Roots of Unity and Their Geometric Properties

The nth roots of unity are the n complex solutions of zⁿ = 1, namely ω_k = cos(2πk/n) + i sin(2πk/n) = e^2πik/n for k = 0, 1, …, n − 1; they are the vertices of a regular n-gon inscribed in the unit circle. In brief: the cube roots of unity are 1, ω = −1/2 + (√3/2)i and ω² = −1/2 − (√3/2)i, and they satisfy 1 + ω + ω² = 0. For every n ≥ 2, the sum of all nth roots of unity is 0, and their product is (−1)ⁿ⁺¹.

Roots of unity are the simplest and most symmetric complex numbers. They connect algebra (the equation zⁿ − 1 = 0) with geometry (regular polygons) and are the basis of the Fast Fourier Transform. This guide describes the roots, their properties (sums, products, powers), primitive roots, how they factor zⁿ − 1 and how they relate to regular polygons, with a table of the roots for small n.

What are the roots of unity?

The equation zⁿ = 1 has n solutions. They are the nth roots of the number 1, which has modulus 1 and argument 0. By the formula, ω_k = cos(2πk/n) + i sin(2πk/n) = e^2πik/n, k = 0, …, n − 1. All have modulus 1, so they lie on the unit circle, and the angle between neighbours is 2π/n. For ω = e^2πi/n, the roots are 1, ω, ω², …, ωⁿ⁻¹, which are the vertices of a regular n-gon with one vertex at 1.

What are their properties?

• Closed under multiplication: ω_jω_k = ω_j+k (the index modulo n). The roots form a group.
• Inverses and conjugates: ω_k⁻¹ = ω̄_k = ω_n−k. The nonreal roots come in conjugate pairs.
• Sum: 1 + ω + ω² + … + ωⁿ⁻¹ = (1 − ωⁿ)/(1 − ω) = 0 for ω ≠ 1. Geometrically, the vertices of a regular polygon have the center of mass at the center.
• Product: the product of all nth roots of unity is (−1)ⁿ⁻¹ (from the constant term of zⁿ − 1 by Vieta’s formulas).
• Powers: the sum of the mth powers is n if n divides m and 0 otherwise.

Example 1: sum and product for n = 4

The fourth roots of unity are 1, i, −1 and −i. Their sum is 1 + i − 1 − i = 0 ✓. Their product is 1 · i · (−1) · (−i) = (i · (−i)) · (−1) = 1 · (−1) = −1, which equals (−1)³ ✓.

Example 2: powers of the cube root

With ω = −1/2 + (√3/2)i: ω³ = 1, ω⁴ = ω and ω¹⁰⁰ = ω¹ (since 100 = 3 · 33 + 1). And 1 + ω + ω² = 0, so ω² + ω = −1. Therefore (1 + ω)² = (−ω²)² = ω⁴ = ω.

What are primitive roots of unity?

A root ω_k is primitive if its powers produce all nth roots of unity, equivalently, if the smallest positive m with ω_k^m = 1 is m = n. The root ω_k is primitive exactly when gcd(k, n) = 1. The number of primitive nth roots is φ(n), Euler’s totient function. For n = 6: k = 1 and 5 are primitive (2 roots), while ω₂ and ω₄ are cube roots of unity, ω₃ = −1 is a square root of unity and ω₀ = 1. For n = 7 (prime): k = 1, …, 6 are all primitive.

How do roots of unity factor polynomials?

The polynomial zⁿ − 1 factors completely over the complex numbers: zⁿ − 1 = (z − 1)(z − ω)(z − ω²)…(z − ωⁿ⁻¹). Over the reals, the conjugate pairs combine into quadratic factors: z⁴ − 1 = (z − 1)(z + 1)(z − i)(z + i) = (z − 1)(z + 1)(z² + 1). The quotient (zⁿ − 1)/(z − 1) = 1 + z + … + zⁿ⁻¹ has the nonreal roots (and all roots other than 1). For example z³ − 1 = (z − 1)(z² + z + 1), and the roots of z² + z + 1 = 0 are ω and ω² = (−1 ± √3 i)/2 by the quadratic formula.

Table of roots for n = 2 to 6

For n = 5: cos 72° = (√5 − 1)/4 ≈ 0.3090 and sin 72° ≈ 0.9511, and cos 144° = −(√5 + 1)/4 ≈ −0.8090 and sin 144° ≈ 0.5878. The appearance of the golden ratio is the reason why the regular pentagon is constructible.

Worked examples

Example 3: solving z⁶ = 64

The solutions are 2ω_k with the sixth roots of unity: 2, 1 + √3 i, −1 + √3 i, −2, −1 − √3 i, 1 − √3 i. They form a regular hexagon with radius 2.

Example 4: a sum with roots of unity

Compute 1 + ω + ω² + … + ω⁶ where ω = e^2πi/7: the first seven terms (ω⁰ to ω⁶) add up to 0, so 1 + ω + … + ω⁶ = 0.

Example 5: a product

(1 − ω)(1 − ω²)…(1 − ωⁿ⁻¹) = n. Setting z = 1 in (zⁿ − 1)/(z − 1) = Π (z − ω^k) gives n. For n = 3: (1 − ω)(1 − ω²) = 1 − ω − ω² + ω³ = 1 + 1 + 1 = 3 ✓.

Example 6: the sum of cosines

cos 72° + cos 144° + cos 216° + cos 288° = −1, because the sum of the real parts of the fifth roots of unity is 0, and the real part of 1 is 1.

Where do they come from?

The roots of unity were studied by de Moivre, Euler and Gauss. In his Disquisitiones Arithmeticae of 1801, Gauss used the roots of unity (the cyclotomic equation) to show that the regular 17-gon can be constructed with a compass and a straightedge. This was a striking result, as no new regular polygon had been found for two thousand years, and the work led to the theory later developed by Galois.

Where are they used today?

The Fast Fourier Transform (FFT), one of the most used algorithms, evaluates polynomials at the nth roots of unity in O(n log n) steps instead of O(n²). It powers audio and image compression, communications and scientific computing. Roots of unity also appear in cryptography, coding theory and in the symmetry analysis of molecules and crystals.

Which mistakes are common?

Forgetting the root 1

The list of nth roots must include 1 (k = 0).

Giving only the real roots

zⁿ = 1 has n complex roots, of which only 1 (and −1 for even n) are real.

Confusing all roots with primitive roots

Only the roots with gcd(k, n) = 1 are primitive. For n = 4, the root −1 is not primitive.

Using degrees in e^2πik/n

The exponent form uses radians: 2πk/n, not 360k/n in degrees.