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https://mathworld.wolfram.com/PiContinuedFraction.html https://mathworld.wolfram.com/PiContinuedFraction.html > The simple continued fraction for pi is given b

https://mathworld.wolfram.com/PiContinuedFraction.html https://mathworld.wolfram.com/PiContinuedFraction.html > The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). > The very large term 292 means that the convergent > [3;7,15,1]=[3,7,16]=(355)/(113)=3.14159292... > is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). You can see how the terms in the sequence jump from 1 to 292. I'm not sure why simple continued fractions are not mentioned here when other continued fractions are not, seems like the author got within a hairs breadth of discovering this connection. I'm not going to say "explanation" because it doesn't explain things, it just connects it to something else. This isn't really a way that you would calculate pi, but you could compare it with, for example, the continued fraction for e: [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] and notice that there is no large numbers here... so there is no fraction for e that is suddenly "very good". Same with sqrt 2 (repeating 2s), same with golden ratio (repeating 1s). In a sense, it's the fact that pi DOESN'T have a neat continued fraction which gives it that nice fractional representation. In a sense, discovering a connection just gives you another question to ask. "Okay, why isn't there a neat little formula for calculating pi as a continued fraction?"

eru 4y ago
To go off on a tangent, the Golden Ratio's simple continued fraction makes it in some sense the most irrational number.
svat 4y ago
To actually quantify how "surprised" we should be, see e.g.: • Gauss-Kuzmin distribution: https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuzmin_distribution&oldid=1098003937 https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuz... • This post: https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.html https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.h... (Look at the tail of the distribution, i.e. sum(p(k)) for k ≥ 292 — that will give the probability that a specific term in the continued fraction of a "random" number is at least 292. This is about 31% I think, so not too surprising.)
rawling 4y ago
> seems like the author got within a hairs breadth of discovering this connection. The fourth comment raises this and the author addresses it in the fifth.
kseistrup 4y ago
These are used by the python module “reals” to provide arbitrary precision for pi and e (among other things): https://github.com/rubenvannieuwpoort/reals https://github.com/rubenvannieuwpoort/reals
jerf 4y ago
"same with golden ratio (repeating 1s)" Mathologer has a good video in which he uses this to justify calling the golden ratio the "most irrational" number because of this expansion: https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg The video also goes over the details of the pi expansion.

eru 4y ago

To go off on a tangent, the Golden Ratio's simple continued fraction makes it in some sense the most irrational number.

svat 4y ago

To actually quantify how "surprised" we should be, see e.g.: • Gauss-Kuzmin distribution: https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuzmin_distribution&oldid=1098003937 https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuz... • This post: https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.html https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.h... (Look at the tail of the distribution, i.e. sum(p(k)) for k ≥ 292 — that will give the probability that a specific term in the continued fraction of a "random" number is at least 292. This is about 31% I think, so not too surprising.)

rawling 4y ago

> seems like the author got within a hairs breadth of discovering this connection. The fourth comment raises this and the author addresses it in the fifth.

kseistrup 4y ago

These are used by the python module “reals” to provide arbitrary precision for pi and e (among other things): https://github.com/rubenvannieuwpoort/reals https://github.com/rubenvannieuwpoort/reals

jerf 4y ago

"same with golden ratio (repeating 1s)" Mathologer has a good video in which he uses this to justify calling the golden ratio the "most irrational" number because of this expansion: https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg The video also goes over the details of the pi expansion.