Roots of Unity: Complex Numbers That Equal One When Powered

The equation x³ = 1 has three solutions, not one.

You know x = 1 works. But in the complex numbers, there are two more: the cube roots of unity. They're evenly spaced on the unit circle, forming an equilateral triangle.

This pattern generalizes. The nth roots of unity are n complex numbers, evenly spaced around the unit circle, that all satisfy xⁿ = 1.

Regular polygons encoded in algebra.

The Definition

The nth roots of unity are the solutions to:

xⁿ = 1

There are exactly n of them (counting multiplicity), given by:

ωₖ = e^(2πik/n) = cos(2πk/n) + i sin(2πk/n)

for k = 0, 1, 2, ..., n-1.

Why There Are n Roots

In the real numbers, x² = 1 has two solutions: ±1. In the complex numbers, every degree-n polynomial has exactly n roots.

For xⁿ - 1 = 0, that's n roots.

The Fundamental Theorem of Algebra guarantees this. What's special about roots of unity is that we can write them all explicitly using Euler's formula.

The Formula

xⁿ = 1 in polar form means:

rⁿ e^(inθ) = 1 · e^(i·0) = e^(i·0)

So rⁿ = 1 (hence r = 1) and nθ = 0 + 2πk for some integer k.

Thus θ = 2πk/n.

The n distinct values occur for k = 0, 1, ..., n-1. After that, the angles repeat (mod 2π).

Examples

Square roots of unity (n = 2):

• ω₀ = e^0 = 1
• ω₁ = e^(iπ) = -1

Two points on the unit circle: 1 and -1.

Cube roots of unity (n = 3):

• ω₀ = 1
• ω₁ = e^(2πi/3) = -1/2 + i√3/2
• ω₂ = e^(4πi/3) = -1/2 - i√3/2

Three points forming an equilateral triangle.

Fourth roots of unity (n = 4):

• 1, i, -1, -i

Four points forming a square: the vertices at compass directions.

Sixth roots of unity (n = 6):

• 1, e^(iπ/3), e^(2iπ/3), -1, e^(4iπ/3), e^(5iπ/3)

Six points forming a regular hexagon.

The Primitive Root

The primitive nth root of unity is:

ω = e^(2πi/n)

All other roots are powers of ω:

ω⁰ = 1, ω¹ = ω, ω² = ω², ..., ωⁿ⁻¹

And ωⁿ = 1, completing the cycle.

ω is called "primitive" because its powers generate all the roots.

The Geometric Picture

The nth roots of unity are the vertices of a regular n-gon inscribed in the unit circle, with one vertex at 1.

• n = 3: equilateral triangle
• n = 4: square
• n = 5: regular pentagon
• n = 6: regular hexagon
• n = 8: regular octagon

Regular polygons are complex number algebra made visible.

Key Properties

Sum equals zero (for n > 1):

ω⁰ + ω¹ + ω² + ... + ωⁿ⁻¹ = 0

Proof: Let S = 1 + ω + ω² + ... + ωⁿ⁻¹. This is a geometric series: S = (1 - ωⁿ)/(1 - ω) = (1 - 1)/(1 - ω) = 0.

Geometrically: the vertices of a regular polygon centered at the origin sum to zero by symmetry.

Product:

ω⁰ · ω¹ · ω² · ... · ωⁿ⁻¹ = ω^(0+1+2+...+(n-1)) = ω^(n(n-1)/2)

For even n, this equals (-1)^(n-1). For odd n, it equals 1 or -1 depending on specifics.

Conjugate pairs:

For real coefficients, complex roots come in conjugate pairs. Indeed, ωₖ and ωₙ₋ₖ are conjugates.

Cyclotomic Polynomials

The polynomial xⁿ - 1 factors over the rationals:

x² - 1 = (x - 1)(x + 1) x³ - 1 = (x - 1)(x² + x + 1) x⁴ - 1 = (x - 1)(x + 1)(x² + 1) x⁶ - 1 = (x - 1)(x + 1)(x² + x + 1)(x² - x + 1)

The cyclotomic polynomial Φₙ(x) is the minimal polynomial for primitive nth roots of unity.

Φ₁(x) = x - 1 Φ₂(x) = x + 1 Φ₃(x) = x² + x + 1 Φ₄(x) = x² + 1 Φ₆(x) = x² - x + 1

These polynomials have integer coefficients and are irreducible over the rationals.

Applications

Discrete Fourier Transform:

The DFT uses nth roots of unity as basis vectors. When you decompose a signal into frequencies, you're projecting onto powers of ω = e^(2πi/n).

This is why the FFT (Fast Fourier Transform) is so efficient — it exploits the algebraic structure of roots of unity.

Solving polynomial equations:

The roots of xⁿ - a are a^(1/n) times the nth roots of unity. Finding one root gives you all of them.

Constructing polygons:

A regular n-gon is constructible by compass and straightedge if and only if n is a product of a power of 2 and distinct Fermat primes. This was proven by Gauss using roots of unity.

The Group Structure

The nth roots of unity form a cyclic group under multiplication.

• Closed: product of two roots is a root
• Identity: 1
• Inverses: ω⁻ᵏ = ωⁿ⁻ᵏ
• Associative: inherited from complex multiplication

This group is isomorphic to ℤ/nℤ (integers mod n under addition).

The generator ω corresponds to 1 in ℤ/nℤ. Adding 1 repeatedly cycles through all elements, just as multiplying by ω cycles through all roots.

Why "Unity"?

"Unity" means 1.

Roots of unity are numbers that, when raised to some power, equal 1.

The term dates to the 18th century when mathematicians first systematically studied these algebraic structures.

The Filter Property

For any nth root of unity ω ≠ 1:

1 + ω + ω² + ... + ωⁿ⁻¹ = 0

But for ω = 1:

1 + 1 + 1 + ... + 1 = n

This "filters" multiples of n: the sum over roots of unity picks out the constant term in Fourier analysis.

Summary

The nth roots of unity are:

ωₖ = e^(2πik/n) for k = 0, 1, ..., n-1.

They form:

• The vertices of a regular n-gon on the unit circle
• A cyclic group under multiplication
• The basis for Fourier analysis
• The algebraic key to polynomial roots

Regular polygons, group theory, and signal processing all meet at the roots of unity.

Further Reading

• Cox, D. Galois Theory. Cyclotomic fields and constructibility.
• Artin, M. Algebra. Group structure of roots of unity.
• Cooley & Tukey. "An Algorithm for the Machine Calculation of Complex Fourier Series." The FFT paper.

This is Part 7 of the Complex Numbers series. Next: "Synthesis: The Completion of Algebra" — why complex numbers make every polynomial solvable.

Part 7 of the Complex Numbers series.

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