A328927 Decimal expansion of (9^2 + (19^2)/22)^(1/4): an approximation for Pi from Srinivasa Ramanujan.

3, 1, 4, 1, 5, 9, 2, 6, 5, 2, 5, 8, 2, 6, 4, 6, 1, 2, 5, 2, 0, 6, 0, 3, 7, 1, 7, 9, 6, 4, 4, 0, 2, 2, 3, 7, 1, 5, 5, 7, 8, 7, 7, 9, 8, 3, 1, 6, 0, 1, 2, 6, 1, 4, 9, 6, 9, 5, 1, 3, 5, 3, 2, 7, 9, 1, 8, 6, 2, 1, 0, 5, 8, 8, 4, 9, 7, 8, 1, 0, 1, 1, 2, 3, 4, 0, 8, 9, 2, 6, 0, 9, 5, 7, 0, 3, 9, 5, 5, 5

Offset: 1

Author

Bernard Schott, Oct 31 2019

Keywords

• nonn
• cons
• easy

Comments

Srinivasa Ramanujan published this curious empirical approximation in 1914 accompanied with a simple geometric construction for Pi based on this value of (9^2 + (19^2)/22)^(1/4) [See Ramanujan link, page 43, section 12, and page 44, Figure 2].

S. Ramanujan found 3.14159265262... as the value for this approximation in 1914 while Maple gives 3.14159265258... and Pi = 3.14159265358...

This approximation is correct to 10^(-8).

Gardner (1985) wrote: "A more astounding discovery is that: 22 pi^4 = 2143. A few multiplications, and the 10 million-plus decimals of pi have vanished. (Can this remarkable relationship mirror some as yet undiscovered facet of physical reality?)" In the Postscript to the 1999 reprint he writes "Divide 2143 (the first four counting numbers) by 22 and hit the square-root button twice. You will get pi to eight decimals", and credits this discovery to Srinivasa Ramanujan. The MathOverflow page also mentions this and the near-integer 10*Pi^4 - 1/11 ≈ 974.0000012... See A352548 for 22*Pi^4. - M. F. Hasler, Jun 22 2022

Examples

			3.141592652582646125206037179644022371557877983160126149695135327918621058849...
		

References

• Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved 5 June 2013, (4.18), page 58.
• Martin Gardner, "Slicing Pi into Millions", Discover, 6:50, January 1985.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.

Links

• William R. Corliss, The Secret Of It All Is In The Pi, Science Frontiers #37, Jan-Feb 1985.
• Martin Gardner, Slicing Pi into Millions, in: Gardner's Why and Wherefores, Prometheus Books (1999), p.87.
• Srinivasa Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, 43-44.
• Zurab Silagadze, The origin of the Ramanujan's π^4 ≈ 2143/22 identity, MathOverflow.net, Feb 26 2016.
• Wikipedia, Approximations of π: Miscellaneous_approximations.

Crossrefs

Cf. A000796, A068028, A068079, A068089, A352548.

Programs

Maple

evalf((9^2 + (19^2)/22)^(1/4),125);

Mathematica

RealDigits[Surd[9^2 + (19^2)/22, 4], 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

PARI

A328927_first(N)=localprec(N+9); digits(10^N\sqrtn(22/.2143,4)) \\ First N terms of the sequence, i.e., a(1, 0, -1, ..., 2-N). - M. F. Hasler, Jun 22 2022

Formula

Equals (102 - 2222/(22^2))^(1/4) = (2143/22)^(1/4).