Three ways to write the same number: Cartesian z = a + bi, modulus-argument z = r(cos θ + i sin θ), often written r cis θ, and Euler z = reiθ.
r = |z| = √(a² + b²). The argument θ satisfies tan θ = b/a, but take the quadrant from the signs of a and b: a calculator's tan⁻¹ only gives angles between −π/2 and π/2.
Going back: a = r cos θ and b = r sin θ.
Multiplying multiplies the moduli and adds the arguments; dividing divides the moduli and subtracts the arguments. That is why products and powers are easier in polar or Euler form.
The conjugate of z = a + bi is z* = a − bi = re−iθ, and zz* = |z|² = a² + b².
De Moivre's theorem: (r cis θ)ⁿ = rⁿ cis(nθ).
Worked example
Write z = −1 + √3 i in modulus-argument and Euler form, then find z⁶.
r = √((−1)² + (√3)²) = √4 = 2.
a < 0 and b > 0, so z is in the second quadrant: θ = π − π/3 = 2π/3. (tan⁻¹(√3 ÷ −1) on a calculator gives −π/3, the wrong quadrant.)
z = 2 cis(2π/3) = 2e2πi/3.
By De Moivre, z⁶ = 2⁶ cis(6 × 2π/3) = 64 cis(4π) = 64.
A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
On the Complex Numbers questions page: 46 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for complex numbers?
In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Complex numbers guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.