Overview

Roots of unity are equally spaced points on the unit circle. Their symmetry collapses many sums to $0$.

Core Skills

Parameterize the Roots

Write $z_k=e^{2\pi i k/n}$ and use symmetry of the regular $n$-gon.

Use the Geometric Sum

Apply $1+z+\cdots+z^{n-1}=0$ when $z^n=1$ and $z\ne 1$.

Filter by Exponents

If a sum includes $z^{mk}$, reduce $m$ modulo $n$ to see which terms survive.

Key Ideas

• $z^n=1$ gives $z_k = e^{2\pi i k/n}$.
• The roots form a regular $n$-gon centered at the origin.
• $1 + z + z^2 + \cdots + z^{n-1} = 0$ for $z^n=1$, $z\ne 1$.

Worked Example

If $z^5=1$ and $z\ne 1$, compute $1+z+z^2+z^3+z^4$.

This is the full sum of the 5th roots except $1$, so it equals $0$.

More Examples

Example 1: Sum of All Roots

Find the sum of all cube roots of unity.

It is $0$.

Example 2: Power Sum

If $\omega^7=1$ and $\omega\ne 1$, find $1+\omega^2+\omega^4+\omega^6$.

This is half the roots; the sum is $-1$.

Example 3: Geometry Interpretation

What is the sum of the vertices of a regular $n$-gon centered at the origin?

By symmetry, it is $0$.

Strategy Checklist

• Write roots in exponential form.
• Use geometric series for full sums.
• Reduce exponents modulo $n$.

Common Pitfalls

• Forgetting to exclude the root $z=1$.
• Treating the sum as $5$ instead of $0$.
• Forgetting to reduce exponents modulo $n$.

Practice Problems