# Stuart Price and Joshua Bowman’s PIth roots of unity exercise

Saw this amazing post about the $\pi^{th}$ roots of unity yesterday and wanted to use it for a quick extension of our $\pi$ day activity:

Unfortunately, kids up at 11:00 pm, dog up at 1:00 am, cat up at 4:00 am and then everyone up at 6:00 am let to “one of those nights” . . . . So, instead of making use of a really great exercise, I sort of totally butchered it – but it is the idea that counts, right 🙂 Despite stumbling through our project this morning, I can’t recommend Price’s post and Bowman’s Desmos program enough.

For the kids to understand the project a little better, I wanted to do a quick introduction to the complex plan and how the roots of unity show up on the unit circle. We’ve talked a little bit about $i$ before, so the ideas here aren’t totally new to the kids, but a quick re-introduction seemed appropriate:

Next up was a reminder of some of the rational approximate to $\pi$ that we found yesterday in our activity inspired by Evelyn Lamb:

The fractions that we found that approximate $\pi$ are 22/7, 333/106, and 355/113. We reviewed these fractions and also what the similar approximations to $2\pi$ would be.

With the background out of the way, we moved on to Bowman’s Desmos activity. First I just like the kids play around with it and see if they could find a situation in which we nearly had a regular polygon using powers of the $\pi^{th}$ roots of unity. This was a fun “what do you notice” exercise.

Also, sorry for the extra blue screen – don’t know what happened to the camera here. Double also, ignore all of my talking for the first minute, please . . . . I was tired, confused, and incoherent.

Finally, having found the number 44 as a case where the dots where nearly equally spaced and having seen that this approximation was the same number we saw in the numerator of our 44/7 approximate for $2\pi$, we looked to see if we’d see something interesting at 666 and 710. Right around 2:00 is the “wow” moment.

So, a fun project showing a geometric representation of some continued fraction approximations for $\pi$. Definitely one I’d like to have a 2nd, non-exhausted chance at, but oh well. Hopefully the awesome work of Stuart Price and Joshua Bowman shines through over my several stumbles in this project.