The Simple Math Behind the Mighty Roots of Unity — Quanta Magazine

My latest column for Quanta Magazine is about the roots of unity, one of my favorite sets of mathematical objects.

Last week one of my students located the vertex of a parabola in a particularly elegant way. “The vertex is at x = 4,” she said, “because the roots are x = 1 and x = 7, and the roots are symmetric about the vertex.” She used the fact that the parabola is the graph of a quadratic polynomial, and that the roots of that polynomial — the values where it becomes 0 — have a certain structure she could take advantage of.

There is a structure to the roots of every polynomial, and mathematicians study these structures and look for opportunities to capitalize on them, just as my student did with her parabola. And when it comes to the roots of polynomials, none have more structure than the “roots of unity.”

You can read about a few of the fascinating properties of these roots here. And be sure to check out the exercises at the end!