Ramanujan pi approximation

Here’s a mysterious approximation to π from Ramanujan:

pi approx frac{12}{sqrt{190}} logleft( (2sqrt{2} + sqrt{10})(3 + sqrt{10})right)

The approximation is correct to 18 decimal places. I have no idea what inspired it.

Related posts:
A Ramanujan series for calculating pi
Ramanujan’s factorial approximation

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13 comments on “Ramanujan pi approximation
  1. Richard says:

    An interesting note from Wolfram: All Rational Approximations of Pi are Useless .

  2. John says:

    All approximations to pi are useless, but some are interesting. :)

  3. rhalbersma says:

    The first factor (2 Sqrt[2] + Sqrt[10]) inside the logarithm is equal to Sqrt[2] Phi^3 where Phi = (1+Sqrt[5])^/2 is the Golden Ratio.

  4. rhalbersma says:

    The formula itself is probably inspired from the similar formula for the Ramanjuan Constant:

    Exp[Pi Sqrt[163]] ~ 640320^3 + 744

    because there is a similar “identity”

    Exp[Pi Sqrt[190]] ~ (Sqrt[2] Phi^3 (3 + Sqrt[10]))^12 + 24

    Dropping the 24, taking logarithms and “solving” for Pi, gives your posted approximation.

    The algebraic foundations (modular functions and all that) for this kind of magic is explained in http://www.oocities.org/titus_piezas/Ramanujan_a.pdf

  5. John says:

    Thanks. I look forward to reading the paper you linked to.

  6. rhalbersma says:

    Unfortunately, I cannot locate a digital version of Ramanujan’s original 1914 paper: Ramanujan, S. “Modular Equations and Approximations to Pi.” Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.

    The Göttinger Digitalisierungszentrum has only caught up to 1900: http://gdz.sub.uni-goettingen.de/dms/load/toc/?IDDOC=646353, so you might need a goold old-fashioned library if you want to go back to the original source.

  7. John says:

    I’ve read the paper you mentioned, but it was hardly clear from the paper where the approximation came from other than “it has something to do with modular functions.” The paper you linked to first seems more promising.

  8. Craig Stuntz says:

    I have no idea what inspired it.

    Presumably it was inspired by Namagiri. Not sure that was helpful, but there you go…

  9. rhalbersma says:

    There are some blog posts on how to derive this stuff (but it looks very complicated!): http://paramanands.wordpress.com/2012/03/19/modular-equations-and-approximations-to-%CF%80pi/

    One other interesting thing to learn from that is that the error is approximately Exp[-Pi Sqrt[190]], which “explains” the 18 decimals of the approximation.

  10. John Stasko says:

    I think he was having fun. The operand of the logarithm can be rewritten as (sqrt(8)+sqrt(10))*(sqrt(9)+sqrt(10)) . People who like to play with numbers like to be able to do something creative with numbers in succession, with patterns. As for myself, it’ll probably be a long time before I forget this formula…

    pi = (3*4 / sqrt (190)) * log ((sqrt(8)+sqrt(10))*(sqrt(9)+sqrt(10)))

  11. Chloe says:

    I’m sorry but as far as I can see, this equation is inaccurate. If I’m missing something or if there is a trick to this, please, please let me know.

  12. John says:

    Chloe: Maybe you’re trying to take logarithms base 10. Logs on this site are always natural logs.

  13. Chloe says:

    I very much appreciate the clarification