Complex numbers is one of the more self-contained topics in AA HL — it doesn't lean heavily on earlier calculus or algebra content, which makes it a good topic to strengthen independently. Most of the difficulty isn't conceptual; it's about staying organised across the three different forms a complex number can take and knowing when to switch between them.
The three forms, and when to use each
| Form | Written as | Best for |
|---|---|---|
| Cartesian | a + bi | Addition, subtraction |
| Polar | r(cos θ + i sin θ) | Multiplication, division, visualising on Argand diagram |
| Euler | re^(iθ) | Powers and roots (De Moivre's theorem), compact working |
A large share of marks lost in this topic come from doing a calculation in the wrong form — trying to add two numbers in polar form, or multiply two numbers in Cartesian form, both of which are technically possible but far more error-prone than switching form first.
De Moivre's theorem: the single most tested idea
De Moivre's theorem states that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) — in Euler form, this is simply (re^(iθ))ⁿ = rⁿe^(inθ). This is what makes powers and roots of complex numbers manageable: instead of expanding a binomial repeatedly, you just scale the modulus and multiply the argument.
When finding the nth roots of a complex number, there are always exactly n distinct roots, evenly spaced around a circle on the Argand diagram (each separated by 2π/n). A frequent error is finding only one root and stopping — the mark scheme expects all n, so always check the question's mark allocation to see how many are expected.
The Argand diagram: more than just a picture
The Argand diagram (a complex number plotted as a point, with the real part on the horizontal axis and the imaginary part on the vertical axis) isn't just a visual aid — many questions expect you to read information directly from it, such as identifying the modulus (distance from origin) and argument (angle from positive real axis) geometrically, or describing a locus of points satisfying a given condition (like |z − a| = r, a circle of radius r centred at a).
Mistake: sign errors with the argument
The argument of a complex number must be given in the correct range (usually −π < θ ≤ π unless stated otherwise), and its sign depends on which quadrant the number lies in. A common error is computing arctan(b/a) without checking which quadrant a + bi actually falls in — arctan alone doesn't distinguish between a point in the first quadrant and one diagonally opposite in the third. Always sketch the point roughly on an Argand diagram first to sanity-check the sign and range of your angle.
How to practise effectively
- Get fast at converting between all three forms in both directions — this is the single skill that unblocks everything else in the topic.
- Practise full root-finding questions, not just De Moivre's theorem applied to a single power — make sure you're finding and listing every root.
- Sketch every Argand diagram question, even roughly — it catches sign and quadrant errors before they cost you a mark.
Practise complex numbers with fresh questions
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✦ Try it freeFrequently asked questions
Is complex numbers only in AA HL, or also AA SL?
Complex numbers is an HL-only topic in the AA course — it doesn't appear in AA SL, and it isn't part of the AI syllabus at either level.
Do I need to memorise De Moivre's theorem, or is it in the formula booklet?
Check the current formula booklet for your exam session — some core relationships are provided, but you should still be comfortable deriving and applying the theorem quickly without relying on looking it up mid-question.