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Root of Unity — Definition, Formula & Examples

A root of unity is a complex number that equals 1 when raised to some positive integer power. For example, −1-1 is a root of unity because (−1)2=1(-1)^2 = 1.

An nnth root of unity is any complex number ω\omega satisfying ωn=1\omega^n = 1. There are exactly nn distinct nnth roots of unity, given by e2πik/ne^{2\pi i k/n} for k=0,1,2,…,n−1k = 0, 1, 2, \ldots, n-1. The root corresponding to k=1k = 1 is called a primitive nnth root of unity, and all other roots are powers of it.

Key Formula

ωk=e2πik/n=cos⁡ ⁣(2πkn)+isin⁡ ⁣(2πkn)\omega_k = e^{2\pi i k / n} = \cos\!\left(\frac{2\pi k}{n}\right) + i\sin\!\left(\frac{2\pi k}{n}\right)
Where:
  • nn = Positive integer; you are finding the nth roots of unity
  • kk = Integer index running from 0 to n − 1
  • ωk\omega_k = The kth root of unity

How It Works

To find the nnth roots of unity, you express 1 in polar form as e2πi⋅0e^{2\pi i \cdot 0} and then take the nnth root while accounting for all nn possible angles. Each root sits on the unit circle in the complex plane, spaced equally at angles of 2πn\frac{2\pi}{n} radians apart. The primitive root ω=e2πi/n\omega = e^{2\pi i/n} generates all the others: the complete set is {1,ω,ω2,…,ωn−1}\{1, \omega, \omega^2, \ldots, \omega^{n-1}\}. A useful identity is that the nnth roots of unity always sum to zero: 1+ω+ω2+⋯+ωn−1=01 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0.

Worked Example

Problem: Find all four 4th roots of unity.
Apply the formula: Use ωk=e2πik/4\omega_k = e^{2\pi i k/4} for k=0,1,2,3k = 0, 1, 2, 3.
ωk=cos⁡ ⁣(πk2)+isin⁡ ⁣(πk2)\omega_k = \cos\!\left(\frac{\pi k}{2}\right) + i\sin\!\left(\frac{\pi k}{2}\right)
Evaluate each root: Substitute each value of kk and simplify.
ω0=1,ω1=i,ω2=−1,ω3=−i\omega_0 = 1,\quad \omega_1 = i,\quad \omega_2 = -1,\quad \omega_3 = -i
Verify: Check that each result raised to the 4th power gives 1. For instance, i4=(i2)2=(−1)2=1i^4 = (i^2)^2 = (-1)^2 = 1.
i4=1✓i^4 = 1 \checkmark
Answer: The four 4th roots of unity are 1,  i,  −1,  −i1,\; i,\; -1,\; -i.

Why It Matters

Roots of unity appear throughout algebra and engineering. They are the backbone of the Discrete Fourier Transform (DFT), which powers signal processing, audio compression, and image analysis. In abstract algebra, they provide concrete examples of cyclic groups and factor into cyclotomic polynomials used in number theory.

Common Mistakes

Mistake: Forgetting that there are nn distinct nnth roots of unity, not just the real ones.
Correction: The equation ωn=1\omega^n = 1 has exactly nn solutions in C\mathbb{C}. For n>2n > 2, most roots are non-real complex numbers. Always use the exponential or trigonometric formula to find all of them.