Roots of Unity

Lesson notes

Overview

The n solutions of zⁿ = 1 are the corners of a regular n-gon on the unit circle. This topic is rated intermediate and is number 57 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

What you are seeing

The panel above shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

The n solutions of zⁿ = 1 are called the nth roots of unity. They are z = e^(2πik/n) for k = 0 to n − 1, which places them at the corners of a regular n-sided polygon inscribed in the unit circle, with one corner at 1. Their sum is zero, and multiplying two of them gives another. They are central to the discrete Fourier transform, which computers use to analyse signals, and to the solution of polynomial equations.

Reading the relationship

The expression "z = e^(2πik/n)" is a compact summary of the idea. It uses the symbols z, e, π, i, k, n; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and an exponential, which changes by the same factor for every equal step. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.

Try this

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Complex Number Multiplication”, the topic before it in Mathematics. Once you are comfortable, continue with “The Derivative as a Slope”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.