A002193 Decimal expansion of square root of 2.
1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0, 1, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7, 1, 8, 7, 5, 3, 7, 6, 9, 4, 8, 0, 7, 3, 1, 7, 6, 6, 7, 9, 7, 3, 7, 9, 9, 0, 7, 3, 2, 4, 7, 8, 4, 6, 2, 1, 0, 7, 0, 3, 8, 8, 5, 0, 3, 8, 7, 5, 3, 4, 3, 2, 7, 6, 4, 1, 5, 7
Offset: 1
Examples
1.41421356237309504880168872420969807856967187537694807317667...
References
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 182.
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.
- David Flannery, The Square Root of 2, Copernicus Books Springer-Praxis Pub. 2006.
- Martin Gardner, Gardner's Workout, Chapter 2 "The Square Root of 2=1.414213562373095..." pp. 9-19 A. K. Peters MA 2002.
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §4.4 Powers and Roots, pp. 84, 145.
- Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 64-67.
- B. Rittaud, Le fabuleux destin de sqrt(2), Le Pommier, Paris 2006.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 60, page 605.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, pp. 34-35.
Links
- Harry J. Smith, Table of n, a(n) for n = 1..20000
- D. and J. Ensley, Review of "The Square Root of 2" by D. Flannery.
- Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020.
- M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., 22 (1968), 234-235.
- I. Khavkine, PlanetMath.org, square root of 2 is irrational.
- Jason Kimberley, Index of expansions of sqrt(d) in base b.
- C. E. Larson, (Avoiding) Proof by Contradiction: sqrt(2) is Not Rational, arXiv:2005.03878 [math.HO], 2020.
- Robert Nemiroff and Jerry Bonnell, The Square Root of Two to 1 Million Digits.
- Robert Nemiroff and Jerry Bonnell, The Square Root of Two to 5 million digits.
- Robert Nemiroff and Jerry Bonnell, The first 10 million digits of the square root of 2.
- Simon Plouffe, Plouffe's Inverter, The square root of 2 to 10 million digits.
- Simon Plouffe, Generalized expansion of real constants.
- M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.
- Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 3.
- Horace S. Uhler, Many-Figure Approximations To Sqrt(2), And Distribution Of Digits In Sqrt(2) And 1/Sqrt(2), Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 63-67.
- Eric Weisstein's World of Mathematics, Pythagoras's Constant.
- Eric Weisstein's World of Mathematics, Square Root.
- Index entries for algebraic numbers, degree 2.
Programs
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Haskell
-- After Michael B. Porter's PARI program. a002193 n = a002193_list !! (n-1) a002193_list = w 2 0 where w x r = dig : w (100 * (x - (20 * r + dig) * dig)) (10 * r + dig) where dig = head (dropWhile (\d -> (20 * r + d) * d < x) [0..]) - 1 -- Reinhard Zumkeller, Nov 22 2013
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Maple
Digits:=100; evalf(sqrt(2)); # Wesley Ivan Hurt, Dec 04 2013
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Mathematica
RealDigits[N[2^(1/2), 128]] (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
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Maxima
fpprec: 100$ ev(bfloat(sqrt(2))); /* Martin Ettl, Oct 17 2012 */
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PARI
default(realprecision, 20080); x=sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002193.txt", n, " ", d)); \\ Harry J. Smith, Apr 21 2009 -
PARI
r=0; x=2; /* Digit-by-digit method */ for(digits=1,100,{d=0;while((20*r+d)*d <= x,d++); d--; /* while loop overshoots correct digit */ print(d);x=100*(x-(20*r+d)*d);r=10*r+d}) \\ Michael B. Porter, Oct 20 2009 -
PARI
\\ Works in v2.15.0; n = 100 decimal places my(n=100); digits(floor(10^n*quadgen(8))) \\ Michal Paulovic, Mar 22 2023
Formula
Sqrt(2) = 14 * Sum_{n >= 0} (A001790(n)/2^A005187(floor(n/2)) * 10^(-2n-1)) where A001790(n) are numerators in expansion of 1/sqrt(1-x) and the denominators in expansion of 1/sqrt(1-x) are 2^A005187(n). 14 = 2*7, see A010503 (expansion of 1/sqrt(2)). - Gerald McGarvey, Jan 01 2005
Limit_{n -> +oo} (1/n)*(Sum_{k = 1..n} frac(sqrt(1+zeta(k+1)))) = 1/(1+sqrt(2)). - Yalcin Aktar, Jul 14 2005
sqrt(2) = Product_{l=0..k-1} 2*cos((2*l+1)*Pi/(4*k)) = (Product_{l=0..k-1} R(2*l+1,rho(4*k)) - 1), identical for k >= 1, with the row polynomials R(n, x) from A127672 and rho(4*k) := 2*cos(Pi/(4*k)) is the length ratio (smallest diagonal)/side in a regular (4*k)-gon. From the product formula given in a Oct 21 2013 formula contribution to A056594, with n -> 2*k, using cos(Pi-alpha) = - cos(alpha) to obtain 2 for the square of the present product. - Wolfdieter Lang, Oct 22 2013
If x = sqrt(2), 1/log(x - 1) + 1/log(x + 1) = 0. - Kritsada Moomuang, Jul 10 2020
From Amiram Eldar, Jul 25 2020: (Start)
Equals Product_{k>=0} (1 + (-1)^k/(2*k + 1)).
Equals Sum_{k>=0} binomial(2*k,k)/8^k. (End)
Equals i^(1/2) + i^(-1/2). - Gary W. Adamson, Jul 11 2022
Equals (sqrt(2) + (sqrt(2) + (sqrt(2) + ...)^(1/3))^(1/3))^(1/3). - Michal Paulovic, Mar 22 2023
Equals 1 + Sum_{k>=1} (-1)^(k-1)/(2^(2*k)*(2*k - 1))*binomial(2*k,k) [Newton]. - Stefano Spezia, Oct 15 2024
From Antonio Graciá Llorente, Dec 19 2024: (Start)
Equals Sum_{k>=0} 2*k*binomial(2*k,k)/8^k.
Equals Product_{k>=2} k/sqrt(k^2 + 1).
Equals Product_{k>=0} (6*k + 3)/((6*k + 3) - (-1)^k).
Equals Product_{k>=1} (2*k + 1)/((2*k + 1) + (-1)^k).
Equals Product_{k>=0} ((4*k + 3)*(4*k + 1 + (-1)^k))/((4*k + 1)*(4*k + 3 + (-1)^k)). (End)
Equals hypergeom([1/2, 1/2], [1/2], 1/2). - Stefano Spezia, Jan 05 2025
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