Creating digits of pi
02-21-2018, 11:01 AM (This post was last modified: 02-21-2018 04:39 PM by EdS2.)
Post: #61
 EdS2 Member Posts: 79 Joined: Apr 2014
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.
02-23-2018, 01:34 PM
Post: #62
 pier4r Senior Member Posts: 1,336 Joined: Nov 2014
RE: Creating digits of pi
(02-11-2018 12:12 AM)TASP Wrote:  Yeah, not helpful at all, but for some random digit of Pi in base 2, just guess.

50/50 chance of being right is good enough for me and I don't need to put the batteries back in my HP41 to do it.

For a single digit, yes. For multiple digits you need to be increasingly lucky.

For 2 digits a guess may be correct 25% of the time. 3 digits, 12,5% of the time. 4 digits 6.25% etc...

Wikis are great, Contribute :)
02-24-2018, 09:08 AM
Post: #63
 Mike (Stgt) Senior Member Posts: 331 Joined: Jan 2014
RE: Creating digits of pi
(02-21-2018 11:01 AM)EdS2 Wrote:  I can't resist [...]

If it is only about approximation the following is quite close:
Code:
/* This is Rexx */ numeric digits 45 s = 0 do k = 1 to 500000    s += (2 * (k // 2) - 1) / (k + k - 1) end numeric digits 40 say 4 * s
(The '//' is the modulo function.)

In the result only every now and then a digit (4 altogether) differs from Pi.

Ciao.....Mike
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