Another Ramanujan Trick

[Gerson W. Barbosa]

(09-29-2019 09:55 AM)Massimo Gnerucci Wrote:  

(09-29-2019 06:45 AM)ttw Wrote:  Just for fun, I came across with another Ramanujan trick:

Pi=Sqrt(Sqrt(2142/22))

It's better than 355/113 but at the cost of a couple of square roots (or a fourth root.)

Isn't it Pi=Sqrt(Sqrt(2143/22)) ?

Ciao, Massimo,

Yes, you are right.

Let us take our HP-32S or 33S and do some “reverse engineering”:

pi x^2 x^2 -> 97.409091034
99 * -> 96435.50001237
ENTER + IP -> 19287
198 / FDISP -> 97 9/22
1/x -> 22/2143

=> pi ~ sqrt(sqrt(2143/22)) = 3.14159265258

This fourth power oddity is common to both ln(pi) and e^x as well:

(ln(pi))^4 = 1.7171652254

and

(e^pi)^4 = 286751.313138,

which lead to

e^(sqrt(sqrt(170/99))) = 3.14159605245

and

ln(sqrt(sqrt(28388380/99))) = 3.14159265359

These don’t make for efficient pi approximations, though.

P.S.:

You might want to check this out:

\[\ln\left(\sqrt[4]{\frac{{305}^{3}+{25}^{3}+{5}^{3}+5+\frac{33}{{9}^{5}+{5}^{5}+{3}^{5}+3}}{99}} \right )\]

P.P.S.:

Or that one:


\[\sqrt[4]{\frac{2143+{\left(6+\sqrt{\frac{6}{1+{6}^{-6}}}\right)}^{-6}}{22}}\]