Convert the complex number $z = -1 + i\sqrt{3}$ to Euler form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \le \pi$.

1. Calculate the modulus $r$: $r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$. 2. Calculate the reference angle: $\alpha = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$. 3. Identify the correct quadrant: Since the real part is negative and the imaginary part is positive, $z$ lies in the second quadrant. 4. Find the principal argument $\theta$: $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$. 5. The Euler form is $z = 2e^{i\frac{2\pi}{3}}$ .

Given the complex number $z = 2e^{i\frac{\pi}{6}}$, use De Moivre's Theorem to find the exact value of $z^5$ in Cartesian form.

1. Apply De Moivre's Theorem to the modulus and argument: $z^5 = (2)^5 e^{i(5 \times \frac{\pi}{6})}$. 2. Evaluate the new modulus and argument: $z^5 = 32e^{i\frac{5\pi}{6}}$. 3. Convert the Euler form into trigonometric form: $32\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)$. 4. Substitute the exact values from the unit circle for the second quadrant: $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ and $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$. 5. Expand the brackets: $32\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) =$ $-16\sqrt{3} + 16i$ .

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IB Maths AI HL · Unit 1: Number and Algebra

IB Maths AI HL Complex Numbers and Argand Diagrams Questions

Q: How many IB Maths AI HL complex numbers and Argand diagrams questions are there?

There are 31 exam-style complex numbers and Argand diagrams questions in the AI HL question bank (Paper 1: 31), graded 10 easy, 15 medium, 6 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is complex numbers and Argand diagrams on Paper 1 or Paper 2?

In the question bank these questions are set as Paper 1 questions.

Q: Where can I get the mark schemes?

Open the AI 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 AI HL complex numbers and Argand diagrams 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.

31 questions
Paper 1: 31
10 easy
15 medium
6 hard
3 worked examples

Practise Complex Numbers and Argand Diagrams questions →

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What you need to know

z = a + bi in Cartesian, |z|·(cosθ + i·sinθ) in polar. HL AI uses complex numbers for AC circuit analysis and signal modelling. Complex numbers and Argand diagrams overview →

What's examined in AI HL complex numbers and Argand diagrams

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

Complex Number Algebra (22)
Argand Diagrams and Loci (5)
De Moivre's Theorem and Roots (4)

Key formulas

Modulus of a complex number: $| z_{1} z_{2} | = | z_{1} | \cdot | z_{2} |, | \frac{z_{1}}{z_{2}} | = \frac{| z_{1} |}{| z_{2} |}$
Complex numbers — Cartesian: $z = a + b i, | z | = \sqrt{a^{2} + b^{2}}, \arg z = \arctan (\frac{b}{a})$

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

Complex Numbers and Argand Diagrams worked examples

Worked example 1: Converting Cartesian to Euler Form · easy

Convert the complex number $z = - 1 + i \sqrt{3}$ to Euler form $r e^{i θ}$ , where $r > 0$ and $- π < θ \leq π$ .

Solution

5 steps

Examiner tip: Always sketch a quick Argand diagram to visually confirm which quadrant your complex number lies in, preventing argument errors.

Worked example 2: Solving Quadratics with Complex Roots · medium

Find the complex roots of the quadratic equation $x^{2} + 4 x + 13 = 0$ . Give your answers in Cartesian form $a + b i$ .

Solution

5 steps

Examiner tip: Notice that for polynomials with real coefficients, complex roots will always appear in conjugate pairs.

Worked example 3: Applying De Moivre's Theorem · hard

Given the complex number $z = 2 e^{i \frac{π}{6}}$ , use De Moivre's Theorem to find the exact value of $z^{5}$ in Cartesian form.

Solution

5 steps

Examiner tip: Operations involving high integer powers or roots are significantly faster and less prone to algebraic error when performed in Euler or polar form rather than Cartesian form.

Try these IB Maths AI HL complex numbers and Argand diagrams 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 · 4 marks · Paper 1

Given the complex numbers

z_{1} = 2 + 3 i

and

z_{2} = 1 - i

, find

z_{1} z_{2}

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

Let $z = 3 + i$ .

(a) Write down the complex conjugate of $z$ . [1 mark]

(b) Given that $w = \frac{z^{*}}{1 + i}$ , express $w$ in the form $p + q i$ , where $p, q \in R$ . [3 marks]

Full worked solution Free, no sign-in

(a)

$z^{*} = 3 - i$ A1
(b)

Substitute $z^{*} = 3 - i$ into the expression for $w$ :
$w = \frac{3 - i}{1 + i}$

Multiply numerator and denominator by the conjugate of the denominator:
$w = \frac{3 - i}{1 + i} \times \frac{1 - i}{1 - i}$ M1

Expand numerator and denominator and simplify:
$w = \frac{3 - 3 i - i + i^{2}}{1^{2} - i^{2}} = \frac{3 - 4 i - 1}{1 - (- 1)} = \frac{2 - 4 i}{2}$ A1

Simplify to the form $p + q i$ :
$w = 1 - 2 i$ A1

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

Question 3 · hard · 7 marks · Paper 1

Solve the following system of simultaneous linear equations for the complex variables $z$ and $w$ : $\begin{aligned} (2 + i) z + w & = 5 i \\ z - i w & = 3 \end{aligned}$ Give your final answers for $z$ and $w$ in Cartesian form.

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FAQ

How many IB Maths AI HL complex numbers and Argand diagrams questions are there?

There are 31 exam-style complex numbers and Argand diagrams questions in the AI HL question bank (Paper 1: 31), graded 10 easy, 15 medium, 6 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is complex numbers and Argand diagrams on Paper 1 or Paper 2?

In the question bank these questions are set as Paper 1 questions.

Where can I get the mark schemes?

Open the AI 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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Free ready-to-teach lessons (slides and teacher notes): Complex numbers — Cartesian form · Modulus, argument & polar form.

IB Maths AI HL Complex Numbers and Argand Diagrams Questions

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What you need to know

What's examined in AI HL complex numbers and Argand diagrams

Key formulas

Complex Numbers and Argand Diagrams worked examples

Worked example 1: Converting Cartesian to Euler Form · easy

Solution

Worked example 2: Solving Quadratics with Complex Roots · medium

Solution

Worked example 3: Applying De Moivre's Theorem · hard

Solution

Try these IB Maths AI HL complex numbers and Argand diagrams questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise complex numbers and Argand diagrams for the exam

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