TOPICS
Search

Pi


CirclePi

The constant pi, denoted pi, is a real number defined as the ratio of a circle's circumference C to its diameter d=2r,

pi=C/d
(1)
=C/(2r)
(2)

pi has decimal expansion given by

 pi=3.141592653589793238462643383279502884197...
(3)

(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for pi.

A brief history of notation for pi is given by Castellanos (1988ab). pi is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch pi calculator. The symbol pi was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing 6·2^n-gons on a circle using the Archimedes algorithm. Using n=4 (a 96-gon), Archimedes obtained

 3+(10)/(71)<pi<3+1/7
(4)

(Wells 1986, p. 49; Shanks 1993, p. 140; Borwein et al. 2004, pp. 1-3).

pi is known to be irrational (Lambert 1761; Legendre 1794; Hermite 1873; Nagell 1951; Niven 1956; Struik 1969; Königsberger 1990; Schröder 1993; Stevens 1999; Borwein and Bailey 2003, pp. 139-140). In 1794, Legendre also proved that pi^2 is irrational (Wells 1986, p. 76). pi is also transcendental (Lindemann 1882). An immediate consequence of Lindemann's proof of the transcendence of pi also proved that the geometric problem of antiquity known as circle squaring is impossible. A simplified, but still difficult, version of Lindemann's proof is given by Klein (1955).

It is also known that pi is not a Liouville number (Mahler 1953), but it is not known if pi is normal to any base (Stoneham 1970). The following table summarizes progress in computing upper bounds on the irrationality measure for pi. It is likely that the exponent can be reduced to 2+epsilon, where epsilon is an infinitesimally small number (Borwein et al. 1989).

upper boundreference
20Mahler (1953), Le Lionnais (1983, p. 50)
14.65Chudnovsky and Chudnovsky (1984)
8.0161Hata (1992)
7.606308Salikhov (2008)
7.10320534Zeilberger and Zudilin (2020)

It is not known if pi+e, pi/e, or lnpi are irrational. However, it is known that they cannot satisfy any polynomial equation of degree <=8 with integer coefficients of average size 10^9 (Bailey 1988ab, Borwein et al. 1989).

J. H. Conway has shown that there is a sequence of fewer than 40 fractions F_1, F_2, ... with the property that if you start with 2^n and repeatedly multiply by the first of the F_i that gives an integer result until a power of 2 (say, 2^k) occurs, then k is the nth decimal digit of pi.

pi crops up in all sorts of unexpected places in mathematics besides circles and spheres. For example, it occurs in the normalization of the normal distribution, in the distribution of primes, in the construction of numbers which are very close to integers (the Ramanujan constant), and in the probability that a pin dropped on a set of parallel lines intersects a line (Buffon's needle problem). Pi also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river (Stølum 1996, Singh 1997).

The Bible contains two references (I Kings 7:23 and Chronicles 4:2) which give a value of 3 for pi (Wells 1986, p. 48). It should be mentioned, however, that both instances refer to a value obtained from physical measurements and, as such, are probably well within the bounds of experimental uncertainty. I Kings 7:23 states, "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits in height thereof; and a line thirty cubits did compass it round about." This implies pi=C/d=30/10=3. The Babylonians gave an estimate of pi as 3+1/8=3.125, while the Egyptians gave 2^8/3^4=3.1605... in the Rhind papyrus and 22/7 elsewhere. The Chinese geometers, however, did best of all, rigorously deriving pi to 6 decimal places.

pi appeared in Alfred Hitchcock's insipid and poorly acted 1966 film Torn Curtain, including in one particularly strange but memorable scene where Paul Newman (Professor Michael Armstrong) draws a pi symbol in the dirt with his foot at the door of a farmhouse. In this film, the symbol pi is the pass-sign of an underground East German network that smuggles fugitives to the West.

The 1998 film Pi is a dark, strange, and hyperkinetic movie about a mathematician who is slowly going insane searching for a pattern to the Stock Market. Both a Hasidic cabalistic sect and a Wall Street firm learn of his investigation and attempt to seduce him. Unfortunately, the film has essentially nothing to do with real mathematics. 314159, the first six digits of pi, is the combination to Ellie's office safe in the novel Contact by Carl Sagan.

On Sept. 15, 2005, Google offered exactly 14159265 shares of Class A stock, which is the same as the first eight digits or pi after the decimal point (Markoff 2005).

The formula for the volume of a cylinder leads to the mathematical joke: "What is the volume of a pizza of thickness a and radius z?" Answer: pi z z a. This result is sometimes known as the second pizza theorem.

The 2005 album Aerial features a song called "Pi" in which the first digits of pi are interspersed (unfortunately incorrectly) with lyrics.

There are many, many formulas for pi, from the simple to the very complicated.

Ramanujan (1913-1914) and Olds (1963) give geometric constructions for 355/113. Gardner (1966, pp. 92-93) gives a geometric construction for 3+16/113=3.1415929.... Dixon (1991) gives constructions for 6/5(1+phi)=3.141640... and sqrt(4+[3-tan(30 degrees)]^2)=3.141533.... Constructions for approximations of pi are approximations to circle squaring (which is itself impossible).


See also

Almost Integer, Archimedes Algorithm, BBP Formula, Brent-Salamin Formula, Buffon-Laplace Needle Problem, Buffon's Needle Problem, Circle, Circumference, Diameter, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, e, Euler-Mascheroni Constant, Maclaurin Series, Machin's Formula, Machin-Like Formulas, Normal Distribution, Pi Approximations, Pi Continued Fraction, Pi Digits, Pi Formulas, Pi Wordplay, Radius, Relatively Prime, Riemann Zeta Function, Sphere, Trigonometry Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/Constants/Pi/

Explore with Wolfram|Alpha

References

Almkvist, G. and Berndt, B. "Gauss, Landen, Ramanujan, and Arithmetic-Geometric Mean, Ellipses, pi, and the Ladies Diary." Amer. Math. Monthly 95, 585-608, 1988.Almkvist, G. "Many Correct Digits of pi, Revisited." Amer. Math. Monthly 104, 351-353, 1997.Arndt, J. "Cryptic Pi Related Formulas." http://www.jjj.de/hfloat/pise.dvi.Arndt, J. and Haenel, C. Pi: Algorithmen, Computer, Arithmetik. Berlin: Springer-Verlag, 1998.Arndt, J. and Haenel, C. Pi--Unleashed, 2nd ed. Berlin: Springer-Verlag, 2001.Assmus, E. F. "Pi." Amer. Math. Monthly 92, 213-214, 1985.Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving pi, e, and Euler's Constant." Math. Comput. 50, 275-281, 1988a.Bailey, D. H. "The Computation of pi to 29360000 Decimal Digit using Borwein's' Quartically Convergent Algorithm." Math. Comput. 50, 283-296, 1988b.Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 55 and 274, 1987. Beck, G. and Trott, M. "Calculating Pi from Antiquity to Modern Times." http://library.wolfram.com/infocenter/Demos/107/.Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, 1989.Beeler, M. et al. Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item140.Berggren, L.; Borwein, J.; and Borwein, P. Pi: A Source Book. New York: Springer-Verlag, 1997.Bellard, F. "Fabrice Bellard's Pi Page." http://www-stud.enst.fr/~bellard/pi/.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Blatner, D. The Joy of Pi. New York: Walker, 1997.Blatner, D. "The Joy of Pi." http://www.joyofpi.com/.Borwein, J. M. "Ramanujan Type Series." http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/omlink9/html/node1.html.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987a.Borwein, J. M. and Borwein, P. B. "Ramanujan's Rational and Algebraic Series for 1/pi." Indian J. Math. 51, 147-160, 1987b.Borwein, J. M. and Borwein, P. B. "More Ramanujan-Type Series for 1/pi." In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, pp. 359-374, 1988.Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for 1/pi." J. Comput. Appl. Math. 46, 281-290, 1993.Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi." Amer. Math. Monthly 96, 201-219, 1989.Borwein, P. B. "Pi and Other Constants." http://www.cecm.sfu.ca/~pborwein/PISTUFF/Apistuff.html.Calvet, C. "First Communication. A) Secrets of Pi: Strange Things in a Mathematical Train." http://www.terravista.pt/guincho/1219/1a_index_uk.html.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988a.Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988b.Chan, J. "As Easy as Pi." Math Horizons, pp. 18-19, Winter 1993.Choong, K. Y.; Daykin, D. E.; and Rathbone, C. R. "Rational Approximations to pi." Math. Comput. 25, 387-392, 1971.Chudnovsky, D. V. and Chudnovsky, G. V. Padé and Rational Approximations to Systems of Functions and Their Arithmetic Applications. Berlin: Springer-Verlag, 1984.Chudnovsky, D. V. and Chudnovsky, G. V. "Approximations and Complex Multiplication According to Ramanujan." In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). Boston, MA: Academic Press, pp. 375-472, 1987.Conway, J. H. and Guy, R. K. "The Number pi." In The Book of Numbers. New York: Springer-Verlag, pp. 237-239, 1996.David, Y. "On a Sequence Generated by a Sieving Process." Riveon Lematematika 11, 26-31, 1957.Dixon, R. "The Story of Pi (pi)." §4.3 in Mathographics. New York: Dover, pp. 44-49 and 98-101, 1991.Dunham, W. "A Gem from Isaac Newton." Ch. 7 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 106-112 and 155-183, 1990.Exploratorium. "pi Page." http://www.exploratorium.edu/learning_studio/pi/.Finch, S. R. "Archimedes' Constant." §1.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 17-28, 2003.Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.Gardner, M. "Memorizing Numbers." Ch. 11 in The Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon and Schuster, p. 103, 1959.Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 91-102, 1966.Gosper, R. W. Table of Simple Continued Fraction for pi and the Derived Decimal Approximation. Stanford, CA: Artificial Intelligence Laboratory, Stanford University, Oct. 1975. Reviewed in Math. Comput. 31, 1044, 1977.Gourdon, X. and Sebah, P. "The Constant pi." http://numbers.computation.free.fr/Constants/Pi/pi.html.Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1952.Hata, M. "Improvement in the Irrationality Measures of pi and pi^2." Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-286, 1992.Havermann, H. "180000000 Terms of the Continued Fraction Expansion of Pi." http://odo.ca/~haha/j/seq/cfpi/.Hermite, C. "Sur quelques approximations algébriques." J. reine angew. Math. 76, 342-344, 1873. Reprinted in Oeuvres complètes, Tome III. Paris: Hermann, pp. 146-149, 1912.Hobson, E. W. Squaring the Circle. New York: Chelsea, 1988.Klein, F. Famous Problems. New York: Chelsea, 1955.Knopp, K. §32, 136, and 138 in Theory and Application of Infinite Series. New York: Dover, p. 238, 1990.Königsberger, K. Analysis 1. Berlin: Springer-Verlag, 1990.Laczkovich, M. "On Lambert's Proof of the Irrationality of pi." Amer. Math. Monthly 104, 439-443, 1997.Lambert, J. H. "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques." Mémoires de l'Academie des sciences de Berlin 17, 265-322, 1761.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 22 and 50, 1983.Legendre, A. M. Eléments de géométrie. Paris, France: Didot, 1794.Lindemann, F. "Über die Zahl pi." Math. Ann. 20, 213-225, 1882.Lopez, A. "Indiana Bill Sets the Value of pi to 3." http://db.uwaterloo.ca/~alopez-o/math-faq/node45.html.MacTutor Archive. "Pi Through the Ages." http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html.Mahler, K. "On the Approximation of pi." Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15, 30-42, 1953.Markoff, J. "14,159,265 New Slices of Rich Technology." The New York Times. Aug. 19, 2005.MathPages. "Rounding Up to Pi." http://www.mathpages.com/home/kmath001.htm.Nagell, T. "Irrationality of the numbers e and pi." §13 in Introduction to Number Theory. New York: Wiley, pp. 38-40, 1951.Niven, I. "A Simple Proof that pi is Irrational." Bull. Amer. Math. Soc. 53, 509, 1947.Niven, I. M. Irrational Numbers. New York: Wiley, 1956.Ogilvy, C. S. "Pi and Pi-Makers." Ch. 10 in Excursions in Mathematics. New York: Dover, pp. 108-120, 1994.Olds, C. D. Continued Fractions. New York: Random House, pp. 59-60, 1963.Pappas, T. "Probability and pi." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 18-19, 1989.Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 178-186, 1990.Pickover, C. A. Keys to Infinity. New York: Wiley, p. 62, 1995.Plouffe, S. "Table of Current Records for the Computation of Constants." http://pi.lacim.uqam.ca/eng/records_en.html.Update a linkPlouffe, S. "1 Billion Digits of Pi." http://pi.lacim.uqam.ca/eng/Plouffe, S. "A Few Approximations of Pi." http://pi.lacim.uqam.ca/eng/approximations_en.html.Plouffe, S. "PiHex: A Distributed Effort to Calculate Pi." http://www.cecm.sfu.ca/projects/pihex/.Plouffe, S. "The pi Page." http://www.cecm.sfu.ca/pi/.Plouffe, S. "Table of Computation of Pi from 2000 BC to Now." http://oldweb.cecm.sfu.ca/projects/ISC/Pihistory.html.Preston, R. "Mountains of Pi." New Yorker 68, 36-67, Mar. 2, 1992. http://www.lacim.uqam.ca/~plouffe/Chudnovsky.html.Project Mathematics. "The Story of Pi." Videotape. http://www.projectmathematics.com/storypi.htm.Rabinowitz, S. and Wagon, S. "A Spigot Algorithm for the Digits of pi." Amer. Math. Monthly 102, 195-203, 1995.Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.Rivera, C. "Problems & Puzzles: Puzzle 050-The Best Approximation to Pi with Primes." http://www.primepuzzles.net/puzzles/puzz_050.htm.Rudio, F. "Archimedes, Huygens, Lambert, Legendre." In Vier Abhandlungen über die Kreismessung. Leipzig, Germany, 1892.Sagan, C. Contact. Pocket Books, 1997.Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008.Schröder, E. M. "Zur Irrationalität von pi^2 und pi." Mitt. Math. Ges. Hamburg 13, 249, 1993.Shanks, D. "Dihedral Quartic Approximations and Series for pi." J. Number. Th. 14, 397-423, 1982.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, pp. 17-18, 1997.Sloane, N. J. A. Sequences A000796/M2218, A001203/M2646, A001901, A002485/M3097, A002486/M4456, A006784, A007509/M2061, A025547, A032510, A032523 A033089, A033090, A036903, and A046126 in in "The On-Line Encyclopedia of Integer Sequences."Smith, D. E. "The History and Transcendence of pi." Ch. 9 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 388-416, 1955.Stevens, J. "Zur Irrationalität von pi." Mitt. Math. Ges. Hamburg 18, 151-158, 1999.Stølum, H.-H. "River Meandering as a Self-Organization Process." Science 271, 1710-1713, 1996.Stoneham, R. "A General Arithmetic Construction of Transcendental Non-Liouville Normal Numbers from Rational Functions." Acta Arith. 16, 239-253, 1970.Stoschek, E. "Modul 33: Algames with Numbers" http://marvin.sn.schule.de/~inftreff/modul33/task33.htm.Struik, D. A Source Book in Mathematics, 1200-1800. Cambridge, MA: Harvard University Press, 1969.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 159, 1991.Viète, F. Uriorum de rebus mathematicis responsorum, liber VIII, 1593.Wagon, S. "Is pi Normal?" Math. Intel. 7, 65-67, 1985.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 48-55 and 76, 1986.Whitcomb, C. "Notes on Pi (pi)." http://witcombe.sbc.edu/earthmysteries/EMPi.html.Woon, S. C. "Problem 1441." Math. Mag. 68, 72-73, 1995.Zeilberger, D. and Zudilin, W. "The Irrationality Measure of pi is at Most 7.103205334137...." 8 Jan 2020. https://arxiv.org/abs/1912.06345.

Referenced on Wolfram|Alpha

Pi

Cite this as:

Weisstein, Eric W. "Pi." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pi.html

Subject classifications