Further Maths · Beta

Roots of unity

Choose a root on the Argand circle and follow its powers.

Select a labelled root, then explore its powers. Every point satisfies zⁿ = 1.

zk=e2πik/nz_k=e^{2\pi ik/n}
Select a labelled root, then explore its powers. Every point satisfies zⁿ = 1.0.375-0.3750.75-0.751.125-1.1251.5-1.5ReIm01234

Tab to a ring or outcome, then use the arrow keys. Sliders also work with touch and keyboard.

The powers of this root visit every root.

Selected root
z1≈0.309017+0.951057iz_{1}\approx 0.309017 + 0.951057i
Angle
2π⋅15\frac{2\pi\cdot1}{5}
Powers: root indices
0 → 1 → 2 → 3 → 4 → 0

There are exactly n distinct roots, all modulus 1 and separated by 2π/n. Index n would repeat root 0. The displayed coordinates are approximate.

Predict, test and explain

Step 1 of 3

Predict

What is the angle between consecutive fifth roots of unity?

InvestigateWhy do all roots have modulus 1?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is the angle between consecutive fifth roots of unity?

angular gap for z5=1\text{angular gap for }z^5=1

Hint

All roots of zⁿ = 1 have modulus 1.

Reveal answer

A full turn is 2π, shared equally among five roots.

2π5\frac{2\pi}{5}

Build an explanation

Hint 1

All roots of zⁿ = 1 have modulus 1.

Hint 2

Their arguments are 2πk/n for k = 0, …, n − 1.

Hint 3

Multiplication adds indices modulo n.

Worked example

This example uses fixed values, separate from the diagram controls.

List the cube roots of unity.

  1. Use arguments 0, 2π/3 and 4π/3.
  2. Evaluate cos θ + i sin θ at those angles.
  3. Index 3 would repeat index 0.

Answer: 1, −1/2 + (√3/2)i, −1/2 − (√3/2)i.

Connect this idea

Watch out: Use a full turn of 2π, rather than a half turn.