

A002193


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


269



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
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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
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
 Clark Kimberling, Oct 12 2017
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


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 SpringerPraxis Pub. 2006.
Martin Gardner, Gardner's Workout, Chapter 2 "The Square Root of 2=1.414213562373095..." pp. 919 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), 234235.
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, Retroanalytical Reasoning IQ tests for the High Range, 2013.
Horace S. Uhler, ManyFigure Approximations To Sqrt(2), And Distribution Of Digits In Sqrt(2) And 1/Sqrt(2), Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 6367.
Eric Weisstein's World of Mathematics, Pythagoras's Constant
Eric Weisstein's World of Mathematics, Square Root


FORMULA

Sqrt(2) = 14 * Sum_{n >= 0} (A001790(n)/2^A005187(floor(n/2)) * 10^(2n1)) where A001790(n) are numerators in expansion of 1/sqrt(1x) and the denominators in expansion of 1/sqrt(1x) are 2^A005187(n). 14 = 2*7, see A010503 (expansion of 1/sqrt(2)).  Gerald McGarvey, Jan 01 2005
Limit(n > +infinity) of (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(n1)/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..k1} 2*cos((2*l+1)*Pi/(4*k)) = (Product_{l=0..k1} 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(Pialpha) =  cos(alpha) to obtain 2 for the square of the present product.  Wolfdieter Lang, Oct 22 2013
a(n) = 10*floor(2^((3/2) + n)*5^(2 + n)) + floor(2^((1/2) + n)*5^(1 + n)) for n > 0.  Mariusz Iwaniuk, Apr 30 2017
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)


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=(xd)*10; write("b002193.txt", n, " ", d)); \\ Harry J. Smith, Apr 21 2009
(PARI) r=0; x=2; /* Digitbydigit 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
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2))); /* Martin Ettl, Oct 17 2012 */
(Haskell)  After Michael B. Porter's PARI program.
a002193 n = a002193_list !! (n1)
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: A348971 A297420 A156896 * A020807 A188582 A230077
Adjacent sequences: A002190 A002191 A002192 * A002194 A002195 A002196


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane


STATUS

approved



