BTW, when thinking about fractions and decimal expansions, I always want to mention the Egyptian numeral system, and in particular the excellent book "Count Like An Egyptian":
https://www.google.com/books/edition/Count_Like_an_Egyptian/a7PzAgAAQBAJ?hl=en
The problem with our "p/q" notation for rationals is that it's hard to compare and approximate them. Say: which is bigger, 17/43 or 11/29? Hard to see, right?
But if I ask the same question for 0.3953488 and 0.3793103, it's easy.
Now think of approximation. Think of 17/43 in your typical quotitive (I think) model: you divide a circle into 43 equal sectors and you have 17 of them. But who can divide a pizza into 43 slices?? I want a smaller denominator that gets me close to 17/43.
Egyptian fractions make both tasks easy. I forget the details of how 17/43 would be done, but it turns out 17/43 is very close to 3/8 + 1/50. And 11/29 is very close to 3/8 + 1/250. So you can:
* see which is bigger;
* look at the denominators of the second-order terms and see how good your approximation is
Nice! But let's talk about infinity and Lovecraft.
3/n