Complex Numbers
The Complex Numbers domain tests algebraic manipulation of a + ib, modulus and argument, polar form, De Moivre's theorem, and the roots of unity.
Core rules
For z = a + ib, |z| = √(a² + b²) and z z̄ = |z|². The argument satisfies tan θ = b/a but must be placed in the correct quadrant; the principal argument lies in (−π, π]. In polar form z = r(cos θ + i sin θ), so z₁z₂ has modulus r₁r₂ and argument θ₁ + θ₂, while z₁/z₂ subtracts arguments. De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, and the nth roots are r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], k = 0, 1, …, n − 1.
Cube roots of unity
They are 1, ω = (−1 + i√3)/2 and ω², with 1 + ω + ω² = 0 and ω³ = 1. These identities collapse long powers into a single term.
Worked example
Evaluate (1 + i)⁸. Here r = √2 and θ = π/4, so (1 + i)⁸ = (√2)⁸(cos 2π + i sin 2π) = 16 × 1 = 16. Repeated multiplication is unnecessary once the polar form is written.
Traps
Writing the argument as tan⁻¹(b/a) without adjusting the quadrant; using De Moivre with a non-integer exponent and reporting one answer instead of n; assuming |z₁ + z₂| = |z₁| + |z₂|; forgetting that |z| is real and non-negative.