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%I #155 Feb 20 2024 10:35:18
%S 6,3,6,6,1,9,7,7,2,3,6,7,5,8,1,3,4,3,0,7,5,5,3,5,0,5,3,4,9,0,0,5,7,4,
%T 4,8,1,3,7,8,3,8,5,8,2,9,6,1,8,2,5,7,9,4,9,9,0,6,6,9,3,7,6,2,3,5,5,8,
%U 7,1,9,0,5,3,6,9,0,6,1,4,0,3,6,0,4,5,5,2,1,1,0,6,5,0,1,2,3,4,3,8,2,4,2,9,1
%N Decimal expansion of Buffon's constant 2/Pi.
%C The probability P(l,d) that a needle of length l will land on a line, given a floor with equally spaced parallel lines at a distance d (>=l) apart, is (2/Pi)*(l/d). - _Benoit Cloitre_, Oct 14 2002
%C Lim_{n->infinity} z(n)/log(n) = 2/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n with real coefficients chosen from a standard Gaussian distribution (cf. Finch reference). - _Benoit Cloitre_, Nov 02 2003
%C Also the ratio of the average chord length when two points are chosen at random on a circle of radius r to the maximum possible chord length (i.e., diameter) = A088538*r / (2*r) = 2/Pi. Is there a (direct or obvious) relationship between this fact and that 2/Pi is the "magic geometric constant" for a circle (see MathWorld link)? - _Rick L. Shepherd_, Jun 22 2006
%C Blatner (1997) says that Euler found a "fascinating infinite product" for Pi involving the prime numbers, but the number he then describes does not match Pi. Switching the numerator and the denominator results in this number. - _Alonso del Arte_, May 16 2012
%C 2/Pi is also the height (the ordinate y) of the geometric centroid of each arbelos (see the references and links given under A221918) with a large radius r=1 and any small ones r1 and r2 = 1 - r1, for 0 < r1 < 1. Use the integral formula given, e.g., in the MathWorld or Wikipedia centroid reference, for the two parts of the arbelos (dissected by the vertical line x = 2*r1), and then use the decomposition formula. The heights y1 and y2 of the centroids of the two parts satisfy: F1(r1)*y1(r1) = 2*r1^2*(1-r1) and F2(1-r1)*y2(1-r1) = 2*(1-r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1-r1). (F1 and F2 are rather complicated but their explicit formulas are not needed here.) The r1 dependent horizontal coordinate x with origin at the left tip of the arbelos is x = r1 + 1/2. - _Wolfdieter Lang_, Feb 28 2013
%C Construct a quadrilateral of maximal area inside a circle. The quadrilateral is necessarily an inscribed square (with diagonals that are diameters). 2/Pi is the ratio of the square's area to the circle's area. - _Rick L. Shepherd_, Aug 02 2014
%C The expected number of real roots of a real polynomial of degree n varies as this constant times the (natural) logarithm of n, see Kac, when its coefficients are chosen from the standard uniform distribution. This may be related to Rick Shepherd's comment. - _Charles R Greathouse IV_, Oct 06 2014
%C 2/Pi is also the minimum value, at x = 1/2, on (0,1) of 1/(Pi*sqrt(x*(1-x))), the nonzero piece of the probability density function for the standard arcsine distribution. - _Rick L. Shepherd_, Dec 05 2016
%C The average distance from the center of a unit-radius circle to the midpoints of chords drawn between two points that are uniformly and independently chosen at random on the circumference of the circle. - _Amiram Eldar_, Sep 08 2020
%C 2/Pi <= sin(x)/x < 1 for 0 < |x| <= Pi/2 is Jordan's inequality, also known as (2/Pi) * x <= sin(x) <= x for 0 <= x <= Pi/2; this inequality was named after the French mathematician Camille Jordan (1838-1922). - _Bernard Schott_, Jan 07 2023
%C This constant 2/Pi was named after the needle experiment, described in 1777 by the French naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). Note that the parrot Buffon's macaw and the antelope Buffon's kob were named also after Buffon. - _Bernard Schott_, Jan 10 2023
%D David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
%D G. Buffon, Essai d'arithmétique morale. Supplément à l'Histoire Naturelle, Vol. 4, 1777.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 7, eq. (1.2) and p. 105 eq. (7.4.2) with s=1/2.
%D Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.
%D Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.
%D Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976.
%D Robert M. Young, Excursions in Calculus, An Interplay of the Continuous and the Discrete. Dolciani Mathematical Expositions Number 13. MAA.
%H Harry J. Smith, <a href="/A060294/b060294.txt">Table of n, a(n) for n = 0..20000</a>
%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath669/kmath669.htm">MathPages: The Algebra of an Infinite Grid of Resistors</a>
%H G. Buffon, <a href="http://www.buffon.cnrs.fr/ice/ice_page_detaila3c7.html">Essai d'arithmétique morale</a>, Supplément à l'Histoire Naturelle, Vol. 4, 1777.
%H Encyclopedia of Mathematics, <a href="https://www.encyclopediaofmath.org/index.php/Arcsine_distribution">Arcsine distribution</a>
%H Boris Gourevitch, <a href="http://www.pi314.net/">L'univers de Pi</a>
%H Mark Kac, <a href="http://projecteuclid.org/euclid.bams/1183505112">On the average number of real roots of a random algebraic equation</a>, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314-320.
%H MacTutor History of Mathematics archive, <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Buffon/">Georges-Louis Leclerc, Comte de Buffon</a>.
%H Veikko Nevanlinna, <a href="https://www.acadsci.fi/mathematica/1973/year1973.html">On constants connected with the prime number theorem for arithmetic progressions</a>, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
%H Da-Wei Niu, Jian Cao, and Feng Qi, <a href="https://www.scientificbulletin.upb.ro/rev_docs_arhiva/full3105.pdf">Generalizations of Jordan's inequality and concerned relations</a>, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010
%H Herbert Solomon, <a href="https://archive.org/stream/GeometricProbability/Geometric_Probability-Solomon#page/n161/mode/2up">Geometric Probability</a>, SIAM, 1978, p. 152. [See average chord length comment]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BuffonsNeedleProblem.html">Buffon's needle problem</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MagicGeometricConstants.html">Magic Geometric Constants</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeometricCentroid.html">Geometric Centroid</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JordansInequality.html">Jordan's Inequality</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Buffon's_needle_problem">Buffon's needle problem</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Centroid">Centroid</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan's_inequality">Jordan's inequality</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F 2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... - _Jason Earls_ [formula corrected by _Paul D. Hanna_, Mar 23 2013]
%F The preceding formula is 2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. - _Alexander R. Povolotsky_, Mar 24 2013. [See the Hardy reference. - _Wolfdieter Lang_, Nov 13 2016]
%F 2/Pi = Product_{n>=2} (p(n) + 2 - (p(n) mod 4))/p(n), where p(n) is the n-th prime. - _Alonso del Arte_, May 16 2012
%F 2/Pi = Sum_{k>=0} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan). - _L. Edson Jeffery_, Mar 23 2013
%F Equals sinc(Pi/2). - _Peter Luschny_, Oct 04 2019
%F From _A.H.M. Smeets_, Apr 11 2020: (Start)
%F Equals Product_{i > 0} cos(Pi/2^(i+1)).
%F Equals Product_{i > 0} f_i(2)/2, where f_0(2) = 0, f_(i+1)(2) = sqrt(2+f_i(2)) for i >= 0; a formula by François Viète (16th century).
%F Note that cos(Pi/2^(i+1)) = f_i(2)/2, i >= 0. (End)
%F Equals lim_{n->infinity} (1/n) * Sum_{k=1..n} abs(sin(k * m)) for all nonzero integers m (conjectured). Works with cos also. - _Dimitri Papadopoulos_, Jul 17 2020
%F From _Amiram Eldar_, Sep 08 2020: (Start)
%F Equals Product_{k>=1} (1 - 1/(2*k)^2).
%F Equals lim_{k->oo} (2*k+1)*binomial(2*k,k)^2/2^(4*k).
%F Equals Sum_{k>=0} binomial{2*k,k)^2/((2*k+2)*2^(4*k)). (End)
%F Equals Sum_{k>=0} mu(4*k+1)/(4*k+1) (Nevanlinna, 1973). - _Amiram Eldar_, Dec 21 2020
%F 2/Pi = 1 - Sum_{n >= 1} (1/16^n) * binomial(2*n, n)^2 * 1/(2*n - 1). See Young, p. 264. - _Peter Bala_, Feb 17 2024
%e 2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...
%p Digits:=100: evalf(2/Pi); # _Wesley Ivan Hurt_, Aug 02 2014
%t RealDigits[ N[ 2/Pi, 111]][[1]]
%o (PARI) default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b060294.txt", n, " ", d)); \\ _Harry J. Smith_, Jul 03 2009
%o (Magma) R:= RealField(100); 2/Pi(R); // _G. C. Greubel_, Mar 09 2018
%Y Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).
%Y Cf. A076668 (sqrt(2/Pi)).
%K cons,nonn
%O 0,1
%A _Jason Earls_, Mar 28 2001
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