A010467 Decimal expansion of square root of 10.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

Continued fraction expansion is 3 followed by {6} repeated. - Harry J. Smith, Jun 02 2009

In 1594, Joseph Scaliger claimed Pi = sqrt(10), but Ludolph van Ceulen immediately knew this to be wrong. - Alonso del Arte, Jan 17 2013

The 7th-century Hindu mathematician Brahmagupta used this constant as value of Pi. - Stefano Spezia, Jul 08 2022

Examples

			3.162277660168379331998893544432718533719555139325216826857504852792594...
		

References

• Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.
• Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 112.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 238.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.
• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, pp. 89, 91.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 55.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Josep M. Brunat and Joan-Carles Lario, A problem on concatenated integers, arXiv:2103.05306 [math.NT], 2021. For 1/sqrt(10).
• Jason Kimberley, Index of expansions of sqrt(d) in base b.
• Robert Nemiroff and Jerry Bonnell, The first 1 million digits of square root of 10.
• Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, The first 1 million digits of square root of 10.
• J. J. O'Connor and E. F. Robertson, Ludolph Van Ceulen.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A000032, A040006 (continued fraction).

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(10); // Vincenzo Librandi, Feb 15 2020

Mathematica

RealDigits[N[Sqrt[10],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

PARI

default(realprecision, 20080); x=sqrt(10); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010467.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Formula

Sqrt(10) = sqrt(1 + i*sqrt(15)) + sqrt(1 - i*sqrt(15)) = sqrt(1/2 + 2*i*sqrt(5)) + sqrt(1/2 - 2*i*sqrt(5)), where i = sqrt(-1). - Bruno Berselli, Nov 20 2012

Equals 1/sqrt(10), with offset 0. - Michel Marcus, Mar 10 2021

Equals 2 + Sum_{k>=1} Lucas(k)*binomial(2*k,k)/8^k. - Amiram Eldar, Jan 17 2022

a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(10^((2/3)*k+(1/3))))) mod 10. Will give the k-th decimal digit of sqrt(10). A005875 : number of ways to write n as sum of 3 squares. - Simon Plouffe, Dec 30 2023