's theorem and simple loci. Marks hinge on correct quadrant work and on treating the argument as an angle defined modulo 2π.
Arithmetic and conjugates
- i² = −1; add and subtract componentwise; multiply out fully and replace i² by −1.
- The conjugate of z = a + bi is z* = a − bi, and zz* = a² + b² = |z|².
- To divide, multiply numerator and denominator by the conjugate of the denominator.
Modulus–argument form
- |z| = √(a² + b²); arg z = arctan(b/a) adjusted for the quadrant.
- z = r(cos θ + i sin θ) = re^(iθ).
- Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.
- De Moivre: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
Roots and loci
- zⁿ = w has n distinct roots with modulus |w|^(1/n) and arguments separated by 2π/n.
- |z − a| = r is a circle with centre a and radius r.
- |z − a| = |z − b| is the perpendicular bisector of the segment ab.
- arg(z − a) = θ is a half-line starting at a, excluding the point a itself.
Worked example
Find (1 + i)⁸. Here |1 + i| = √2 and arg(1 + i) = π/4. By De Moivre, (1 + i)⁸ = (√2)⁸(cos 2π + i sin 2π) = 16(1 + 0i) = 16.
Traps
- Taking arctan(b/a) without adjusting the quadrant: arg(−1 + i) is 3π/4, not −π/4.
- Quoting arguments outside the principal range (−π, π] when the question demands it.
- Forgetting to add multiples of 2π when extracting all n roots of zⁿ = w.
- Losing the i when expanding products of two complex numbers.