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Pretty good Mathologer video about continued fractions: https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg One neat trivi

by uoaei 3y ago

Pretty good Mathologer video about continued fractions: https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg One neat trivium I like to quiz the mathematically-minded on is: what number is the most irrational, i.e., toward which number do rational approximations converge the slowest? The answer can be determined from continued fractions and is discussed in the video near the end.

bitshiftfaced 3y ago
Also got me thinking about what fraction is most "pi efficient"? That is, what fraction is closest to pi compared to how many digits you must use to represent the fraction (or perhaps only digits in the numerator or denominator)?
- uoaei 3y ago
  Seems they answered that question in TFA.
  - bitshiftfaced 3y ago
    Using continued fractions in Mathematica. Maybe there's enough information in the article to say that this is enough to have answered the question. I don't know enough about all of the methods to say, personally. Also, I never outlined what was the best metric to evaluate how well accuracy compares to the number of digits needed.
    - ninepoints 3y ago
      The answer is already there. If you extend the sequence given in the article longer, the ratio of the number of digits in the rational representation to the decimal representation gets worse, not better, so you know you reached the inflection point early.
    - ykonstant 3y ago
      As others have said, the convergents of the simple continued fraction representation give the best rational approximation [1] relative to the size of the fraction, in the strongest relevant metric: that of absolute error bound. Proofs can be found in any book on Diophantine Approximation; the classic (and readable) reference is Khinchin's Continued Fractions book. [1] https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...

by uoaei 3y ago

bitshiftfaced 3y ago

Also got me thinking about what fraction is most "pi efficient"? That is, what fraction is closest to pi compared to how many digits you must use to represent the fraction (or perhaps only digits in the numerator or denominator)?

uoaei 3y ago
Seems they answered that question in TFA.
- bitshiftfaced 3y ago
  Using continued fractions in Mathematica. Maybe there's enough information in the article to say that this is enough to have answered the question. I don't know enough about all of the methods to say, personally. Also, I never outlined what was the best metric to evaluate how well accuracy compares to the number of digits needed.
  - ninepoints 3y ago
    The answer is already there. If you extend the sequence given in the article longer, the ratio of the number of digits in the rational representation to the decimal representation gets worse, not better, so you know you reached the inflection point early.
  - ykonstant 3y ago
    As others have said, the convergents of the simple continued fraction representation give the best rational approximation [1] relative to the size of the fraction, in the strongest relevant metric: that of absolute error bound. Proofs can be found in any book on Diophantine Approximation; the classic (and readable) reference is Khinchin's Continued Fractions book. [1] https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...