Quality of rational approximations

[Vanadium 50]

22/7 is a very good approximation for π. Sqrt(2) doesn’t do that well until 99/70 and e doesn’t do that well until 193/71. 355/113 is even better.

Is there some reason for this? Perhaps geometrical? Why do the ratios of small integers work better for π than other numbers? Or is it just coincidence? ("Gotta be something...")

 

[fresh_42]

The pattern of continued fractions?

 

[Vanadium 50]

Sure, but e has one too. What's special, if anythnig, about pi?

 

[martinbn]

If you want to keep the denominators of the rationals somewhat small, then algebraic numbers are not well approximated by rationals.
https://en.wikipedia.org/wiki/Roth's_theorem

ps. Interesting that Dyson had worked on these matters.