Complex numbers and Argand diagrams (IB Math AI HL)

Key ideas

• Find the argument from a sketch on the Argand diagram: arctan(b/a) only gives the right angle in the first and fourth quadrants.
• Product: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂. Quotient: divide moduli and subtract arguments.
• Powers: (re^iθ)ⁿ = rⁿe^inθ, so z⁴ for z = 2e^iπ/6 is 16e^i2π/3.
• Sinusoids of the same frequency: a sin t + b sin(t + φ) is the imaginary part of (a + be^iφ)e^it. Add the complex amplitudes, then read off the modulus (amplitude) and argument (phase shift).
• In AC circuits, impedances in series add; in parallel, 1/Z = 1/Z₁ + 1/Z₂.

Worked example

Write 3 sin t + 4 cos t in the form R sin(t + α), using complex numbers.

1. 4 cos t = 4 sin(t + π/2), so the two waves have complex amplitudes 3 and 4e^iπ/2 = 4i.
2. Add them: 3 + 4i.
3. Modulus: R = |3 + 4i| = √(9 + 16) = 5.
4. Argument: α = arctan(4/3) ≈ 0.927 (first quadrant, so no adjustment).
5. So 3 sin t + 4 cos t = 5 sin(t + 0.927).

Answer: 5 sin(t + 0.927)

Exam-style HL AI questions on this topic

• Complex Numbers and Argand Diagrams questions — 31 questions with mark schemes, plus free worked examples
• All HL AI Number and Algebra questions by topic

Key formulas for complex numbers and Argand diagrams

Modulus of a complex number

|z₁ z₂| = |z₁| · |z₂|, |z₁/z₂| = (|z₁|)/(|z₂|)

Complex numbers — Cartesian

z = a + bi, |z| = √(a² + b²), arg z = arctan (b/a)

In the notation of the IB formula booklet. All HL AI formulas · see them used in worked examples

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Complex numbers on your calculator
• Try it: complex number calculator (free, shows the working)

Related in Number and Algebra

• Sequences, series, and the sum to infinity
• Exponential models for real-world decay
• Matrices — multiplication, inverse, and determinant

Revise Complex numbers and Argand diagrams for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Is mathematics discovered or invented?

Stretch yourself: original number theory problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S47 · Problem S06 · Problem S46 · Problem S05. All 22 →

Sibling topic in other IB Maths courses