Roots of Unity

An $n$ -th root of unity is any complex number $z$ satisfying

$z^n = 1.$

Because $1 = e^{2\pi i k}$ for every integer $k$ , taking the $n$ -th root gives

$\omega_k = e^{2\pi i k / n}, \qquad k = 0, 1, \ldots, n-1.$

There are exactly $n$ distinct values (the $k = n$ case wraps back to $k = 0$ ), and each one sits on the unit circle at angle $2\pi k / n$ from the positive real axis. They are perfectly evenly spaced: neighbouring roots differ in angle by exactly $2\pi / n$ .

Visualising the roots

For $n = 4$ the four roots are

$1,\quad i,\quad -1,\quad -i$

at angles $0°$ , $90°$ , $180°$ , and $270°$ — the four compass points on the unit circle. For $n = 3$ the three cube roots sit at $0°$ , $120°$ , and $240°$ .

The primitive $n$ -th root of unity is $\omega = e^{2\pi i / n}$ (the $k = 1$ case). Every other $n$ -th root is a power of $\omega$ :

$\omega_k = \omega^k.$

Why roots of unity matter in quantum mechanics

Two places they appear immediately:

Discrete Fourier transform (DFT) and QFT. The quantum Fourier transform maps $|j\rangle \mapsto \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \omega^{jk} |k\rangle$ , where $\omega = e^{2\pi i / N}$ is the primitive $N$ -th root of unity. The entire algorithm is built from nothing but these phase factors.
Symmetry and selection rules. When a physical system has discrete rotational symmetry by angle $2\pi/n$ , its energy eigenstates carry eigenvalues that are $n$ -th roots of unity. Quantum numbers reflect which root labels each state.

A key identity: the sum is zero

For $n \geq 2$ , the sum of all $n$ -th roots of unity vanishes:

$\sum_{k=0}^{n-1} \omega^k = \frac{\omega^n - 1}{\omega - 1} = \frac{1 - 1}{\omega - 1} = 0.$

(The geometric-series formula applies because $\omega \neq 1$ for $n \geq 2$ .) This orthogonality is the reason Fourier analysis — classical and quantum — works: each frequency component picks out exactly one mode and cancels all others.

Rectangular form of a root

Using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ , the $k$ -th $n$ -th root is

$\omega_k = \cos\!\left(\frac{2\pi k}{n}\right) + i\sin\!\left(\frac{2\pi k}{n}\right).$

For the primitive cube root ( $n = 3$ , $k = 1$ ):

= -\tfrac{1}{2} + i\,\tfrac{\sqrt{3}}{2}.$$ <Callout type="tip"> Every $n$-th root of unity lies on the unit circle, so its modulus is 1 and its square modulus is 1. That means it is always a valid quantum phase factor: multiplying a state by $\omega_k$ cannot change any measurement probability. </Callout> ## Try it This is a numerical exercise — `return` a number. The first non-trivial cube root of unity is $\omega_1 = e^{2\pi i / 3}$. What is its real part? Derive it using $\operatorname{Re}(e^{i\theta}) = \cos\theta$. <RunnableExample />