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this post was submitted on 08 Nov 2023
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No Stupid Questions
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There are infinitely many. Any sequence of rational numbers converging to pi contains infinitely many. 22/7 and 355/113 are just particularly good ones for their small denominators. You can find such good approximations ("the best rational approximations of a given size") by truncating the continued fraction representation of pi:
These approximations yield:
and so on. Note that by pure coincidence one of the first few terms is quite large (292), so the difference between the corresponding partial expansions is small (1 + 1/292 is close to 1). That's why 355/113 is an unusually good approximation for such a small denominator.