Creating digits of pi

[Gerson W. Barbosa]

(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 \(\pi ^{4}\) = 97.0409091034..., very close to 97.0409090909..., which when multiplied by 990 this gives 96435. Thus, \(\pi \approx \sqrt[4]{\frac{96435}{990}}\), or \(\pi \approx \sqrt[4]{\frac{2143}{22}}\).

Likewise, \((e^{\pi })^{4}\) = \(e^{4\pi }\) = 286751.31313665..., which when multiplied by 99 gives 28388380.0005287... However, \(\frac{28388380}{99}\) is irreducible. Still, \(\frac{\ln \left ( \frac{28388380}{99} \right )}{4 }\) = 3.141592653585137 is an approximation to \(\pi\), although not nearly as good as Ramanujan's, as 10 digits and two operations are used to produce only 12 digits.

For \(e^{\pi }\) I would suggest these two approximations:

\(\frac{16\ln 878}{\ln\left ( 16\ln 878 \right )}\) = 23.14069263691337

and

\(\frac{64146}{2772}\) = 23.14069264069264

the latter being a palindromic approximation.

Of course, the natural logarithm of these are also approximations to \(\pi\).

Gerson.

Edited to fix a typo.

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PS:

Like the ones for \(\pi\) and for \(e ^{\pi }\), the fourth power of \(\ln \pi\), 1.71716522553..., will make for another approximation. So now we have


\(\pi \approx \sqrt[4]{\frac{96435}{990}}\), or \(\pi \approx \sqrt[4]{\frac{2143}{22}}\) = 3.141592652582646 (3589793)

\(e^{\pi }\approx \sqrt[4]{\frac{28388380}{99}}\) = 23.14069263267153 (277926)

\(\ln \pi \approx \sqrt[4]{\frac{170}{99}}\) = 1.14473096774 (2988585)

The latter can be improved to obtain yet another approximation for \(\pi\), albeit a not so good one:

\(\pi \approx e ^{\sqrt[4]{\frac{170-\frac{400}{622401}}{99}}}\) = 3.1415926535897932121 (384)