A002193 Decimal expansion of square root of 2. (Formerly M3195 N1291)

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

1,2

Comments

Sometimes called Pythagoras's constant.

Its continued fraction expansion is [1; 2, 2, 2, ...] (see A040000). - Arkadiusz Wesolowski, Mar 10 2012

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

From Clark Kimberling, Oct 12 2017: (Start)

In the first million digits,

0 occurs 99814 times;

1 occurs 99925 times;

2 occurs 100436 times;

3 occurs 100190 times;

4 occurs 100024 times;

5 occurs 100155 times;

6 occurs 99886 times;

7 occurs 100008 times;

8 occurs 100441 times;

9 occurs 100121 times. (End)

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

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

Also aspect ratio of the ISO 216 standard for paper sizes. - Stefano Spezia, Feb 24 2021

The standard deviation of a roll of a 5-sided die. - Mohammed Yaseen, Feb 23 2023

From Michal Paulovic, Mar 22 2023: (Start)

The length of a unit square diagonal.

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)

References

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.

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).

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

D. & 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.

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

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) = 2 + n*A167199(n-1)/A167199(n) as n -> infinity (conjecture). - Mats Granvik, Oct 30 2009

sqrt(2) = limit as n goes to infinity of A179807(n+1)/A179807(n) - 1. - Mats Granvik, Feb 15 2011

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

Example

1.41421356237309504880168872420969807856967187537694807317667...

Maple

Digits:=100; evalf(sqrt(2)); # Wesley Ivan Hurt, Dec 04 2013

Mathematica

RealDigits[N[2^(1/2), 128]] (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)

Prog

(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
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2))); /* Martin Ettl, Oct 17 2012 */
(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

Crossrefs

Cf. A020807, A010503, A001790, A005187.

Cf. A004539 (binary version).

Sequence in context: A297420 A156896 A364711 * A020807 A188582 A230077

Adjacent sequences: A002190 A002191 A002192 * A002194 A002195 A002196

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved