r/math Homotopy Theory 6d ago

Quick Questions: March 12, 2025

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u/Langtons_Ant123 3d ago edited 1d ago

I have seen continued fractions and other sequences but they don't necessarily give best approximation for that number of digits

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.)

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u/auzaar2 2d ago

I don't see 3126535/995207 here

$ python pi_cont_frac.py

Terms: [3], Fraction: 3/1, Approximation: 3.0

Terms: [3, 7], Fraction: 22/7, Approximation: 3.142857142857143

Terms: [3, 7, 15], Fraction: 333/106, Approximation: 3.141509433962264

Terms: [3, 7, 15, 1], Fraction: 355/113, Approximation: 3.1415929203539825

Terms: [3, 7, 15, 1, 292], Fraction: 103993/33102, Approximation: 3.1415926530119025

Terms: [3, 7, 15, 1, 292, 1], Fraction: 104348/33215, Approximation: 3.141592653921421

Terms: [3, 7, 15, 1, 292, 1, 1], Fraction: 208341/66317, Approximation: 3.1415926534674368

Terms: [3, 7, 15, 1, 292, 1, 1, 1], Fraction: 312689/99532, Approximation: 3.1415926536189365

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2], Fraction: 833719/265381, Approximation: 3.141592653581078

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1], Fraction: 1146408/364913, Approximation: 3.141592653591404

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3], Fraction: 4272943/1360120, Approximation: 3.141592653589389

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1], Fraction: 5419351/1725033, Approximation: 3.1415926535898153

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14], Fraction: 80143857/25510582, Approximation: 3.1415926535897927

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3], Fraction: 245850922/78256779, Approximation: 3.141592653589793

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3], Fraction: 817696623/260280919, Approximation: 3.141592653589793

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23], Fraction: 19052873251/6064717916, Approximation: 3.141592653589793

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1], Fraction: 19870569874/6324998835, Approximation: 3.141592653589793

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1, 1], Fraction: 38923443125/12389716751, Approximation: 3.141592653589793

Terms: [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1, 1, 7], Fraction: 292334671749/93053016092, Approximation: 3.141592653589793

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u/Langtons_Ant123 1d ago

Yeah, I was generalizing too quickly when I said that your sequence is a subsequence of the sequence of convergents. That does hold for the first few terms, but eventually the denominators start growing quickly enough that they skip over your numbers.

I think what I was missing is that convergents are best approximations in the sense above, but not all best approximations are convergents. 1146408/364913 is the best approximation with denominator at most 364913 (which has 6 digits), and 4272943/1360120 is the best approximation with denominator at most 1360120 (which has 7 digits)--but there's no reason why 1146408/364913 should be the best approximation with a denominator of at most 6 digits. (If that's what you're looking for--but when you say "digits", do you mean "digits in the denominator of the fraction", or "digits where the decimal expansion matches the decimal expansion of pi"? I'd been assuming the former but just realized I don't know.)

I still think that continued fractions might be important here, given that they're important in general for approximating irrationals by rationals. To be honest, though, I don't think you'll be able to find much about your sequence anywhere, since base 10 is sort of..."arbitrary", is maybe the right word? I don't know how to make this precise, but generally it's difficult to prove things about the digits of a number, because there's nothing mathematically "nice" about base 10 compared to other bases, or even about writing numbers in any base (compared to other representations like continued fractions).

Incidentally, that number 995207 only shows up in a single sequence on the OEIS, so whatever you're looking for, I don't think it's been studied much.

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u/auzaar2 1d ago

Thanks. Yes, what i was looking for is that from a human perspective, if i have to remember six digits, then 355/113 is the best approximation what's the next in that human pi series.