Concept

Roots of unity

The complex numbers whose nth power is one, sitting at the corners of a regular polygon on the unit circle. They form a cyclic group under multiplication, they add to zero, and their orthogonality is what makes discrete Fourier analysis work.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

algebra · complex numbers

The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

algebra · roots of unity

Named alongside it

The objects these essays reach for when they reach for this one.

Argument Complex exponential Complex numbers Cyclic group Cyclotomic polynomial Factorisation Imaginary unit Modulus Pi Primitive element Regular polygon Totient

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