A game with $\pi$

Here's an image of something I wrote down, took a photo of, and posted here for you.

It's a little game you can play with any irrational number. I took $\pi$ as an example.

You just learned about an important math concept/process called continued fraction expansions.

With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish.

As an example, if you truncate the above last expansion where the 292 appears (so you omit the "1 over 292" part) you get the rational number $\frac{355}{113}$ which approximates $\pi$ to 6 decimal places. (Better than $\frac{22}{7}$.)

You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers.

Exercise: for the square root of 2, show that the sequence you get is
1, 2, 2, 2, 2, ...
(all 2's after the 1). For the square root of 3 the continued fraction sequence is
1, 1, 2, 1, 2, 1, 2, 1, 2, ... 
(so it starts with 1 and then the pair "1, 2" repeat periodically forever).