A002193 - OEIS

%I M3195 N1291 #255 Dec 30 2024 17:21:54

%S 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,

%T 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,

%U 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

%N Decimal expansion of square root of 2.

%C Sometimes called Pythagoras's constant.

%C Its continued fraction expansion is [1; 2, 2, 2, ...] (see A040000). - _Arkadiusz Wesolowski_, Mar 10 2012

%C The discovery of irrational numbers is attributed to Hippasus of Metapontum, who may have proved that sqrt(2) is not a rational number; thus sqrt(2) is often regarded as the earliest known irrational number. - _Clark Kimberling_, Oct 12 2017

%C From _Clark Kimberling_, Oct 12 2017: (Start)

%C In the first million digits,

%C   0 occurs  99814 times;

%C   1 occurs  99925 times;

%C   2 occurs 100436 times;

%C   3 occurs 100190 times;

%C   4 occurs 100024 times;

%C   5 occurs 100155 times;

%C   6 occurs  99886 times;

%C   7 occurs 100008 times;

%C   8 occurs 100441 times;

%C   9 occurs 100121 times. (End)

%C Diameter of a sphere whose surface area equals 2*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - _Omar E. Pol_, Nov 10 2018

%C Sqrt(2) = 1 + area of region bounded by y = sin x, y = cos x, and x = 0. - _Clark Kimberling_, Jul 03 2020

%C Also aspect ratio of the ISO 216 standard for paper sizes. - _Stefano Spezia_, Feb 24 2021

%C The standard deviation of a roll of a 5-sided die. - _Mohammed Yaseen_, Feb 23 2023

%C From _Michal Paulovic_, Mar 22 2023: (Start)

%C The length of a unit square diagonal.

%C The infinite tetration (power tower) sqrt(2)^(sqrt(2)^(sqrt(2)^(...))) equals 2 from the identity (x^(1/x))^((x^(1/x))^((x^(1/x))^(...))) = x where 1/e <= x <= e. (End)

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.

%D David Flannery, The Square Root of 2, Copernicus Books Springer-Praxis Pub. 2006.

%D Martin Gardner, Gardner's Workout, Chapter 2 "The Square Root of 2=1.414213562373095..." pp. 9-19 A. K. Peters MA 2002.

%D B. Rittaud, Le fabuleux destin de sqrt(2), Le Pommier, Paris 2006.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, pp. 34-35.

%H Harry J. Smith, <a href="/A002193/b002193.txt">Table of n, a(n) for n = 1..20000</a>

%H D. and J. Ensley, <a href="https://web.archive.org/web/20100613153947/http://www.maa.org/reviews/roottwo.html">Review of "The Square Root of 2" by D. Flannery</a>.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020.

%H M. F. Jones, <a href="http://www.jstor.org/stable/2004806">22900D approximations to the square roots of the primes less than 100</a>, Math. Comp., 22 (1968), 234-235.

%H I. Khavkine, PlanetMath.org, <a href="https://planetmath.org/proofthatsqrt2isirrational">square root of 2 is irrational</a>.

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/sqrt_base">Index of expansions of sqrt(d) in base b</a>.

%H C. E. Larson, <a href="https://arxiv.org/abs/2005.03878">(Avoiding) Proof by Contradiction: sqrt(2) is Not Rational</a>, arXiv:2005.03878 [math.HO], 2020.

%H Robert Nemiroff and Jerry Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt2.1mil">The Square Root of Two to 1 Million Digits</a>.

%H Robert Nemiroff and Jerry Bonnell, <a href="https://web.archive.org/web/20070930182428/http://www.ibiblio.org/pub/docs/books/gutenberg/etext94/2sqrt10a.txt">The Square Root of Two to 5 million digits</a>.

%H Robert Nemiroff and Jerry Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt2.10mil">The first 10 million digits of the square root of 2</a>.

%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/sqrt2.txt">The square root of 2 to 10 million digits</a>.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/141421.html">Generalized expansion of real constants</a>.

%H M. Ripa and G. Morelli, <a href="http://www.iqsociety.org/general/documents/Retro_analytical_Reasoning_IQ_tests_for_the_High_Range.pdf">Retro-analytical Reasoning IQ tests for the High Range</a>, 2013.

%H Horace S. Uhler, <a href="https://doi.org/10.1073/pnas.37.1.63">Many-Figure Approximations To Sqrt(2), And Distribution Of Digits In Sqrt(2) And 1/Sqrt(2)</a>, Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 63-67.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareRoot.html">Square Root</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F 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

%F Limit_{n -> +oo} (1/n)*(Sum_{k = 1..n} frac(sqrt(1+zeta(k+1)))) = 1/(1+sqrt(2)). - _Yalcin Aktar_, Jul 14 2005

%F sqrt(2) = 2 + n*A167199(n-1)/A167199(n) as n -> infinity (conjecture). - _Mats Granvik_, Oct 30 2009

%F sqrt(2) = limit as n goes to infinity of A179807(n+1)/A179807(n) - 1. - _Mats Granvik_, Feb 15 2011

%F 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

%F If x = sqrt(2), 1/log(x - 1) + 1/log(x + 1) = 0. - _Kritsada Moomuang_, Jul 10 2020

%F From _Amiram Eldar_, Jul 25 2020: (Start)

%F Equals Product_{k>=0} (1 + (-1)^k/(2*k + 1)).

%F Equals Sum_{k>=0} binomial(2*k,k)/8^k. (End)

%F Equals i^(1/2) + i^(-1/2). - _Gary W. Adamson_, Jul 11 2022

%F Equals (sqrt(2) + (sqrt(2) + (sqrt(2) + ...)^(1/3))^(1/3))^(1/3). - _Michal Paulovic_, Mar 22 2023

%F Equals 1 + Sum_{k>=1} (-1)^(k-1)/(2^(2*k)*(2*k - 1))*binomial(2*k,k) [Newton]. - _Stefano Spezia_, Oct 15 2024

%F From _Antonio Graciá Llorente_, Dec 19 2024: (Start)

%F Equals Sum_{k>=0} 2*k*binomial(2*k,k)/8^k.

%F Equals Product_{k>=2}  k/sqrt(k^2 + 1).

%F Equals Product_{k>=0} (6*k + 3)/((6*k + 3) - (-1)^k).

%F Equals Product_{k>=1} (2*k + 1)/((2*k + 1) + (-1)^k).

%F Equals Product_{k>=0} ((4*k + 3)*(4*k + 1 + (-1)^k))/((4*k + 1)*(4*k + 3 + (-1)^k)). (End)

%e 1.41421356237309504880168872420969807856967187537694807317667...

%p Digits:=100; evalf(sqrt(2)); # _Wesley Ivan Hurt_, Dec 04 2013

%t RealDigits[N[2^(1/2), 128]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008 *)

%o (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

%o (PARI) r=0; x=2; /* Digit-by-digit method */

%o for(digits=1,100,{d=0;while((20*r+d)*d <= x,d++);

%o d--; /* while loop overshoots correct digit */

%o print(d);x=100*(x-(20*r+d)*d);r=10*r+d}) \\ _Michael B. Porter_, Oct 20 2009

%o (PARI) \\ Works in v2.15.0; n = 100 decimal places

%o my(n=100); digits(floor(10^n*quadgen(8))) \\ _Michal Paulovic_, Mar 22 2023

%o (Maxima) fpprec: 100$ ev(bfloat(sqrt(2))); /* _Martin Ettl_, Oct 17 2012 */

%o (Haskell) -- After _Michael B. Porter_'s PARI program.

%o a002193 n = a002193_list !! (n-1)

%o a002193_list = w 2 0 where

%o w x r = dig : w (100 * (x - (20 * r + dig) * dig)) (10 * r + dig)

%o where dig = head (dropWhile (\d -> (20 * r + d) * d < x) [0..]) - 1

%o -- _Reinhard Zumkeller_, Nov 22 2013

%Y Cf. A020807, A010503, A001790, A005187.

%Y Cf. A004539 (binary version).

%K nonn,cons,changed

%O 1,2

%A _N. J. A. Sloane_