Ramanujan pi approximation

Here’s a mysterious approximation to π from Ramanujan:

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

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13 thoughts on “Ramanujan pi approximation”

1. All approximations to pi are useless, but some are interesting. 🙂

2. rhalbersma

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.

3. rhalbersma

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

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

5. rhalbersma

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.

6. 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.

7. I have no idea what inspired it.

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