4 ms·

“As of yet undiscovered” Well by the author, maybe. [1] I had to do some more digging on the sequence stated by the author of that post. The idea is to constr

by zwkrt 4y ago

“As of yet undiscovered” Well by the author, maybe. [1] I had to do some more digging on the sequence stated by the author of that post. The idea is to construct a continued fraction, always choosing the next largest denominator you can that doesn’t make the resulting partial evaluation larger than pi. [1] https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/Todd-Trimble-1?ch=15&oid=236476195&share=c7238daa&target_type=answer https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/T...

roywiggins 4y ago
A comment on the original post brings that exact continued fraction up, the author responds: "the continued fraction sequence 3 7 15... is the continued fraction analogy to the decimal sequence 3 1 4 1 5 9 2... etc: they are both guaranteed-to-exist sequences of successively better approximations for pi that come from the real value of pi. Neither is a way to derive or compute pi if you don't already have the exact value." So it's not a sequence like the Wallis rational expression or Taylor series approximation, so it's not very helpful in saying "where 355/113 comes from." 355/113 is a good approximation of Pi, and if you engineer your continued fraction to include the best approximation without going over at each step, it will be in there. Doesn't tell you why there's a nice accurate Pi that has such a high "quality" measure. That is, the 292 showing up in the continued fraction is still (arguably) a surprise, or at least left unexplained.
- civilized 4y ago
  "Why does pi have this awesome continued fraction" is exactly how a professional mathematician would ask the question.
  - roywiggins 4y ago
    Yeah, I changed my comment around to try and say that- the question just becomes "huh, why does the big integer 292 show up like that?" which is really the same question as before. Maybe the answer is "that's not actually that surprising", I don't know.
  - akomtu 4y ago
    More like "why do transcendent numbers have so awesome continued fractions?"
    - eru 4y ago
      Or at least some of them.
- zwkrt 4y ago
  You’re not wrong, I just feel like the way the post was written it was a little bit disingenuous of the author not to at least include the series and discuss it. They make 355/113 seem mysterious, or that it might point to a new series. In fact, it comes from a boring/inevitable series. Maybe it’s my physics training coming through, but asking “why” pi is close to this cute fraction is the wrong question. I would be much more interested in knowing “given some transcendental number, what is the likelihood that there exists some fraction with at most $x digits that approximates it to $y decimal places?” For instance, 577/408 approximates sqrt(2) even more closely than 355/113. Is this usual? Can I come up with a number between 1 and 10 that has no “good” 3 decimal approximation? My very basic understanding of measure theory is that it is actually hard to find numbers that are not well approximated by /some/ integer fraction. After all there are a lot of fractions…
  - rawling 4y ago
    > They make 355/113 seem mysterious, or that it might point to a new series. In fact, it comes from a boring/inevitable series. It might! Just because it's in the boring series, doesn't mean there might not be a new series that it's in. ... unless it does?
quickthrower2 4y ago
Sequence hacking!? What about the sequence where the denominator starts at 113 and keeps doubling, the numerator each time being the one giving the least error.

by zwkrt 4y ago

roywiggins 4y ago

A comment on the original post brings that exact continued fraction up, the author responds: "the continued fraction sequence 3 7 15... is the continued fraction analogy to the decimal sequence 3 1 4 1 5 9 2... etc: they are both guaranteed-to-exist sequences of successively better approximations for pi that come from the real value of pi. Neither is a way to derive or compute pi if you don't already have the exact value." So it's not a sequence like the Wallis rational expression or Taylor series approximation, so it's not very helpful in saying "where 355/113 comes from." 355/113 is a good approximation of Pi, and if you engineer your continued fraction to include the best approximation without going over at each step, it will be in there. Doesn't tell you why there's a nice accurate Pi that has such a high "quality" measure. That is, the 292 showing up in the continued fraction is still (arguably) a surprise, or at least left unexplained.

civilized 4y ago
"Why does pi have this awesome continued fraction" is exactly how a professional mathematician would ask the question.
- roywiggins 4y ago
  Yeah, I changed my comment around to try and say that- the question just becomes "huh, why does the big integer 292 show up like that?" which is really the same question as before. Maybe the answer is "that's not actually that surprising", I don't know.
- akomtu 4y ago
  More like "why do transcendent numbers have so awesome continued fractions?"
  - eru 4y ago
    Or at least some of them.
zwkrt 4y ago
You’re not wrong, I just feel like the way the post was written it was a little bit disingenuous of the author not to at least include the series and discuss it. They make 355/113 seem mysterious, or that it might point to a new series. In fact, it comes from a boring/inevitable series. Maybe it’s my physics training coming through, but asking “why” pi is close to this cute fraction is the wrong question. I would be much more interested in knowing “given some transcendental number, what is the likelihood that there exists some fraction with at most $x digits that approximates it to $y decimal places?” For instance, 577/408 approximates sqrt(2) even more closely than 355/113. Is this usual? Can I come up with a number between 1 and 10 that has no “good” 3 decimal approximation? My very basic understanding of measure theory is that it is actually hard to find numbers that are not well approximated by /some/ integer fraction. After all there are a lot of fractions…
- rawling 4y ago
  > They make 355/113 seem mysterious, or that it might point to a new series. In fact, it comes from a boring/inevitable series. It might! Just because it's in the boring series, doesn't mean there might not be a new series that it's in. ... unless it does?

quickthrower2 4y ago

Sequence hacking!? What about the sequence where the denominator starts at 113 and keeps doubling, the numerator each time being the one giving the least error.