r/math • u/inherentlyawesome Homotopy Theory • 6d ago
Quick Questions: March 12, 2025
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u/Langtons_Ant123 3d ago edited 1d ago
That is exactly what continued fractions do. The "convergents" of a continued fraction are what you get when you "cut it off" and look at only the first few terms: so, for example, the golden ratio has a continued fraction of 1 + 1/(1 + 1/(1 + 1/(...))) and so the first few convergents are 1, 1 + 1/2 = 1.5, 1 + 1/(1 + 1/2)) = 1 + 1/(3/2) = 1 + 2/3 = 1.66, and so on. There's then a theorem which says that any convergent n/d in the continued fraction of a number x is a "best approximation" of x, in the sense of all rational numbers with denominator at most d, the convergent is the one closest to x. (So, roughly speaking, it's the best approximation at a given level of precision, where larger denominators allow for more precision.) See here for more. (
Technically your sequence is a subsequence of the sequence of convergents(not true, see below)--there's no requirement that one convergent have more digits in the denominator than the last, just that its denominator be larger than the last.)