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Creating digits of pi
02-21-2018, 11:01 AM (This post was last modified: 02-21-2018 04:39 PM by EdS2.)
Post: #43
RE: Creating digits of pi
(02-17-2018 03:02 PM)Gerson W. Barbosa Wrote:  
(02-17-2018 12:27 PM)EdS2 Wrote:  I just came across this nice approximation, by Ramanujan (of course)
∜(2143/22) = 3.14159265258...

That's probably the only Ramanujan approximation that doesn't rely on any of his astounding theories. He just noticed that \(\pi ^{4}\) = 97.409091034..., very close to 97.409090909..., which when multiplied by 990 this gives 96435.

Hmm, would a mathematical genius look at a decimal expansion? I would hope for some rather more sophisticated source of the insight - but do we know, or can we ever know, where this approximation came from?

I can't resist sharing this other one from Ramanujan, which agrees to 18 digits apparently but with only 12 digits in the expression:

\(\pi \approx \frac{12}{\sqrt{190}}\log\big((2\sqrt{2}+\sqrt{10})(3+\sqrt{10})\big)\)

As continued fractions:
3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...
vs
3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 1, 2, ...

Edit: found this one in Ramanujan's papers - see TABLE II here.
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Messages In This Thread
Creating digits of pi - brickviking - 02-06-2018, 12:24 AM
RE: Creating digits of pi - Joe Horn - 02-06-2018, 01:38 AM
RE: Creating digits of pi - toml_12953 - 02-06-2018, 09:24 PM
RE: Creating digits of pi - brickviking - 02-06-2018, 10:38 PM
RE: Creating digits of pi - brickviking - 02-08-2018, 04:22 AM
RE: Creating digits of pi - toml_12953 - 02-08-2018, 01:02 PM
RE: Creating digits of pi - TASP - 02-07-2018, 10:30 PM
RE: Creating digits of pi - pier4r - 02-09-2018, 02:59 PM
RE: Creating digits of pi - emece67 - 02-10-2018, 09:14 PM
RE: Creating digits of pi - TASP - 02-11-2018, 12:12 AM
RE: Creating digits of pi - ttw - 02-12-2018, 03:59 AM
RE: Creating digits of pi - EdS2 - 02-17-2018, 12:27 PM
RE: Creating digits of pi - EdS2 - 02-21-2018 11:01 AM
RE: Creating digits of pi - pier4r - 02-17-2018, 04:25 PM
RE: Creating digits of pi - brickviking - 02-17-2018, 09:27 PM
RE: Creating digits of pi - pier4r - 02-23-2018, 01:34 PM
RE: Creating digits of pi - SlideRule - 05-13-2018, 05:09 PM



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