Notes on the complex roots of unity

The roots of unity begin as a middle-school question — which numbers solve x² = 1? Over the reals the answer is just 1 and -1. For odd-degree roots of 1 the only real answer is 1; for even-degree roots it is either 1 or -1. Extending to the complex numbers turns a triviality into a rich structure.

Definition and the polar route

By the formal definition, a root of unity is any complex number that yields 1 when raised to a positive integer power. It is usually more useful to fix an n and discuss the n-th roots of unity: the complex numbers z with zⁿ = 1. For n = 2 the roots are 1 and -1, both with zero imaginary part. For n = 4 there are four roots: 1, -1, i and -i. For n = 3 the three roots are 1 and the two primitive values. Checking that these cube to 1 is easy; deriving them in the first place takes polar coordinates.

A complex number of magnitude 1 can be written in polar form as z = cos θ + i sin θ. Raising it to the n-th power leaves its magnitude at 1 — exactly what a root of unity requires — and equating zⁿ to 1 means matching angles. Writing 1 in polar form, the angle of 1 is any integer multiple of 2π, i.e. 2πk for some integer k. The n-th roots of unity are therefore the numbers of the form e^(2πik/n).

Rotation structure and conjugates

Because each root has magnitude 1, multiplying by it rotates a point along the unit circle by that root's angle. Take the cube root with angle 2π/3: multiplying it by itself moves to 4π/3, and one more multiplication lands on 1, as it must for a third root. Some roots return to 1 under a power smaller than n — for example, a certain root when n = 6; that observation resurfaces in the group-theoretic picture.

The reciprocal of a root is also a root. If z = e^(2πik/n), its powers orbit counter-clockwise until reaching 1 at the n-th power, while the powers of z⁻¹ orbit clockwise. And z⁻¹ is exactly the complex conjugate of z. So if z₀ … zⁿ⁻¹ are the n-th roots, so is each of their conjugates — in fact the same set, merely enumerated in the opposite order.

Roots, regular polygons, and the vanishing sum

The roots sit evenly spaced around the unit circle. Joining each root to its immediate neighbors gives a regular polygon for every n. Connect each root to the origin and you get congruent triangles: every side from the origin is length 1 and every central angle is 2π/n, so the polygon's sides and angles are all equal.

The even spacing also explains the sum of all n-th roots. With the standard notation it is a geometric series with first term 1 and common ratio z, and it totals zero; viewed as vectors around the circle, the components cancel. Raising the roots to an integer power m gives the same kind of series. When m is not divisible by n the sum is 0; when m is divisible by n each power equals 1 and the n terms sum to n.

The cyclic group and its generators

The set of n-th roots of unity, under multiplication, is a group. Closure: if z₁ and z₂ are roots, then z₁ z₂ raised to n gives 1. Identity: 1 is one of the roots. Associativity follows from complex multiplication. Inverse: the conjugate section already showed that z⁻¹ is a root whenever z is. Since the operation is complex multiplication, the group is also commutative, hence Abelian.

Writing the group as 1, z, …, zⁿ⁻¹ makes its cyclic nature obvious: every element is an integer power of z, so z is a generator. But it is not the only one. Any power z^k generates the group provided k and n are coprime. In the group of six elements, z² fails: its powers repeat every three steps and reach only three of the six elements. For n = 6 the generators are z and z⁵. Such generating roots are called primitive roots.

Powers as an orthogonal basis

Collect the powers of the roots into vectors: first all roots raised to power 0, then power 1, power 2, and so on up to power n-1. Assembling these vectors as the rows of an n×n matrix F allows an inner-product calculation. Over the complex numbers the inner product conjugates the second element, so the entry for row a against row b is a sum over k — precisely the geometric sum from the previous section. The result is zero whenever a ≠ b, which means the vectors are mutually orthogonal. Since there are n orthogonal vectors in n-dimensional complex space, they form a basis, and every vector in the space is a linear combination of them. This is a critical ingredient of the Discrete Fourier Transform, and F is also known as the Fourier matrix.

One further observation: F is symmetric, so its columns are orthogonal as well.