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IB DP Maths AI · Topic 1 Number and algebra · AHL 1.12, 1.13 · ~6 min read

Complex numbers

HL complex numbers for Maths AI: Cartesian form and the Argand diagram, modulus and argument, polar and Euler forms, multiplying and dividing, solving quadratics with complex roots, and adding sinusoidal waves (phasors) with complex numbers.

i² = −1Cartesian form a + biArgand diagramModulus & argumentPolar & Euler formMultiplying & dividingPowersComplex roots of quadratics
RSby Riyaz Sir
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • A complex number z = a + bi has real part a and imaginary part b, where i2 = −1. Plot it as the point (a, b) on an Argand diagram.
  • Modulus r = ǀzǀ = √(a2 + b2); argument θ = arg z, the angle from the positive real axis (in radians, usually −π < θ ≤ π).
  • Three forms: z = a + bi = r(cos θ + i sin θ) = reiθ. Multiply by multiplying moduli and adding arguments; divide by dividing moduli and subtracting arguments.
  • Why AI cares: waves with the same frequency (AC voltages, sound) add like complex numbers. This turns a messy trig sum into one arrow addition.

By the end you can

  • Calculate with complex numbers in Cartesian form, including conjugates and division. HL
  • Represent complex numbers on an Argand diagram and find modulus and argument. HL
  • Convert between Cartesian, polar and Euler forms. HL
  • Multiply, divide and find powers in polar and Euler form, and interpret them geometrically. HL
  • Solve quadratic equations with complex roots. HL
  • Use complex numbers to add sinusoidal functions with the same frequency. HL
Syllabus noteFrom first exams in 2029, complex numbers leave the AI HL syllabus. If you sit exams up to November 2028, you still need everything in this note.

1Cartesian form HL

  • Adding: add real parts and imaginary parts. Multiplying: expand the brackets and replace i2 with −1.
  • Conjugate: z* = a − bi. z z* = a2 + b2 is always real, which is how you divide.
Worked example

z = 3 + 4i and w = 1 − 2i. Find zw and zw.

  1. zw = 3 − 6i + 4i − 8i2 = 11 − 2i M1A1
  2. zw = (3 + 4i)(1 + 2i)(1 − 2i)(1 + 2i) = −5 + 10i5 M1
  3. = −1 + 2i A1
Worked example

Solve z2 + 4z + 13 = 0.

  1. z = −4 ± √(16 − 52)2 = −4 ± √(−36)2 M1
  2. z = −2 ± 3i A1
Exam tipQuadratics with real coefficients have complex roots in conjugate pairs: if 2 + 5i is a root, so is 2 − 5i.

2Modulus, argument and the Argand diagram HL

ReImz = 3 + 4iw = 1 − 2iθr = 5r = |z| = 5θ = arg z = 0.927z = 5e^(0.927i)= 5 cis 0.927
The modulus is the length of the arrow; the argument is its angle from the positive real axis.
ǀzǀ = √(a2 + b2), arg z = θ with tan θ = ba FORMULA BOOKLET
check the quadrant
z = r(cos θ + i sin θ) = reiθ = r cis θ FORMULA BOOKLET
polar and Euler forms
Worked example

Write z = 3 + 4i in polar and Euler form.

  1. r = √(9 + 16) = 5 A1
  2. θ = arctan(43) = 0.927 (first quadrant) A1
  3. z = 5(cos 0.927 + i sin 0.927) = 5e0.927i A1
Common mistakeTrusting arctan for every quadrant. For z = −1 − i, arctan(1) = π/4, but z is in the third quadrant, so arg z = −3π4. Sketch the point first.
Worked example

Write 2eiπ/3 in Cartesian form.

  1. 2(cos π3 + i sin π3) M1
  2. = 1 + √3 i A1

3Multiplying, dividing and powers HL

r1eiθ1 × r2eiθ2 = r1r2ei(θ1 + θ_{2)}
multiply moduli, add arguments
  • Geometry: multiplying by i rotates a point 90° anticlockwise about the origin. Multiplying by reiθ stretches by r and rotates by θ.
  • Powers: (reiθ)n = rneinθ (De Moivre's theorem).
Worked example

Find (1 + i)4.

  1. 1 + i = √2 eiπ/4 M1
  2. (√2)4 eiπ = 4eiπ M1
  3. = −4 A1
GDC (calculator) methodMost GDCs work in complex mode: set it to a + bi or reθi, and use the abs( ) and angle( ) functions. Use radians.

4Adding sinusoidal waves HL

53e^(iπ/3)7e^(0.380i)5 sin(ωt)+ 3 sin(ωt + π/3)= 7 sin(ωt + 0.380)add the arrows(phasors)Waves of the same frequency add like complex numbers: amplitudes and phases come out together.
Represent each wave by an arrow (a phasor), add the arrows, then read off the length and angle.
  • Method: write a sin(ωt + α) as the complex number aeiα. Add the complex numbers. The modulus is the new amplitude and the argument is the new phase shift.
Worked example

Two AC voltages are V1 = 5 sin(ωt) and V2 = 3 sin(ωt + π3). Write V1 + V2 in the form R sin(ωt + φ).

  1. 5 + 3eiπ/3 = 5 + 1.5 + 2.598i = 6.5 + 2.598i M1A1
  2. R = ǀ6.5 + 2.598iǀ = 7 A1
  3. φ = arg = 0.380, so V = 7 sin(ωt + 0.380) A1
Common mistakeAdding amplitudes directly: 5 + 3 = 8. The waves are out of phase, so their peaks don't line up and the combined amplitude is smaller (here 7).
Linking questionThe argument uses the radian measure and trig values from Radians, vectors and matrix transformations, and rotation by multiplying complex numbers matches rotation matrices. Maths AA covers complex numbers with more algebra and proof: see Complex numbers.
Idea for your IA / explorationModel the sound of two tuning forks or an AC circuit by adding sinusoids with complex numbers, and compare the predicted amplitude with a measurement from a phone sound app or an oscilloscope simulation.

Test yourself — cover the answers first

Quick check

  1. HL Simplify i3.
  2. HL Find (2 + i)(2 − i).
  3. HL Find ǀ−5 + 12iǀ.
  4. HL What is the argument of z = 4i?
  5. HL Write 3eiπ in Cartesian form.

Exam-style

  1. HL z = 2eiπ/6 and w = 3eiπ/3. Find zw in Euler and Cartesian form, and describe the effect of multiplying by w.
  2. HL Write 4 cos(t) + 4 sin(t) in the form R sin(t + φ).
Show answers
  1. −i
  2. 5
  3. 13
  4. π/2
  5. −3
  6. zw = 6eiπ/2 = 6i
    multiplying by w stretches by a factor of 3 and rotates by π3 anticlockwise
  7. 4 sin t ↔ 4, and 4 cos t = 4 sin(t + π/2) ↔ 4i
    4 + 4i = 4√2 eiπ/4
    = 4√2 sin(t + π4)
Practise this chapter
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Next chapterMatrices and eigenvalues →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.