1 · IB Maths AA
De Moivre's theorem
Set z = r(cos θ + i sin θ) and watch its powers zⁿ spiral round the Argand diagram, or switch mode to see the n nth roots of a number form a regular polygon.
Key equation
Drag the large point on the diagram to change r and θ. Pause to see the full pattern at once.
Try this
- Set r = 1.00, θ = 45° and n = 8. Predict where z⁸ lands using nθ, then check the diagram.
- Compare r = 1.10 with r = 0.90 for n = 12: which way does the spiral go? Work out |z⁶| by hand and compare with the readout.
- In roots mode set r = 1.00, θ = 0° and step n from 3 to 6: confirm the roots form a regular polygon and sum to zero. Then set θ = 90° and note each root turns by only 90°/n.
Exam tip: when finding nth roots, write the argument as θ + 2πk before dividing by n, then list exactly n distinct roots with arguments in the range the question asks for.