A nice exercise about cos(72) from Martin Weissman’s An Illustrated Theory of Numbers

My younger son is working through Martin Weissman’s An Illustrated Theory Numbers and came across this exercise last week:

Today we finally got to doing a project on the problem. We worked through the first 4 parts and will save the last part for another project.

Here’s the first part of the problem which is mostly a discussion of how you can think about points on the unit circle using complex numbers:

The next part of the problem asked to show that if x is a 5th root of unity then $1 + x + x^2 + x^3 + x^4 = 0$. I forgot to zoom out after we zoomed in on the problem, but I do finally remember to zoom out around 1:30 – sorry about that:

Part c was the part that gave my younger son a lot of trouble, but luckily my older son was able to help out with the ideas about sums and products of roots require to get through this step:

Finally, we find the roots of the quadratic polynomial from the last part and find the exact value for cos(72). What a fun project!