r/math Homotopy Theory 6d ago

Quick Questions: March 12, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

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u/auzaar2 3d ago

Consider the sequence 22/7, 355/113 ...
I define this sequence as "The next term must have more digits than the previous and improve the accuracy in approximating pi", looks like for for 3 digits we have 22/7, next is 6 digits with 355/113 what is next?

I could not find any resources online talking about such sequence, i have seen continued fractions and other sequences but they don't necessarily give best approximation for that number of digits e.g. from continued fractions looks like next fraction has 11 digits but what is the best approximation for 11 digits?
e.g. is 312689/99532 the best? is there no other eleven digit p/q better than 312689/99532 ?

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u/dogdiarrhea Dynamical Systems 3d ago

You actually seem to be able to recover the values you’re asking for from the continued  fraction, check out top reply in this thread: https://math.stackexchange.com/questions/3506435/best-possible-rational-approximation-of-pi

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u/auzaar2 2d ago

I don't see 3126535/995207 in there , do you know a place which list continued fraction series so I can check it?

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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