Express the complex number $z = -1 + \sqrt{3}i$ in the polar form $r\text{cis}\theta$, where $-\pi < \theta \le \pi$.

1. Calculate the modulus $r = |z| = \sqrt{(-1)^2 + (\sqrt{3})^2}$. 2. Evaluate the sum to get $r = \sqrt{1 + 3} = 2$. 3. Determine the reference angle $\alpha = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$. 4. Identify the quadrant on the Argand diagram; since the real part is negative and imaginary part is positive, it lies in Quadrant 2. 5. Compute the principal argument $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$. 6. State the final polar form: $\mathbf{z = 2\text{cis}\left(\frac{2\pi}{3}\right)}$.

Solve the equation $z^3 = 8i$, giving your answers in the Euler form $re^{i\theta}$ where $-\pi < \theta \le \pi$.

1. Express $8i$ in Euler form. Its modulus is $8$ and it lies on the positive imaginary axis, so $8i = 8e^{i\frac{\pi}{2}}$. 2. Set up the general equation for roots by adding $2k\pi i$ to the exponent to account for full rotations: $z^3 = 8e^{i\left(\frac{\pi}{2} + 2k\pi\right)}$. 3. Take the cube root of both sides to find the general root: $z = 8^{1/3} e^{i\left(\frac{\pi/2 + 2k\pi}{3}\right)} = 2e^{i\left(\frac{\pi}{6} + \frac{2k\pi}{3}\right)}$. 4. Substitute $k = 0$ to find the first root: $z_1 = \mathbf{2e^{i\frac{\pi}{6}}}$. 5. Substitute $k = 1$ to find the second root: $z_2 = 2e^{i\left(\frac{\pi}{6} + \frac{4\pi}{6}\right)} = \mathbf{2e^{i\frac{5\pi}{6}}}$. 6. Substitute $k = -1$ to ensure the principal argument domain $[-\pi, \pi]$ for the third root: $z_3 = 2e^{i\left(\frac{\pi}{6} - \frac{4\pi}{6}\right)} = \mathbf{2e^{-i\frac{\pi}{2}}}$.

Given $z_1 = 4\operatorname{cis}\left(\frac{\pi}{3}\right)$ and $z_2 = 2\operatorname{cis}\left(\frac{\pi}{6}\right)$ , find the product $z_1 z_2$ , leaving your answer in the simplest Cartesian form $a + bi$ .

Multiply the moduli and add the arguments: $z_1 z_2 = (4 \times 2)\operatorname{cis}\left(\frac{\pi}{3} + \frac{\pi}{6}\right)$ (M1) Simplify: $z_1 z_2 = 8\operatorname{cis}\left(\frac{\pi}{2}\right)$ A1 Use $\cos\frac{\pi}{2} = 0,\ \sin\frac{\pi}{2} = 1$ : $z_1 z_2 = 8i$ A1

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IB Maths AA HL · Unit 1: Number and Algebra

IB Maths AA HL Complex Numbers Questions

Q: How many IB Maths AA HL complex numbers questions are there?

There are 46 exam-style complex numbers questions in the AA HL question bank (Paper 1: 41 · Paper 2: 5), graded 10 easy, 20 medium, 16 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is complex numbers on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 41 · Paper 2: 5. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Q: Where can I get the mark schemes?

Open the AA HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

Updated October 2026

Exam-style IB Maths AA HL complex numbers questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

46 questions
Paper 1: 41
Paper 2: 5
10 easy
20 medium
16 hard
3 worked examples

Practise Complex Numbers questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

z = a + bi, r·e^iθ, r·(cosθ + i·sinθ). HL AA expects fluent conversion between all three and De Moivre's theorem. Complex numbers — Cartesian, polar, and Euler forms overview →

(cosθ + i·sinθ)ⁿ = cos(nθ) + i·sin(nθ). Use to find nth roots of unity, expand cos(3θ), or prove trig identities. De Moivre's theorem and roots of unity overview →

What's examined in AA HL complex numbers

The question bank covers these complex numbers question types (number of questions in brackets):

De Moivre's Theorem & Roots (20)
Complex Arithmetic & Forms (17)
Loci & Polynomials (9)

Key formulas

Complex numbers — polar (Cartesian → polar): $z = a + b i = r cis θ = r e^{i θ}$
Modulus & argument: $r = | z | = \sqrt{a^{2} + b^{2}}, θ = \arg z$

In the same notation as the IB formula booklet. All AA HL formulas →

Where students lose marks on complex numbers

Examiner's noteComplex numbers and polynomials: conjugate roots, all the roots, general solutions

Where marks are lost: May 2024: many knew the conjugate root was needed but did not use all the roots in the sum and product relations. November 2025: manipulation of a number in cis form was unconvincing and general solutions of a trigonometric equation were often wrong; on Paper 2, candidates had to discard an inappropriate root.

Examiners flagged this in 2 exam series (May 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Complex Numbers worked examples

Worked example 1: Finding the modulus and argument · easy

Express the complex number $z = - 1 + \sqrt{3} i$ in the polar form $r cis θ$ , where $- π < θ \leq π$ .

Solution

6 steps · 1 to fill in

Examiner tip: A common mistake is simply plugging

\arctan (- \sqrt{3} / 1)

into the calculator and blindly accepting the negative fourth-quadrant angle without checking the coordinates on an Argand diagram sketch.

Worked example 2: Applying De Moivre's theorem to powers · medium

Use De Moivre's Theorem to find the exact value of $(1 - i)^{8}$ .

Solution

6 steps

Examiner tip: Students often try to expand high powers of complex binomials using Pascal's triangle, which is incredibly time-consuming and error-prone compared to immediately converting to polar form and utilizing De Moivre's theorem.

Worked example 3: Finding complex roots of equations · hard

Solve the equation $z^{3} = 8 i$ , giving your answers in the Euler form $r e^{i θ}$ where $- π < θ \leq π$ .

Solution

6 steps

Examiner tip: Forgetting to add

2 k π

to the argument before dividing by

n

is the most frequent error, resulting in students only finding one single complex root instead of finding all

n

distinct roots symmetrically spaced around the origin.

Try these IB Maths AA HL complex numbers questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 2 marks · Paper 1

Consider the complex number $z = 3 - 4 i$ . Find its complex conjugate $z^{*}$ , and calculate its exact modulus $| z |$ .

Attempt it and see the mark scheme →

Question 2 · medium · 3 marks · Paper 1

Given $z_{1} = 4 cis (\frac{π}{3})$ and $z_{2} = 2 cis (\frac{π}{6})$ , find the product $z_{1} z_{2}$ , leaving your answer in the simplest Cartesian form $a + b i$ .

Full worked solution Free, no sign-in

Multiply the moduli and add the arguments: $z_{1} z_{2} = (4 \times 2) cis (\frac{π}{3} + \frac{π}{6})$ (M1)
Simplify: $z_{1} z_{2} = 8 cis (\frac{π}{2})$ A1
Use $\cos \frac{π}{2} = 0, \sin \frac{π}{2} = 1$ : $z_{1} z_{2} = 8 i$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

Find the three distinct complex cube roots of $w = 27 cis (\frac{3 π}{4})$ . Give your answers in the Euler form $r e^{i θ}$ , where $- π < θ \leq π$ .

Attempt it and see the mark scheme →

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All 46 complex numbers questions with mark schemes →

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Revise complex numbers for the exam

FAQ

How many IB Maths AA HL complex numbers questions are there?

There are 46 exam-style complex numbers questions in the AA HL question bank (Paper 1: 41 · Paper 2: 5), graded 10 easy, 20 medium, 16 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is complex numbers on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 41 · Paper 2: 5. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Complex numbers — Cartesian form · Modulus, argument & polar form · De Moivre & Euler form (roots of unity).

IB Maths AA HL Complex Numbers Questions

Try it first

What you need to know

What's examined in AA HL complex numbers

Key formulas

Where students lose marks on complex numbers

Complex Numbers worked examples

Worked example 1: Finding the modulus and argument · easy

Solution

Worked example 2: Applying De Moivre's theorem to powers · medium

Solution

Worked example 3: Finding complex roots of equations · hard

Solution

Try these IB Maths AA HL complex numbers questions

Question 1 · easy · 2 marks · Paper 1

Question 2 · medium · 3 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise complex numbers for the exam

FAQ

What else is free here

Teaching this topic?

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