

A002193


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


289



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.
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. (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 5sided die.  Mohammed Yaseen, Feb 23 2023
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 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



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 > +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..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
Equals Product_{k>=0} (1 + (1)^k/(2*k + 1)).
Equals Sum_{k>=0} binomial(2*k,k)/8^k. (End)
Equals (sqrt(2) + (sqrt(2) + (sqrt(2) + ...)^(1/3))^(1/3))^(1/3).  Michal Paulovic, Mar 22 2023


EXAMPLE

1.41421356237309504880168872420969807856967187537694807317667...


MAPLE



MATHEMATICA



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 */
(PARI) \\ Works in v2.15.0; n = 100 decimal places
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2))); /* Martin Ettl, Oct 17 2012 */
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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



