π-th roots of unity

Earlier today, I saw this post from @thalesdisciple about visualising the irrationality of π:

thales

He links to this excellent Desmos graph which uses sliders and lists to plot the set of points he describes. Then I had a flashback to a conversation I had over 10 years ago when I was mentoring Maths undergraduates.

What are the π-th roots of unity?

That was the question that a first year asked in one of our meetings. I think it was intended partly out of natural curiosity and partly as a general distraction from the kind of maths we should have been doing. It turned out to be more interesting than he perhaps hoped.

Writing \sqrt[\pi]{1} hides some rather interesting and important facts. Notably, as with all roots of unity, we should expect multiple values. Clearly, 1^\pi=1 so z=1 is a solution of z^\pi-1=0, but what are the others? And how many of them are there?

The Second Solution

While many argue that e^{i\pi}=-1 is the most beautiful formula in mathematics, if we square both sides then we see e^{2i\pi}=1 and thus z=e^{2i} is another π-th root of unity.

But how can we make sense of this number e^{2i}? Is it real? No. Can we plot it on an Argand diagram? Yes.

Using the form e^{i\theta}=\cos\theta+i\sin\theta, we can write

e^{2i}=\cos 2 + i\sin 2.

That’s not particularly impressive, but it tells us the location of this complex number: it lies on the unit circle at an angle 2 radians to the positive real axis. (2 radians is just shy of 120 degrees.) Here are the solutions we have found so far, plotted on the unit circle:

2roots

On  a roll

Now we are in a position to generalise what we have found so far. Noting that e^{2ni\pi}=1 then z=e^{2ni} is a π-th root of unity for ever natural number n. We now have (countably) infinitely many points of the form

z=\cos(2n)+i\sin(2n).

So let’s plot some more and see what happens.

3points

The first three points are approximately equidistant around the circle.

6points

When six points are plotted, they appear to cluster in three sets.

22points

After plotting 22 points they are again (approximately) equally spread.

There is something behind this ‘clustering’ and (approximately) even distribution of the points. When we plot precisely 355 points, they once again look evenly distributed:

150points

With only 150 points plotted, the 22 ‘clusters’ are still visible

355 points

With 355 points plotted, they are evenly spread.

And there is even more to discover: zooming in on the 355 points shows them to be clearly spread and separated. However, the next 355 stick right beside them! Plotting 710 points actually creates 355 clusters of double points.

355zoom

A zoom of the 355 plotted points – evenly spread.

doubledot

A zoom of 710 plotted points – they pair up to make 355 clusters!

 Rational Approximations

Truth be told, I cannot really explain with mathematical rigour why there is the apparent spreading and clustering that we see in the above plots. However, there is an observation to be made about the numbers of clusters. To recap, the pattern was 3 evenly spread; then 22; then 355. These are the numerators of successively better rational approximations to the value of π:

\frac{3}{1},\quad \frac{22}{7},\quad \frac{355}{113},\quad\dots

Think, Pair, Share

  • Play with the Demos sheet that I created, based on Joshua Bowman’s.
  • Try the homework I set my student: to prove that the (countably infinite) set of π-th roots of unity forms a group under multiplication. [Edit: n can be any integer, not just a natural number.]
  • Can you prove that this set of numbers is dense in the unit circle? [Joshua has this video on YouTube which does a great job of explaining the concept of a ‘dense’ subset.]
  • Can you explain the connection between the ‘clustering’ and the rational approximations?
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