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Comment by BoiledCabbage | Hacker News Reader

I never knew about continued fractions.

Impressive the very simple representations of sqrt of 2, sqrt of 3 and sqrt of 5.

There seems to be a continued fraction for pi[1] as well:

  pi = 3 + (1^2)/6 + (3^2)/6 + (5^2)/6 + (7^2)/6 + (9^2)/6 + (11^2)/6 + ...

[1]: https://www.maa.org/sites/default/files/pdf/pubs/amm_supplem...

Mathematicians use words differently. What you're talking about is what the article calls a generalized continued fraction. Generalized continued fractions are not unique, and there are actually many of them for pi, which have different rates of convergence. The article here contains a (different) nice generalized continued fraction for pi, and an algorithm for converting from generalized continued fractions to standard ones, thereby computing the exact standard continued fraction for pi as a result.

(These were later edits to the article, so if you read it early on, you may not have read those parts.)

There are multiple representations of pi as stated by the other commentor. One way of deriving it is given in the link, very elegant using the simple series formula. (Euler came up with it)

[1]: https://golem.ph.utexas.edu/category/2020/08/eulers_continue...

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