1,1

Sometimes called Archimedes's constant.

Ratio of a circle's circumference to its diameter.

Also area of a circle with radius 1.

Also surface area of a sphere with diameter 1.

A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ...

Also ratio of surface area of sphere to one of the faces of the circumscribed cube. Also ratio of volume of a sphere to one of the six inscribed pyramids in the circumscribed cube. - Omar E. Pol, Aug 09 2012

Also surface area of a quarter of a sphere of radius 1. - Omar E. Pol, Oct 03 2013

Also the area under the peak-shaped even function f(x)=1/cosh(x). Proof: for the upper half of the integral, write f(x) = (2*exp(-x))/(1+exp(-2x)) = 2*Sum_{k>=0} (-1)^k*exp(-(2k+1)*x) and integrate term by term from zero to infinity. The result is twice the Gregory series for Pi/4. - Stanislav Sykora, Oct 31 2013

A curiosity: a 144 X 144 magic square of 7th powers was recently constructed by Toshihiro Shirakawa. The magic sum = 3141592653589793238462643383279502884197169399375105, which is the concatenation of the first 52 digits of Pi. See the MultiMagic Squares link for details. - Christian Boyer, Dec 13 2013 [Comment revised by N. J. A. Sloane, Aug 27 2014]

x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 25 2013

Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Jan 13 2014

From Daniel Forgues, Mar 20 2015: (Start)

An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:

  358 0

  359 3

  360 6

  361 0

  362 0

And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...

(End)

Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610) who calculated up to 35 digits of Pi in the late 16th century. - Martin Renner, Sep 07 2016

As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - Harvey P. Dale, Jan 23 2019

On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - David Radcliffe, Apr 10 2019

Also volume of three quarters of a sphere of radius 1. - Omar E. Pol, Aug 16 2019

Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.

P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.

J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.

P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.

Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.

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).

J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.

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

Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020.

Dave Andersen, Pi-Search Page

Anonymous, A million digits of Pi

Anonymous, Liste de quelques milliers de decimales du nombre de pi

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.

Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.

Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009, pp. 903-914.

D. H. Bailey, On Kanada's computation of 1.24 trillion digits of Pi [archived page]

D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Volume 52, Number 5,  May 2005, pp. 502-514.

Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.

J. M. Borwein, Talking about Pi

J. M. Borwein and M. Macklem, The (Digital) Life of Pi, The Australian Mathematical Society Gazette, Volume 33, Number 5, Sept. 2006, pp. 243-248.

Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.

Christian Boyer, MultiMagic Squares

J. Britton, Mnemonics For The Number Pi [archived page]

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.

Jonas Castillo Toloza, Fascinating Method for Finding Pi

E. S. Croot, Pade Approximations and the Transcendence of pi

L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.

L. Euler, De summis serierum reciprocarum, E41.

Eureka, Tout pi or not tout pi

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

Jeremy Gibbons, Unbounded Spigot Algorithms for the Digits of Pi

GJ, 10 million digits of Pi

X. Gourdon, Pi to 16000 decimals [archived page]

Xavier Gourdon, A new algorithm for computing Pi in base 10

X. Gourdon and P. Sebah, Archimedes' constant Pi

B. Gourevitch, L'univers de Pi

L. Grebelius, Approximation of Pi: First 1000000 digits

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270. Preprint: arXiv:math/0506319 [math.NT] (2005-2006).

Carl-Johan Haster, Pi from the sky -- A null test of general relativity from a population of gravitational wave observations, arXiv:2005.05472 [gr-qc], 2020.

H. Havermann, Simple Continued Fraction for Pi [archived page]

M. D. Huberty et al., 100000 Digits of Pi

ICON Project, Pi to 50000 places [archived page]

Emma Haruka Iwao, Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes’ constant on Google Cloud

P. Johns, 120000 Digits of Pi [archived page]

Yasumasa Kanada, 1.24 trillion digits of Pi

Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi [archived page]

Literate Programs, Pi with Machin's formula (Haskell) [archived page]

Johannes W. Meijer, Pi everywhere poster, Mar 14 2013

J. Moyer, First 10000 digits of pi

NERSC, Search Pi [broken link]

Remco Niemeijer, The Digits of Pi, programmingpraxis

Steve Pagliarulo, Stu's pi page [archived page]

I. Peterson, A Passion for Pi

G. M. Phillips, Table of contents of "Pi: A source Book"

Simon Plouffe, 10000 digits of Pi

D. Pothet, Chronologie du calcul des decimales de pi [broken link]

M. Z. Rafat and D. Dobie, Throwing Pi at a wall, arXiv:1901.06260 [physics.class-ph], 2020.

S. Ramanujan, Modular equations and approximations to \pi, Quart. J. Math. 45 (1914), 350-372.

H. Ricardo, Review of "The Number Pi" by P. Eymard & J.-P. Lafon

M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.

Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019).

Daniel B. Sedory, The Pi Pages

D. Shanks and J. W. Wrench, Jr., Calculation of pi to 100,000 decimals, Math. Comp. 16 1962 76-99.

Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI

Sizes, pi

A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.

J. Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004.

D. Surendran, Can I have a small container of coffee? [archived page]

Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002.

G. Vacca, A new analytical expression for the number pi, and some historical considerations, Bull. Amer. Math. Soc. 16 (1910), 368-369.

Stan Wagon, Is Pi Normal?

Eric Weisstein's World of Mathematics, Pi and Pi Digits

Wikipedia, Bailey-Borwein-Plouffe formula, Normal Number, and Pi

Alexander J. Yee & Shigeru Kondo, 5 Trillion Digits of Pi - New World Record

Alexander J. Yee & Shigeru Kondo, Round 2... 10 Trillion Digits of Pi

Index entries for sequences related to the number Pi

Index entries for "core" sequences

Index entries for transcendental numbers

Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - N. J. A. Sloane, Feb 27 2013

From Johannes W. Meijer, Mar 10 2013: (Start)

2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593]

2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655]

Pi = 3*sqrt(3)/4 + 24*(1/12 - Sum_{n>=2} (2*n-2)!/((n-1)!^2*(2*n-3)*(2*n+1)*2^(4*n-2))). [Newton, 1666]

Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706]

Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748]

1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914]

1/Pi = 12*Sum_{n>=0} (-1)^n*(6*n)!*(13591409 + 545140134*n)/((3*n)!*(n!)^3*(64032^3)^(n+1/2)). [David and Gregory Chudnovsky, 1989]

Pi = Sum_{n>=0} (1/16^n) * (4/(8*n+1) - 2/(8*n+4) - 1/(8*n+5) - 1/(8*n+6)). [Bailey-Borwein-Plouffe, 1989] (End)

Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - Alexander R. Povolotsky, Dec 25 2008

Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - Alexander R. Povolotsky, Jan 25 2009

Pi = Integral_{x=-infinity..infinity} dx/(1+x^2). - Mats Granvik and Gary W. Adamson, Sep 23 2012

Pi - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + ... [Jonas Castillo Toloza, 2007], that is, Pi - 2 = Sum_{n>=1} (1/((-1)^floor((n-1)/2)*(n^2+n)/2)). - José de Jesús Camacho Medina, Jan 20 2014

Pi = 3 * Product_{t=img(r),r=(1/2+i*t) root of zeta function} (9+4*t^2)/(1+4*t^2) <=> RH is true. - Dimitris Valianatos, May 05 2016

From Ilya Gutkovskiy, Aug 07 2016: (Start)

Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.

Pi = 2*Product_{k>=2} sec(Pi/2^k).

Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)

Pi = 2^{k + 1}*arctan(sqrt(2 - a_{k - 1})/a_k) at k >= 2, where a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2). - Sanjar Abrarov, Feb 07 2017

a(n) = -10*floor(Pi*10^(-2 + n)) + floor(Pi*10^(-1 + n)) for n > 0. - Mariusz Iwaniuk, Apr 28 2017

Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - Arkadiusz Wesolowski, Nov 20 2017

Pi = lim_{n->infinity} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - Dimitri Papadopoulos, May 31 2019

From Peter Bala, Oct 29 2019: (Start)

Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.

More generally, Pi = 4^x*x!/(2*x)! * Sum_{n >= 0} 2^(n+1)*(n+x)!*(n+2*x)!/(2*n+2*x+1)! = 2*4^x*x!^2/(2*x+1)! * hypergeom([2*x+1,1], [x+3/2], 1/2), valid for complex x not in {-1/2,-1,-3/2,-2,...}. Here, x! is shorthand notation for the function Gamma(x+1). This identity may be proved using Gauss's second summation theorem.

Setting x = 3/4 and x = -1/4 (resp. x = 1/4 and x = -3/4) in the above identity leads to series representations for the constant A085565 (resp. A076390). (End)

Pi = Im(log(-i^i)) = log(i^i)*(-2). - Peter Luschny, Oct 29 2019

From Amiram Eldar, Aug 15 2020: (Start)

Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.

Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.

Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.

Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)

3.1415926535897932384626433832795028841971693993751058209749445923078164062\

862089986280348253421170679821480865132823066470938446095505822317253594081\

284811174502841027019385211055596446229489549303819...

Digits := 110: Pi*10^104:

ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 29 2019

RealDigits[ N[ Pi, 105]] [[1]]

(* PROGRAM STARTS *)

Clear[a, k]

k = RandomInteger[{2, 10^3}];

Print["Random integer k = ", k]

a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 1000]

RealDigits[N[2^(k + 1)*ArcTan[Sqrt[2 - a[k - 1]]/a[k]], 100]][[1]]

(* Sanjar Abrarov, Feb 07 2017 *)

(Macsyma) py(x) := if equal(6, 6+x^2) then 2*x else (py(x:x/3), 3*%%-4*(%%-x)^3); py(3.); py(dfloat(%)); block([bfprecision:35], py(bfloat(%))) /* Bill Gosper, Sep 09 2002 */

(PARI) { default(realprecision, 20080); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b000796.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

(Haskell)  see link: Literate Programs

import Data.Char (digitToInt)

a000796 n = a000796_list (n + 1) !! (n + 1)

a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where

   machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)

   unity = 10 ^ (len + 10)

   arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where

     arccot' x unity summa xpow n sign

      | term == 0 = summa

      | otherwise = arccot'

        x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)

      where term = xpow `div` n

-- Reinhard Zumkeller, Nov 24 2012

(Haskell) See Niemeijer link and also Gibbons link.

a000796 n = a000796_list !! (n-1) :: Int

a000796_list = map fromInteger $ piStream (1, 0, 1)

   [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where

   piStream z xs'@(x:xs)

     | lb /= approx z 4 = piStream (mult z x) xs

     | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'

     where lb = approx z 3

           approx (a, b, c) n = div (a * n + b) c

           mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)

-- Reinhard Zumkeller, Jul 14 2013, Jun 12 2013

(MAGMA) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013

(Python) from sympy import pi, N; print(N(pi, 1000)) # David Radcliffe, Apr 10 2019

Cf. A001203 (continued fraction).

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), this sequence (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A224750 (b=26), A224751 (b=27), A060707 (b=60). - Jason Kimberley, Dec 06 2012

Decimal expansions of expressions involving Pi: A002388 (Pi^2), A003881 (Pi/4), A013661 (Pi^2/6), A019692 (2*Pi=tau), A019727 (sqrt(2*Pi)), A059956 (6/Pi^2), A060294 (2/Pi), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A163973 (Pi/log(2)).

Cf. A001901 (Pi/2; Wallis), A002736 (Pi^2/18; Euler), A007514 (Pi), A048581 (Pi; BBP), A054387 (Pi; Newton), A092798 (Pi/2), A096954 (Pi/4; Machin), A097486 (Pi), A122214 (Pi/2), A133766 (Pi/4 - 1/2), A133767 (5/6 - Pi/4), A166107 (Pi; MGL).

See A245770 for an interesting sieve related to this sequence.

Sequence in context: A247385 A253214 A112602 * A212131 A114609 A271452

Adjacent sequences:  A000793 A000794 A000795 * A000797 A000798 A000799

cons,nonn,nice,core,easy

N. J. A. Sloane

Additional comments from William Rex Marshall, Apr 20 2001

approved