February 7, 2013 | 4

Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter

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

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.

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 *n*th 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: i^{2}=-1, so i^{4}=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

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

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

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.

In fact, there are seven 7th roots of unity, and each gold disc in that picture is one of them. We can get an *n*th 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.

Rights & Permissions

Add a Comment

You must sign in or register as a ScientificAmerican.com member to submit a comment.

Some cool rock bands?

Symmetry – First album title? “Again and Again” featuring “I Can See for Miles of Tiles”

The Spherical Harmonics Project – First album title? “Tales of Imaginary Parts” featuring “The Fall of the House of Euler” and “(The System of) Dr. Gauss and Professor Jordan”.

The Anistropics – First album title? “Wavelength” featuring “We Gotta Get Out of This Laplace” (Is Eric Bourdon looking for a new singing gig?)

CCW – First album title? “Around the Bend” featuring “Around the Bend” and “You Won’t Find Us Down On the Corner”

The Beta Distribution – First album title? “Probably Not” featuring “I Don’t Give a Flip”, “Ask Me Again (And I’ll Give a Different Answer)” and “Randomly Uniform”

Link to thisThank you Evelyn! I loved this post. I hadn’t thought about Euler’s identity in a very long time. It really has it all doesn’t it? e, i, pi, 0 and 1. I had a wonderful time with your post and all the tangents it led me to.

Link to thisThank you, Evelyn, for a delightful blog! I shall look forward to fresh installments of such ‘cool stuff’. Would it be possible to keep me informed (at my email Inbox) whenever a new posting comes out?

Best wishes,

Link to thisGSC

I like your blog , roots of unity , keep at it .

Link to this