What Are Roots of Unity?

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

This blog is called "Roots of Unity" because in 2004 I thought it would be an awesome band name.

Not the cover of my band's first album. Image courtesy of Gregory A. Moore.

I encountered the term in a college math class, and I had no illusions that I would be in a band at any time in the near future, but it seemed prudent to have some good band names in my back pocket, just in case. You never know when an awesome band will be looking for a recreational violist who already has a nerdy band name picked out.

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So what is a root of unity anyway? The word "root" in the term refers to square roots, cube roots, and any other roots you might happen to need. For any integer n, the nth root of a number k is a number that, when multiplied by itself n times, yields k. The word "unity," perhaps a bit anticlimactically, just means "one." So a root of unity is any number which, when multiplied by itself some number of times, yields 1.

At this point, the definition really seems like it's making a mountain out of a molehill: 1 and -1 are the only numbers that seem to fit the bill. But the evocative phrase doesn't usually come up when we're talking about real numbers: we need to work with complex numbers to get any interesting roots of unity.

The complex numbers consist of all numbers that can be written in the form a+bi, where a and b are real numbers and i is the square root of -1. The number i doesn't exist on the real number line because any real number multiplied by itself is positive, so the letter i, standing for "imaginary," is used instead. Imaginary numbers aren't any more imaginary than the numbers 6, 8, or 27,000, but the label has stuck. The number i itself is a root of unity: i²=-1, so i⁴=1, making i a 4th root of unity. Any square, cube, or other roots of i are also roots of unity.

To see what makes roots of unity special, we need to delve a little bit into notation. If you don't like notation, you should probably skip the next three paragraphs.

We can think of the set of complex numbers as a 2-dimensional plane. We specify a complex number with two coordinates the same way we would on a graph: the point (1,1) refers to the number 1+i. If we were standing on the point (0,0), we probably wouldn't want to walk one unit over and then one unit up to get to the point (1,1). Instead, we'd take a diagonal by walking √2 units at an angle of 45 degrees to the x-axis. Notationally, when we use this radial distance and direction, we write a complex number as re^{i θ}, where r is the distance and θ is the direction, usually written in radians rather than degrees. There are 2π radians in a whole circle, so the number 1 is written e^{i 2π}. The angle 45 degrees is π/4 radians, so the point (1,1) that we found above would be written as √2e^{i π/4}.

The complex number 1+i, written in rectangular and polar coordinates.

This method of specifying a point with a length and a direction is called using polar coordinates, and it shows up most beautifully in Euler's identity, e^iπ=-1.

Polar coordinates make it really easy to multiply complex numbers. With a+bi notation, you have to FOIL (remember middle school algebra?), and you end up with a total of four terms you have to add together. But with the polar notation of re^iθ, it's really easy: you multiply the distances together and add the angles. So 5e^iπ/6× 2e^iπ/3=10 e^{i π/2}. So multiplying involves both expansion or contraction (that's the distance part) and rotation (that's the angle part).

I think the most beautiful thing about roots of unity (besides the awesome name) is that they are kind of a balancing point between 0 and infinity. What I mean by that is if we have a number written re^{i θ} which, when multiplied by itself a certain number of times, yields 1, the distance r itself must be 1. If r were larger than 1, say 2, then as we multiplied the number by itself more and more times, its distance from (0,0) would go from 2 to 4 to 8 and on and on, spiraling out to infinity. If r were smaller than 1, say 1/2, the point would spiral in to 0: 1/2, 1/4, 1/8, and so on. 1 is the only radial distance that will stay perfectly balanced, just marching around the circle as we multiply the numbers together.

To make this more concrete, I happen to know that e^i2π/7 is a root of unity. When I raise it to the 7th power, I get e^i2π, which is 1. Each time I multiply e^i2π/7 by itself, I rotate 1/7 of the way around the circle. In fact, as I multiply e^i2π/7 by itself successively, I get this picture, my blog banner.

The 7th roots of unity. Image courtesy of Gregory A. Moore.

In fact, there are seven 7th roots of unity, and each gold disc in that picture is one of them. We can get an nth root of unity for any number n by replacing the 7 in e^{i 2π/7} by n. The pictures for other roots of unity look a lot like that diagram above, they just have a different number of gold discs.

I'm not sure the mathematical definition of a root of unity stands up to the awesomeness of the phrase as a band name, but I do think it's a beautiful idea, and it's very useful in complex analysis. I'm still collecting awesome band names now that Roots of Unity is out. Two current frontrunners are "The Butterfly Assumption," which is related to a friend's area of math research, and "Premeditated Rosemary Theft," which is just related to an unfortunate incident involving my balcony and some missing herbs.