A343393 Decimal expansion of 99^2/1103, an approximation to 2*Pi*sqrt(2) from Ramanujan.

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

Offset: 1

Author

Bernard Schott, Apr 13 2021

Keywords

• nonn
• cons

Comments

Srinivasa Ramanujan produced this curious approximation to 2*Pi*sqrt(2) (A343392) with dividing 99^2 by prime 1103 (see link Prime Curios!). This approximation comes from the 1st term of the series (44) page 47 at the Ramanujan link.

This formula is correct to 5 places exactly with 2*Pi*sqrt(2) = 8.885765... while 99^2/1103 = 8.885766...

Indeed, in the Ramanujan paper, there is 1/(2*Pi*sqrt(2)) = 1103/99^2 + ..., and in the case of these two numbers, the approximation becomes correct to 8 places exactly with 1/(2*Pi*sqrt(2)) = 0.112539539... while 1103/99^2 = 0.112539536... (see David Wells).

Examples

			8.88576609247506799637352674524025385312783318223...
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 3.14159..., page 36.

Links

• Chris K. Caldwell and G. L. Honaker, Jr., 1103, 1st comment, Prime Curios!
• S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 47.

Crossrefs

Cf. A343392.

Programs

Maple

evalf(99^2/1103,120);

Mathematica

RealDigits[99^2/1103, 10, 100][[1]] (* Amiram Eldar, Apr 13 2021 *)

Formula

Equals 99^2/1103.