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{{about|the number {{pi}}|the Greek letter|Pi (letter)|other uses of pi, π, and Π|Pi (disambiguation)}}
{{Use dmy dates|date=July 2012}}
{{Pi box}}
The number '''{{pi}}''' is a [[mathematical constant]], the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]], commonly approximated as 3.14159. It has been represented by the Greek letter "{{pi}}" since the mid-18th century, though it is also sometimes spelled out as "'''pi'''" ({{IPAc-en|p|aɪ}}).

Being an [[irrational number]], {{pi}} cannot be expressed exactly as a [[common fraction|fraction]] (equivalently, its [[decimal representation]] never ends and never [[repeating decimal|settles into a permanent repeating pattern]]). Still, fractions such as 22/7 and other rational numbers are commonly used to [[Approximations of π|approximate]] {{pi}}. The digits appear to be randomly distributed; however, to date, no proof of this has been discovered. Also, {{pi}} is a [[transcendental number]]&nbsp;– a number that is not the [[Root of a polynomial|root]] of any non-zero [[polynomial]] having [[Rational number|rational]] [[coefficient]]s. This transcendence of {{pi}} implies that it is impossible to solve the ancient challenge of [[squaring the circle]] with a [[Compass-and-straightedge construction|compass and straightedge]].

Ancient civilizations needed the value of {{pi}} to be computed accurately for practical reasons.  It was calculated to seven digits, using geometrical techniques, in [[History of mathematics#Chinese mathematics|Chinese mathematics]] and to about five in [[History of mathematics#Indian mathematics|Indian mathematics]] in the 5th century CE. The historically first exact formula for {{pi}}, based on [[Series (mathematics)|infinite series]], was not available until a millennium later, when in the 14th century the [[Leibniz formula for pi|Madhava–Leibniz series]] was discovered in Indian mathematics.<ref>{{Cite book|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=[[Cambridge University Press]]|year=1999|isbn=0-521-78988-5|page=58|postscript=<!--None-->}}</ref><ref>{{Cite journal|first=R. C.|last=Gupta|title=On the remainder term in the Madhava–Leibniz's series|journal=Ganita Bharati|volume=14|issue=1-4|year=1992|pages=68–71|postscript=<!--None-->}}</ref> In the 20th and 21st centuries, mathematicians and [[computer science|computer scientists]] discovered new approaches that, when combined with increasing computational power, extended the decimal representation of {{pi}} to, as of 2015, over 13.3 trillion (10<sup>13</sup>) digits.<ref>[http://www.numberworld.org/y-cruncher/ y-cruncher - A Multi-Threaded Pi Program<!-- Bot generated title -->]</ref> Practically all scientific applications require no more than a few hundred digits of {{pi}}{{discuss|section=Resolving the typical precision required for scientific calculations}}, and many substantially fewer, so the primary motivation for these computations is the human desire to break records.<ref>{{harvnb|Arndt|Haenel|2006|p=17}}</ref><ref>{{citation|author1=David Bailey|author2=Jonathan Borwein|author3=Peter Borwein|author4=Simon Plouffe|title=The Quest for Pi|journal=The Mathematical Intelligencer|year=1997|volume=19|issue=1|pages=50–56|doi=10.1007/bf03024340}}</ref> However, the extensive calculations involved have been used to test [[supercomputer]]s and high-precision multiplication [[algorithms]].

Because its definition relates to the circle, {{pi}} is found in many formulae in [[trigonometry]] and [[geometry]], especially those concerning circles, ellipses or spheres. It is also found in formulae used in other branches of science such as [[cosmology]], [[number theory]], [[statistics]], [[fractal]]s, [[thermodynamics]], [[mechanics]] and [[electromagnetism]]. The ubiquity of {{pi}} makes it one of the most widely known mathematical constants both inside and outside the scientific community: Several books devoted to it have been published, the number is celebrated on [[Pi Day]] and record-setting calculations of the digits of {{pi}} often result in news headlines. Attempts to memorize the value of {{pi}} with increasing precision have led to records of over 67,000 digits.

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==Fundamentals==

===Name===
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase [[Pi (letter)|Greek letter {{pi}}]], sometimes spelled out as ''pi,'' and derived from the first letter of the Geek word ''perimetros,'' meaning circumference.<ref>{{cite journal |last=Boeing |first=Niels |date=14 March 2016 |url=http://www.zeit.de/zeit-wissen/2016/02/pi-tag-mathematik-pi-kreiszahl |title=Die Welt ist Pi |trans_title=The World is Pi |language=de |journal=Zeit Online |quote=''Die Ludolphsche Zahl oder Kreiszahl erhielt nun auch das Symbol, unter dem wir es heute kennen: William Jones schlug 1706 den griechischen Buchstaben π vor, in Anlehnung an perimetros, griechisch für Umfang. Leonhard Euler etablierte π schließlich in seinen mathematischen Schriften.'' [The Ludolphsche number or Kreiszahl now also received the symbol under which we know it today: William Jones proposed in 1706 the Greek letter π, based on perimetros [περίμετρος], Greek for perimeter. Leonhard Euler firmly established π in his mathematical writings.]}}</ref> In English, {{pi}} is [[English pronunciation of Greek letters|pronounced as "pie"]] ( {{IPAc-en|p|aɪ}}, {{respell|paɪ}}).<ref>{{cite web|url=http://dictionary.reference.com/browse/pi?s=t|title=pi|publisher=Dictionary.reference.com|date=2 March 1993|accessdate=18 June 2012}}</ref> In mathematical use, the lowercase letter {{pi}} (or π in [[sans-serif]] font) is distinguished from its capital counterpart {{PI}}, which denotes a [[product of a sequence]].

The choice of the symbol {{pi}} is discussed in the section [[#Adoption of the symbol π|''Adoption of the symbol {{pi}}'']].

===Definition===
[[File:Test.png|alt=A diagram of a circle, with the width labeled as diameter, and the perimeter labeled as circumference test|thumb|right|ssssssssssssssssssssssssssssssssssssssssssssss]]
{{pi}} is commonly defined as the [[ratio]] of a [[circle]]'s [[circumference]] {{MATH|''C''}} to its [[diameter]] {{MATH|''d''}}:<ref name="Arndt">{{harvnb|Arndt|Haenel|2006|p=8}}</ref>
:<math> \pi = \frac{C}{d}</math>
The ratio {{MATH|''C''/''d''}} is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio {{MATH|''C''/''d''}}. This definition of {{pi}} implicitly makes use of [[Euclidean geometry|flat (Euclidean) geometry]]; although the notion of a circle can be extended to any [[Non-Euclidean geometry|curved (non-Euclidean) geometry]], these new circles will no longer satisfy the formula {{MATH|{{pi}} {{=}} ''C''/''d''}}.<ref name="Arndt" />

Here, the circumference of a circle is the [[arc length]] around the perimeter of the circle, a quantity which can be formally defined independently of geometry using [[limit (mathematics)|limit]]s, a concept in [[calculus]].<ref>{{citation|author=[[Tom Apostol]]|title=Calculus, volume 1|publisher=Wiley|edition=2nd|year=1967}}. Page 102: "From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length."  Arc length is introduced on page 529.</ref>  For example, one may compute directly the arc length of the top half of the unit circle given in [[Cartesian coordinates]] by <math>x^2+y^2=1</math>, as the [[integral]]:<ref name="Reinhold Remmert 1991 129">{{citation|first=Reinhold|last=Remmert|chapter=What is {{pi}}?|title=Numbers|publisher=Springer|year=1991|page=129}}</ref>
:<math>\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}.</math>
An integral such as this was adopted as the definition of {{pi}} by [[Karl Weierstrass]], who defined it directly as an integral in 1841.<ref>{{harvtxt|Remmert|1991}}.  The precise integral that Weierstrass used was <math>\pi=\int_{-\infty}^\infty\frac{dx}{1+x^2}.</math></ref>

Definitions of {{pi}} such as these that rely on a notion of circumference, and hence implicitly on concepts of the [[integral calculus]], are no longer common in the literature.  {{harvtxt|Remmert|1991}} explains that this is because in many modern treatments of calculus, [[differential calculus]] typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of {{pi}} that does not rely on the latter.  One such definition, due to [[Richard Baltzer]],<ref>{{citation|author=Richard Baltzer|title=Die Elemente der Mathematik|year=1870|page=195|url=https://archive.org/details/dieelementederm02baltgoog|publisher=Hirzel}}</ref> and popularized by [[Edmund Landau]],<ref>{{citation|author=Edmund Landau|title=Einführung in die Differentialrechnung und Integralrechnung|publisher=Noordoff|year=1934|page=193}}</ref> is the following: {{pi}} is twice the smallest positive number at which the [[cosine|cosine function]] equals 0.<ref name="Arndt" /><ref name="Reinhold Remmert 1991 129"/><ref name="Rudin 1976">{{cite book|last=Rudin|first=Walter|title=Principles of Mathematical Analysis|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X|ref=harv}}, p 183.</ref>  The cosine can be defined independently of geometry as a [[power series]],<ref>{{cite book|last=Rudin|first=Walter|title=Real and complex analysis|publisher=McGraw-Hill|year=1986|ref=harv}}, p 2.</ref> or as the solution of a [[differential equation]].<ref name="Rudin 1976"/>

In a similar spirit, {{pi}} can be defined instead using properties of the [[complex exponential]], {{MATH|exp(''z'')}}, of a [[complex number|complex]] variable {{MATH|''z''}}.  Like the cosine, the complex exponential can be defined in one of several ways.  The set of complex numbers at which {{MATH|exp(''z'')}} is equal to one is then an (imaginary) arithmetic progression of the form:
:<math>\{\dots,-2\pi i, 0, 2\pi i, 4\pi i,\dots\} = \{2\pi ki| k\in\mathbb Z\}</math>
and there is a unique positive real number {{pi}} with this property.<ref name="Reinhold Remmert 1991 129"/><ref>{{citation|author=[[Lars Ahlfors]]|title=Complex analysis|publisher=McGraw-Hill|year=1966|p=46}}</ref>
A more abstract variation on the same idea, making use of sophisticated mathematical concepts of [[topology]] and [[algebra]], is the following theorem:<ref>{{citation|author=[[Nicolas Bourbaki]]|title=Topologie generale|publisher=Springer|year=1981}}, §VIII.2</ref> there is a unique [[continuous function|continuous]] [[isomorphism]] from the [[group (mathematics)|group]] <math>\mathbb R/\mathbb Z</math> of real numbers under addition [[quotient group|modulo]] integers (the [[circle group]]) onto the multiplicative group of [[complex numbers]] of [[absolute value]] one.  The number {{pi}} is then defined as half the magnitude of the derivative of this homomorphism.<ref>{{citation|author=[[Nicolas Bourbaki]]|title=Fonctions d'une variable réelle|publisher=Springer|year=1979}}, §II.3.</ref>

===Properties===
{{pi}} is an [[irrational number]], meaning that it cannot be written as the [[rational number|ratio of two integers]] (fractions such as {{MATH|{{sfrac|22|7}}}} are commonly used to approximate {{pi}}; no [[common fraction]] (ratio of whole numbers) can be its exact value).<ref name="Arndt_i">{{harvnb|Arndt|Haenel|2006|p=5}}</ref> Because {{pi}} is irrational, it has an infinite number of digits in its [[decimal representation]], and it does not settle into an infinitely [[repeating decimal|repeating pattern]] of digits. There are several [[proof that π is irrational|proofs that {{pi}} is irrational]]; they generally require calculus and rely on the [[reductio ad absurdum]] technique. The degree to which {{pi}} can be approximated by [[rational number]]s (called the [[irrationality measure]]) is not precisely known; estimates have established that the irrationality measure is larger than the measure of {{MATH|''e''}} or {{MATH|ln(2)}} but smaller than the measure of [[Liouville number]]s.<ref>{{cite journal|last1=Salikhov|first1=V.|year=2008|title=On the Irrationality Measure of pi|journal=Russian Mathematical Survey|volume=53|issue=3|pages=570–572|ref=harv|doi=10.1070/RM2008v063n03ABEH004543|bibcode = 2008RuMaS..63..570S }}</ref>
[[File:Squaring the circle.svg|thumb|alt=A diagram of a square and circle, both with identical area; the length of the side of the square is the square root of pi|Because {{pi}} is a [[transcendental number]], [[squaring the circle]] is not possible in a finite number of steps using the classical tools of [[Compass-and-straightedge construction|compass and straightedge]].]] More strongly, {{pi}} is a [[transcendental number]], which means that it is not the [[root of a function|solution]] of any non-constant [[polynomial]] with [[rational number|rational]] coefficients, such as&nbsp;{{MATH|{{sfrac|''x''<sup>5</sup>|120}}&nbsp;&minus;&nbsp;{{sfrac|''x''<sup>3</sup>|6}}&nbsp;+&nbsp;''x''&nbsp;{{=}}&nbsp;0}}.<ref name="ttop">{{cite web |first=Steve |last=Mayer |url=http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html |archive-url=https://web.archive.org/web/20000929033317/http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html |archive-date=2000-09-29 |title=The Transcendence of {{pi}} |accessdate=4 November 2007 }} {{Webarchive|url=https://web.archive.org/web/20000929033317/http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html |date=29 September 2000 }}</ref><ref>The polynomial shown is the first few terms of the [[Taylor series]] expansion of the [[sine]] function.</ref>

The transcendence of {{pi}} has two important consequences: First, {{pi}} cannot be expressed using any finite combination of rational numbers and square roots or [[n-th root|''n''-th roots]] such as {{MATH|{{radic|31|3}}}} or {{MATH|{{sqrt|10}}}}. Second, since no transcendental number can be [[Constructible number|constructed]] with [[Compass-and-straightedge construction|compass and straightedge]], it is not possible to "[[squaring the circle|square the circle]]". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.<ref>{{harvnb|Posamentier|Lehmann|2004|p=25}}</ref> Squaring a circle was one of the important geometry problems of the [[classical antiquity]].<ref>{{harvnb|Eymard|Lafon|1999|p=129}}</ref> Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is impossible.<ref>{{harvnb|Beckmann|1989|p=37}}<br />{{cite book|last=Schlager|first=Neil|last2=Lauer|first2=Josh|title=Science and Its Times: Understanding the Social Significance of Scientific Discovery|publisher=Gale Group|year=2001|isbn=0-7876-3933-8|ref=harv}}, p 185.</ref>

The digits of {{pi}} have no apparent pattern and have passed tests for [[statistical randomness]], including tests for [[normal number|normality]]; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.<ref name="random" /> The conjecture that {{pi}} is normal has not been proven or disproven.<ref name="random">{{harvnb|Arndt|Haenel|2006|pp=22–23}}<br />{{cite news|url=http://www.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are The Digits of Pi Random? Lab Researcher May Hold The Key|first=Paul|last=Preuss|authorlink=Paul Preuss (author)|publisher=[[Lawrence Berkeley National Laboratory]]|date=23 July 2001|accessdate=10 November 2007|archive-date=20 October 2007|archive-url=https://web.archive.org/web/20071020010208/http://lbl.gov/Science-Articles/Archive/pi-random.html|url-status=dead}} {{Webarchive|url=https://web.archive.org/web/20071020010208/http://lbl.gov/Science-Articles/Archive/pi-random.html |date=20 October 2007 }}</ref> Since the advent of computers, a large number of digits of {{pi}} have been available on which to perform statistical analysis. [[Yasumasa Kanada]] has performed detailed statistical analyses on the decimal digits of {{pi}} and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to [[statistical significance test]]s, and no evidence of a pattern was found.<ref>{{harvnb|Arndt|Haenel|2006|pp=22, 28–30}}</ref> Despite the fact that {{pi}}'s digits pass statistical tests for randomness, {{pi}} contains some sequences of digits that may appear non-random to non-mathematicians, such as a [[Six nines in pi|sequence of six consecutive 9s]] that begins at the 762nd decimal place of the decimal representation of {{pi}}.<ref>{{harvnb|Arndt|Haenel|2006|p=3}}</ref>

===Continued fractions===
[[File:Matheon2.jpg|thumb|alt=A photograph of the Greek letter pi, created as a large stone mosaic embedded in the ground.|The constant {{pi}} is represented in this [[mosaic]] outside the Mathematics Building at the [[Technical University of Berlin]].]]
Like all irrational numbers, {{pi}} cannot be represented as a [[common fraction]] (also known as a [[Simple fractions|simple]] or [[vulgar fraction]]), by the very definition of "irrational". But every irrational number, including {{pi}}, can be represented by an infinite series of nested fractions, called a [[continued fraction]]:

:<math>
\pi=3+\textstyle \frac{1}{7+\textstyle \frac{1}{15+\textstyle \frac{1}{1+\textstyle \frac{1}{292+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\ddots}}}}}}}</math>
{{OEIS2C|id=A001203}}

Truncating the continued fraction at any point yields a rational approximation for {{pi}}; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to {{pi}} than any other fraction with the same or a smaller denominator.<ref name="Eymard 1999 78">{{harvnb|Eymard|Lafon|1999|p=78}}</ref> Because {{pi}} is known to be transcendental, it is by definition not algebraic and so cannot be a [[quadratic irrational]]. Therefore, {{pi}} cannot have a [[periodic continued fraction]]. Although the simple continued fraction for {{pi}} (shown above) also does not exhibit any other obvious pattern,<ref name="ReferenceA">
{{SloanesRef|sequencenumber=A001203|name=Continued fraction for Pi}} Retrieved 12 April 2012.</ref> mathematicians have discovered several [[generalized continued fraction]]s that do, such as:<ref>{{cite journal|title=An Elegant Continued Fraction for {{pi}}|first=L. J.|last=Lange|journal=[[The American Mathematical Monthly]]|volume=106|issue=5| date=May 1999 |pages=456–458|jstor=2589152|doi=10.2307/2589152|ref=harv}}</ref>
:<math>\pi=\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}
=3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}}
=\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}}</math>

===Approximate value===
Some [[approximations of π|approximations of ''pi'']] include:
* '''Integers''': [[3 (number)|3]]
* '''Fractions''': Approximate fractions include (in order of increasing accuracy) {{sfrac|22|7}}, {{sfrac|333|106}}, {{sfrac|355|113}}, {{sfrac|52163|16604}}, {{sfrac|103993|33102}}, and {{sfrac|245850922|78256779}}.<ref name="Eymard 1999 78" /> (List is selected terms from {{OEIS2C|id=A063674}} and {{OEIS2C|id=A063673}}.)
* '''Decimal''': The first 50 decimal digits are {{gaps|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510...}}<ref>{{harvnb|Arndt|Haenel|2006|p=240}}</ref> {{OEIS2C|id=A000796}}
* '''[[Binary numeral system|Binary]]''': The [[Radix|base]] 2 approximation to 48 digits is {{gaps|11.0010|0100|0011|1111|0110|1010|1000|1000|1000|0101|1010|0011...}}
* '''[[Hexadecimal]]''': The [[Radix|base]] 16 approximation to 20 digits is {{gaps|3.243F|6A88|85A3|08D3|1319...}}<ref>{{harvnb|Arndt|Haenel|2006|p=242}}</ref>
* '''[[Sexagesimal]]''': A base 60 approximation to five sexagesimal digits is 3;8,29,44,0,47<ref>{{citation|title=Abu-r-Raihan al-Biruni, 973-1048|last=Kennedy|first=E. S.|journal=Journal for the History of Astronomy|volume=9|page=65|bibcode=1978JHA.....9...65K|doi=10.1177/002182867800900106}}. [[Ptolemy]] used a three-sexagesimal-digit approximation, and [[Jamshīd al-Kāshī]] expanded this to nine digits; see {{Citation |last= Aaboe |first= Asger |authorlink = Asger Aaboe |year= 1964 |title= Episodes from the Early History of Mathematics |series = New Mathematical Library |volume = 13 |publisher= Random House |publication-place= New York |page=125|url=https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA125}}</ref>

==History== <!-- [[History of pi]] links here -->
{{main|Approximations of π{{!}}Approximations of {{pi}}}}
{{see also|Chronology of computation of π|l1=Chronology of computation of {{pi}}}}

===Antiquity===
The best known approximations to {{pi}} dating [[1st millennium BC|before the Common Era]] were accurate to two decimal places; this was improved upon in [[Chinese mathematics]] in particular by the mid first millennium, to an accuracy of seven decimal places.
After this, no further progress was made until the late medieval period.

Some Egyptologists<ref>Petrie, W.M.F. ''Wisdom of the Egyptians'' (1940)</ref> have claimed that the [[ancient Egypt]]ians used an approximation of {{pi}} as {{sfrac|22|7}}&nbsp;from as early as the [[Old Kingdom]].<ref>Based on the [[Great Pyramid of Giza]], supposedly built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760 [[cubits]] around and 280 cubits in height). Verner, Miroslav. ''The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments.'' Grove Press. 2001 (1997). ISBN 0-8021-3935-3</ref>
This claim has met with skepticism.<ref name=Rossi>Rossi, ''Corinna Architecture and Mathematics in Ancient Egypt,'' Cambridge University Press. 2007. ISBN 978-0-521-69053-9.</ref><ref>Legon, J. A. R. ''On Pyramid Dimensions and Proportions'' (1991) Discussions in Egyptology (20) 25–34 [http://www.legon.demon.co.uk/pyrprop/propde.htm] {{Webarchive|url=https://web.archive.org/web/20110718144356/http://www.legon.demon.co.uk/pyrprop/propde.htm |date=18 July 2011 }}</ref><ref>"We can conclude that although the ancient Egyptians could not precisely define the value of {{pi}}, in practice they used it". {{cite journal|last=Verner|first=M.|title=The Pyramids: Their Archaeology and History|year=2003|ref=harv}}, p. 70.<br />{{cite journal|last=Petrie|first=|title=Wisdom of the Egyptians|year=1940|ref=harv}}, p. 30.<br />See also {{cite journal|last=Legon|first=J. A. R.|title=On Pyramid Dimensions and Proportions|year=1991|journal=Discussions in Egyptology|volume=20|issue=|pages=25–34|url=http://www.legon.demon.co.uk/pyrprop/propde.htm|ref=harv|access-date=20 March 2016|archive-date=18 July 2011|archive-url=https://web.archive.org/web/20110718144356/http://www.legon.demon.co.uk/pyrprop/propde.htm|url-status=dead}}.<br />See also {{cite journal|last=Petrie|first=W. M. F.|year=1925|title=Surveys of the Great Pyramids|journal=Nature |volume=116|issue=2930|pages=942–942|ref=harv|doi=10.1038/116942a0|bibcode=1925Natur.116..942P}}</ref><ref>Egyptologist: Rossi, Corinna, ''Architecture and Mathematics in Ancient Egypt'', Cambridge University Press, 2004, pp 60–70, 200, ISBN 9780521829540.<br>Skeptics: [[Michael Shermer|Shermer, Michael]], ''The Skeptic Encyclopedia of Pseudoscience'', ABC-CLIO, 2002, pp 407–408, ISBN 9781576076538.<br />See also Fagan, Garrett G., ''Archaeological Fantasies: How Pseudoarchaeology Misrepresents The Past and Misleads the Public'', Routledge, 2006, ISBN 9780415305938.<br />For a list of explanations for the shape that do not involve {{pi}}, see {{Cite book|pages=67–77, 165–166|title=The Shape of the Great Pyramid|author=Roger Herz-Fischler|publisher=Wilfrid Laurier University Press|year=2000|isbn=9780889203242|url=https://books.google.com/books?id=066T3YLuhA0C&pg=67|ref=harv|accessdate=2013-06-05}}</ref>

The earliest written approximations of {{pi}} are found in [[Ancient Egypt|Egypt]] and [[Babylon]], both within one percent of the true value. In Babylon, a [[clay tablet]] dated 1900–1600&nbsp;BC has a geometrical statement that, by implication, treats {{pi}} as {{sfrac|25|8}}&nbsp;=&nbsp;3.1250.<ref name="Arndt_d">{{harvnb|Arndt|Haenel|2006|p=167}}</ref> In Egypt, the [[Rhind Papyrus]], dated around 1650&nbsp;BC but copied from a document dated to 1850&nbsp;BC, has a formula for the area of a circle that treats {{pi}} as ({{sfrac|16|9}})<sup>2</sup>&nbsp;≈&nbsp;3.1605.<ref name="Arndt_d" />

Astronomical calculations in the ''[[Shatapatha Brahmana]]'' (ca. 4th century BC) use a fractional approximation of {{sfrac|339|108}}&nbsp;≈&nbsp;3.139 (an accuracy of 9×10<sup>−4</sup>).<ref>Chaitanya, Krishna. [https://books.google.com/books?ei=9h35T9PdIoHQrQeO4ZXhBg&id=hDc8AAAAMAAJ&dq=Satapatha+Brahmana+value+of+pi&q=pi#search_anchor A profile of Indian culture.] Indian Book Company (1975). p.133.</ref> Other Indian sources by about 150&nbsp;BC treat {{pi}} as {{math.|{{sqrt|10}}}}&nbsp;≈&nbsp;3.1622<ref>{{harvnb|Arndt|Haenel|2006|p=169}}</ref>

===Polygon approximation era===
[[File:Archimedes pi.svg|350px|right|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|{{pi}} can be estimated by computing the perimeters of circumscribed and inscribed polygons.]]
The first recorded algorithm for rigorously calculating the value of {{pi}} was a geometrical approach using polygons, devised around 250&nbsp;BC by the Greek mathematician [[Archimedes]].<ref>{{harvnb|Arndt|Haenel|2006|p=170}}</ref> This polygonal algorithm dominated for over 1,000 years, and as a result {{pi}} is sometimes referred to as "Archimedes' constant".<ref>{{harvnb|Arndt|Haenel|2006|pp=175, 205}}</ref> Archimedes computed upper and lower bounds of {{pi}} by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that {{MATH|{{sfrac|223|71}} < {{pi}} < {{sfrac|22|7}}}} (that is {{MATH|3.1408 < {{pi}} < 3.1429}}).<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html#37|title=The Computation of Pi by Archimedes: The Computation of Pi by Archimedes – File Exchange – MATLAB Central|publisher=Mathworks.com|accessdate=2013-03-12|archive-date=25 February 2013|archive-url=https://web.archive.org/web/20130225181030/http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html#37}}</ref> Archimedes' upper bound of {{MATH|{{sfrac|22|7}}}} may have led to a widespread popular belief that {{pi}} is equal to {{MATH|{{sfrac|22|7}}}}.<ref>{{harvnb|Arndt|Haenel|2006|p=171}}</ref> Around 150 AD, Greek-Roman scientist [[Ptolemy]], in his ''[[Almagest]]'', gave a value for {{pi}} of 3.1416, which he may have obtained from Archimedes or from [[Apollonius of Perga]].<ref>{{harvnb|Arndt|Haenel|2006|p=176}}<br />{{harvnb|Boyer|Merzbach|1991|p=168}}<!--may be suspect--></ref> Mathematicians using polygonal algorithms reached 39 digits of {{pi}} in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.<ref name="ArPI">{{harvnb|Arndt|Haenel|2006|pp=15–16, 175, 184–186, 205}}. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.</ref>
[[File:Domenico-Fetti Archimedes 1620.jpg|thumb|upright|alt=A painting of a man studying|[[Archimedes]] developed the polygonal approach to approximating {{pi}}.]]

In [[ancient China]], values for {{pi}} included 3.1547 (around 1 AD), {{MATH|{{sqrt|10}}}} (100 AD, approximately 3.1623), and {{MATH|{{sfrac|142|45}}}} (3rd century, approximately 3.1556).<ref>{{harvnb|Arndt|Haenel|2006|pp=176–177}}</ref> Around 265 AD, the [[Wei Kingdom]] mathematician [[Liu Hui]] created a [[Liu Hui's π algorithm|polygon-based iterative algorithm]] and used it with a 3,072-sided polygon to obtain a value of {{pi}} of&nbsp;3.1416.<ref name="autogenerated202">{{harvnb|Boyer|Merzbach|1991|p=202}}</ref><ref>{{harvnb|Arndt|Haenel|2006|p=177}}</ref> Liu later invented a faster method of calculating {{pi}} and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of&nbsp;4.<ref name="autogenerated202" /> The Chinese mathematician [[Zu Chongzhi]], around 480 AD, calculated that {{MATH|{{pi}} ≈ {{sfrac|355|113}}}} (a fraction that goes by the name ''[[Milü]]'' in Chinese), using [[Liu Hui's π algorithm|Liu Hui's algorithm]] applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of {{pi}} available for the next 800 years.<ref>{{harvnb|Arndt|Haenel|2006|p=178}}</ref>

The Indian astronomer [[Aryabhata]] used a value of 3.1416 in his ''[[Āryabhaṭīya]]'' (499 AD).<ref>{{harvnb|Arndt|Haenel|2006|pp=179}}</ref> [[Fibonacci]] in c.&nbsp;1220 computed 3.1418 using a polygonal method, independent of Archimedes.<ref name="Arndt_e">{{harvnb|Arndt|Haenel|2006|pp=180}}</ref> Italian author [[Dante]] apparently employed the value {{MATH|3+{{sfrac|{{sqrt|2}}|10}} ≈ 3.14142}}.<ref name="Arndt_e" />

The Persian astronomer [[Jamshīd al-Kāshī]] produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×2<sup>28</sup> sides,<ref>{{cite journal| first1=Mohammad K.|last1=Azarian|title=al-Risāla al-muhītīyya: A Summary|journal=Missouri Journal of Mathematical Sciences|volume=22|issue=2|year=2010|pages=64–85|url=http://projecteuclid.org/euclid.mjms/1312233136|ref=harv}}</ref><ref>{{cite web|author=O'Connor, John J.; Robertson, Edmund F.|year=1999|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi|work=[[MacTutor History of Mathematics archive]]|url=http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html|accessdate=August 11, 2012}}</ref> which stood as the world record for about 180 years.<ref name="Arndt_f">{{harvnb|Arndt|Haenel|2006|p=182}}</ref> French mathematician [[François Viète]] in 1579 achieved 9 digits with a polygon of 3×2<sup>17</sup> sides.<ref name="Arndt_f" /> Flemish mathematician [[Adriaan van Roomen]] arrived at 15 decimal places in 1593.<ref name="Arndt_f" /> In 1596, Dutch mathematician [[Ludolph van Ceulen]] reached 20 digits, a record he later increased to 35 digits (as a result, {{pi}} was called the "Ludolphian number" in Germany until the early 20th century).<ref>{{harvnb|Arndt|Haenel|2006|pp=182–183}}</ref> Dutch scientist [[Willebrord Snellius]] reached 34 digits in 1621,<ref name="Arndt_g">{{harvnb|Arndt|Haenel|2006|p=183}}</ref> and Austrian astronomer [[Christoph Grienberger]] arrived at 38 digits in 1630 using 10<sup>40</sup> sides,<ref>{{cite book|first=Christophorus|last=Grienbergerus|authorlink=Christoph Grienberger|language=Latin|year=1630|title=Elementa Trigonometrica|url=http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf|archiveurl=https://web.archive.org/web/20140201234124/http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf|archivedate=2014-02-01|format=PDF}} His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < {{pi}} < 3.14159 26535 89793 23846 26433 83279 50288 4199.</ref> which remains the most accurate approximation manually achieved using polygonal algorithms.<ref name="Arndt_g" />

===Infinite series===
{{comparison_pi_infinite_series.svg}}
The calculation of {{pi}} was revolutionized by the development of [[infinite series]] techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite [[sequence (mathematics)|sequence]].<ref name="Ais">{{harvnb|Arndt|Haenel|2006|pp=185–191}}</ref> Infinite series allowed mathematicians to compute {{pi}} with much greater precision than [[Archimedes]] and others who used geometrical techniques.<ref name="Ais" /> Although infinite series were exploited for {{pi}} most notably by European mathematicians such as [[James Gregory (mathematician)|James Gregory]] and [[Gottfried Wilhelm Leibniz]], the approach was first discovered in [[India]] sometime between 1400 and 1500 AD.<ref>{{harvnb|Roy|1990|pp=101–102}}<br />{{harvnb|Arndt|Haenel|2006|pp=185–186}}</ref> The first written description of an infinite series that could be used to compute {{pi}} was laid out in Sanskrit verse by Indian astronomer [[Nilakantha Somayaji]] in his ''[[Tantrasamgraha]]'', around 1500 AD.<ref name="Roypp">{{harvnb|Roy|1990|pp=101–102}}</ref> The series are presented without proof, but proofs are presented in a later Indian work, ''[[Yuktibhāṣā]]'', from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, [[Madhava of Sangamagrama]], who lived c.&nbsp;1350&nbsp;– c.&nbsp;1425.<ref name="Roypp" /> Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the [[Madhava series]] or [[Leibniz formula for π|Gregory–Leibniz series]].<ref name="Roypp" /> Madhava used infinite series to estimate {{pi}} to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician [[Jamshīd al-Kāshī]], using a polygonal algorithm.<ref>{{harvnb|Joseph|1991|p=264}}</ref>
[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright|alt=A formal portrait of a man, with long hair|[[Isaac Newton]]
used [[infinite series]] to compute {{pi}} to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".<ref name="Newton" />]]

The [[Viète's formula|first infinite sequence discovered in Europe]] was an [[infinite product]] (rather than an [[infinite sum]], which are more typically used in {{pi}} calculations) found by French mathematician [[François Viète]] in 1593:<ref name="Arndt_h">{{harvnb|Arndt|Haenel|2006|p=187}}</ref>

: <math> \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots</math> {{OEIS2C|id=A060294}}

The [[Wallis product|second infinite sequence found in Europe]], by [[John Wallis]] in 1655, was also an infinite product.<ref name="Arndt_h" /> The discovery of [[calculus]], by English scientist [[Isaac Newton]] and German mathematician [[Gottfried Wilhelm Leibniz]] in the 1660s, led to the development of many infinite series for approximating {{pi}}. Newton himself used an arcsin series to compute a 15 digit approximation of {{pi}} in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."<ref name="Newton">{{harvnb|Arndt|Haenel|2006|p=188}}. Newton quoted by Arndt.</ref>

In Europe, Madhava's formula was rediscovered by Scottish mathematician [[James Gregory (mathematician)|James Gregory]] in 1671, and by Leibniz in 1674:<ref>{{harvnb|Arndt|Haenel|2006|pp=188–189}}</ref><ref name="LS" />

: <math>
\arctan z = z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots
</math>

This formula, the Gregory–Leibniz series, equals {{MATH|π/4}} when evaluated with {{MATH|''z''}}&nbsp;=&nbsp;1.<ref name="LS">{{harvnb|Eymard|Lafon|1999|pp=53–54}}</ref> In 1699, English mathematician [[Abraham Sharp]] used the Gregory–Leibniz series to compute {{pi}} to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.<ref>{{harvnb|Arndt|Haenel|2006|p=189}}</ref> The Gregory–Leibniz series is simple, but [[Convergent series|converges]] very slowly (that is, approaches the answer gradually), so it is not used in modern {{pi}} calculations.<ref>{{harvnb|Arndt|Haenel|2006|p=156}}</ref>

In 1706 [[John Machin]] used the Gregory–Leibniz series to produce an algorithm that converged much faster:<ref>{{harvnb|Arndt|Haenel|2006|pp=192–193}}</ref>
:<math> \frac{\pi}{4} = 4 \, \arctan \frac{1}{5} - \arctan \frac{1}{239}</math>
Machin reached 100 digits of {{pi}} with this formula.<ref name="A72n4">{{harvnb|Arndt|Haenel|2006|pp=72–74}}</ref> Other mathematicians created variants, now known as [[Machin-like formula]]e, that were used to set several successive records for calculating digits of {{pi}}.<ref name="A72n4" /> Machin-like formulae remained the best-known method for calculating {{pi}} well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson&nbsp;– the best approximation achieved without the aid of a calculating device.<ref>{{harvnb|Arndt|Haenel|2006|pp=192–196, 205}}</ref>

A record was set by the calculating prodigy [[Zacharias Dase]], who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of German mathematician [[Carl Friedrich Gauss]].<ref name="A194" /> British mathematician [[William Shanks]] famously took 15 years to calculate {{pi}} to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.<ref name="A194">{{harvnb|Arndt|Haenel|2006|pp=194–196}}</ref>

====Rate of convergence====
Some infinite series for {{pi}} [[convergent series|converge]] faster than others. Given the choice of two infinite series for {{pi}}, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate {{pi}} to any given accuracy.<ref name="Aconverge">{{cite journal|last=Borwein|first=J. M.|last2=Borwein|first2=P. B.|title=Ramanujan and Pi|year=1988|journal=Scientific American|volume=256|issue=2|pages=112–117|ref=harv|bibcode=1988SciAm.258b.112B|doi=10.1038/scientificamerican0288-112}}<br />{{harvnb|Arndt|Haenel|2006|pp=15–17, 70–72, 104, 156, 192–197, 201–202}}</ref> A simple infinite series for {{pi}} is the [[Leibniz formula for π|Gregory–Leibniz series]]:<ref>{{harvnb|Arndt|Haenel|2006|pp=69–72}}</ref>
:<math> \pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots</math>
As individual terms of this infinite series are added to the sum, the total gradually gets closer to {{pi}}, and&nbsp;– with a sufficient number of terms&nbsp;– can get as close to {{pi}} as desired. It converges quite slowly, though&nbsp;– after 500,000 terms, it produces only five correct decimal digits of {{pi}}.<ref>{{cite journal|last=Borwein|first=J. M.|last2=Borwein|first2=P. B.|last3=Dilcher|first3=K.|year=1989|title=Pi, Euler Numbers, and Asymptotic Expansions|journal=American Mathematical Monthly|volume=96|issue=8|pages=681–687|doi=10.2307/2324715|ref=harv }}</ref>

An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:<ref>{{harvnb|Arndt|Haenel|2006|p=223}}, (formula 16.10). Note that (''n''&nbsp;−&nbsp;1)''n''(''n''&nbsp;+&nbsp;1) = ''n''<sup>3</sup>&nbsp;−&nbsp;''n''.<br />{{cite book|last=Wells|first=David|page=35|title=The Penguin Dictionary of Curious and Interesting Numbers|edition=revised|publisher=Penguin|year=1997|isbn=978-0-140-26149-3|ref=harv}}</ref>

: <math> \pi = 3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + \cdots</math>

The following table compares the convergence rates of these two series:
<center>
{|class="wikitable" style="text-align: center; "
|-
! Infinite series for {{pi}} !! After 1st term !! After 2nd term !! After 3rd term !! After 4th term !! After 5th term !! Converges to:
|-
| <math>\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} \cdots.</math>
||4.0000||2.6666...||3.4666...||2.8952...||3.3396...||rowspan=2| {{pi}} = 3.1415...
|-
| <math>\pi = {{3}} + \frac{{4}}{2\times3\times4} - \frac{{4}}{4\times5\times6} + \frac{{4}}{6\times7\times8} \cdots. </math>
||3.0000||3.1666...||3.1333...||3.1452...||3.1396...
|}
</center>
After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of {{pi}}, whereas the sum of Nilakantha's series is within 0.002 of the correct value of {{pi}}. Nilakantha's series converges faster and is more useful for computing digits of {{pi}}. Series that converge even faster include [[Machin-like formula|Machin's series]] and [[Chudnovsky algorithm|Chudnovsky's series]], the latter producing 14 correct decimal digits per term.<ref name="Aconverge" />

===Irrationality and transcendence===
{{See also|Proof that π is irrational{{!}}Proof that {{pi}} is irrational|Proof that π is transcendental{{!}}Proof that {{pi}} is transcendental}}
Not all mathematical advances relating to {{pi}} were aimed at increasing the accuracy of approximations. When Euler solved the [[Basel problem]] in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between {{pi}} and the [[prime number]]s that later contributed to the development and study of the [[Riemann zeta function]]:<ref name="Posamentier">{{harvnb|Posamentier|Lehmann|2004|pp=284}}</ref>

:<math> \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots</math>

Swiss scientist [[Johann Heinrich Lambert]] in 1761 proved that {{pi}} is [[irrational number|irrational]], meaning it is not equal to the quotient of any two whole numbers.<ref name="Arndt_i" /> [[Proof that π is irrational|Lambert's proof]] exploited a continued-fraction representation of the tangent function.<ref>Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in {{harvnb|Berggren|Borwein|Borwein|1997|pp=129–140}}</ref> French mathematician [[Adrien-Marie Legendre]] proved in 1794 that {{pi}}<sup>2</sup> is also irrational. In 1882, German mathematician [[Ferdinand von Lindemann]] proved that {{pi}} is [[transcendental number|transcendental]], confirming a conjecture made by both [[Adrien-Marie Legendre|Legendre]] and Euler.<ref>{{harvnb|Arndt|Haenel|2006|p=196}}.</ref><ref>Hardy and Wright 1938 and 2000:177 footnote §11.13-14 references Lindemann's proof as appearing at ''Math. Ann''. 20 (1882), 213-25.</ref> Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".<ref> cf Hardy and Wright 1938 and 2000:177 footnote §11.13-14. The proofs that e and π are transcendental can be found on pages 170-176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pages 417-419 for full citations.</ref>

===Adoption of the symbol {{pi}}===
[[File:Leonhard Euler.jpg|thumb|upright|[[Leonhard Euler]] popularized the use of the Greek letter {{pi}} in works he published in 1736 and 1748.]]
The earliest known use of the Greek letter {{pi}} to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[William Jones (mathematician)|William Jones]] in his 1706 work ''Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics''.<ref>{{harvnb|Arndt|Haenel|2006|p=165}}. A facsimile of Jones' text is in {{harvnb|Berggren|Borwein|Borwein|1997|pp=108–109}}</ref> The Greek letter first appears there in the phrase "1/2 Periphery ({{pi}})" in the discussion of a circle with radius one. Jones may have chosen {{pi}} because it was the first letter in the Greek spelling of the word ''periphery''.<ref>See {{harvnb|Schepler|1950|p=220}}: [[William Oughtred]] used the letter {{pi}} to represent the periphery (i.e., circumference) of a circle.</ref> However, he writes that his equations for {{pi}} are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that [[John Machin|Machin]] may have employed the Greek letter before Jones.<ref name="Arndt_a">{{harvnb|Arndt|Haenel|2006|p=166}}</ref> It had indeed been used earlier for geometric concepts.<ref name="Arndt_a" /> [[William Oughtred]] used {{pi}} and δ, the Greek letter equivalents of p and d, to express ratios of periphery and diameter in the 1647 and later editions of ''Clavis Mathematicae''.

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until [[Euler]] started using it, beginning with his 1736 work ''[[Mechanica]]''. Before then, mathematicians sometimes used letters such as ''c'' or ''p'' instead.<ref name="Arndt_a" /> Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.<ref name="Arndt_a" /> In 1748, Euler used {{pi}} in his widely read work ''[[Introductio in analysin infinitorum]]'' (he wrote: "for the sake of brevity we will write this number as {{pi}}; thus {{pi}} is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the [[Western world]].<ref name="Arndt_a" />

==Modern quest for more digits==

===Computer era and iterative algorithms===
[[File:JohnvonNeumann-LosAlamos.gif|thumb|upright|alt=Formal photo of a balding man wearing a suit|[[John von Neumann]] was part of the team that first used a digital computer, [[ENIAC]], to compute {{pi}}.]]
{{quote box|quote=
The [[Gauss–Legendre algorithm|Gauss–Legendre iterative algorithm]]:<br />Initialize
:<math>\scriptstyle a_0 = 1 \quad b_0 = \frac{1}{\sqrt 2} \quad t_0 = \frac{1}{4} \quad p_0 = 1</math>
Iterate
:<math>\scriptstyle a_{n+1} = \frac{a_n+b_n}{2} \quad \quad b_{n+1} = \sqrt{a_n b_n}</math>
:<math>\scriptstyle t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad p_{n+1} = 2 p_n</math>
Then an estimate for {{pi}} is given by
:<math>\scriptstyle \pi \approx \frac{(a_n + b_n)^2}{4 t_n}</math>
|fontsize=90%|qalign=left}}

The development of computers in the mid-20th century again revolutionized the hunt for digits of {{pi}}. American mathematicians [[John Wrench]] and Levi Smith reached 1,120 digits in 1949 using a desk calculator.<ref>{{harvnb|Arndt|Haenel|2006|pp=205}}</ref> Using an [[inverse tangent]] (arctan) infinite series, a team led by George Reitwiesner and [[John von Neumann]] that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the [[ENIAC]] computer.<ref>{{harvnb|Arndt|Haenel|2006|p=197}}. See also {{harvnb|Reitwiesner|1950}}.</ref> The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973.<ref>{{harvnb|Arndt|Haenel|2006|p=197}}</ref>

Two additional developments around 1980 once again accelerated the ability to compute {{pi}}. First, the discovery of new [[iterative algorithm]]s for computing {{pi}}, which were much faster than the infinite series; and second, the invention of [[Multiplication algorithm#Fast multiplication algorithms for large inputs|fast multiplication algorithms]] that could multiply large numbers very rapidly.<ref>{{harvnb|Arndt|Haenel|2006|pp=15–17}}</ref> Such algorithms are particularly important in modern {{pi}} computations, because most of the computer's time is devoted to multiplication.<ref>{{harvnb|Arndt|Haenel|2006|pp=131}}</ref> They include the [[Karatsuba algorithm]], [[Toom–Cook multiplication]], and [[FFT multiplication#Fourier transform methods|Fourier transform-based methods]].<ref>{{harvnb|Arndt|Haenel|2006|pp=132, 140}}</ref>

The iterative algorithms were independently published in 1975–1976 by American physicist [[Eugene Salamin (mathematician)|Eugene Salamin]] and Australian scientist [[Richard Brent (scientist)|Richard Brent]].<ref name="Arndt_j">{{harvnb|Arndt|Haenel|2006|p=87}}</ref> These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by [[Carl Friedrich Gauss]], in what is now termed the [[AGM method|arithmetic–geometric mean method]] (AGM method) or [[Gauss–Legendre algorithm]].<ref name="Arndt_j" /> As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally ''multiply'' the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers [[Jonathan Borwein|John]] and [[Peter Borwein]] produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.<ref>{{harvnb|Arndt|Haenel|2006|pp=111 (5 times); pp. 113–114 (4 times)}}.<br />See {{harvnb|Borwein|Borwein|1987}} for details of algorithms.</ref> Iterative methods were used by Japanese mathematician [[Yasumasa Kanada]] to set several records for computing {{pi}} between 1995 and 2002.<ref name="Background" /> This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.<ref name="Background">{{cite web|last=Bailey|first=David H.|url=http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/dhb-kanada.pdf|title=Some Background on Kanada’s Recent Pi Calculation|date=16 May 2003|accessdate=12 April 2012}}</ref>

===Motivations for computing {{pi}}===
[[File:Record pi approximations.svg|thumb|400px|right|As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of {{pi}} increased dramatically. Note that the vertical scale is [[logarithm]]ic.]]
For most numerical calculations involving {{pi}}, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most [[cosmological]] calculations, because that is the accuracy necessary to calculate the circumference of the [[observable universe]] with a precision of one atom.<ref name="Arndt_b">{{harvnb|Arndt|Haenel|2006|p=17}}. <!--quote from p. 17:-->"39 digits of {{pi}} are sufficient to calculate the volume of the universe to the nearest atom."<br /> Accounting for additional digits needed to compensate for computational [[round-off error]]s, Arndt concludes that a few hundred digits would suffice for any scientific application.</ref> Despite this, people have worked strenuously to compute {{pi}} to thousands and millions of digits.<ref>{{harvnb|Arndt|Haenel|2006|pp=17–19}}</ref> This effort may be partly ascribed to the human compulsion to break records, and such achievements with {{pi}} often make headlines around the world.<ref name="msnbc.msn.com">{{cite news|title=John W. Wrench, Jr.: Mathematician Had a Taste for Pi|first=Matt|last=Schudel|newspaper=The Washington Post|date=25 March 2009|page=B5}}</ref><ref name="independent.co.uk">{{cite news|title=The Big Question: How close have we come to knowing the precise value of pi?|url=http://www.independent.co.uk/news/science/the-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html|newspaper=The Independent|date=8 January 2010|accessdate=14 April 2012|location=London|first=Steve|last=Connor}}</ref> They also have practical benefits, such as testing [[supercomputer]]s, testing numerical analysis algorithms (including [[Multiplication algorithm#Fast multiplication algorithms for large inputs|high-precision multiplication algorithms]]); and within pure mathematics itself, providing data for evaluating the randomness of the digits of {{pi}}.<ref name="Arndt_c">{{harvnb|Arndt|Haenel|2006|p=18}}</ref>

===Rapidly convergent series===
[[File:Srinivasa Ramanujan - OPC - 1.jpg|thumb|upright|alt=Photo portrait of a man| [[Srinivasa Ramanujan]], working in isolation in India, produced many innovative series for computing {{pi}}.]]
Modern {{pi}} calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.<ref name="Background" /> The fast iterative algorithms were anticipated in 1914, when the Indian mathematician [[Srinivasa Ramanujan]] published dozens of innovative new formulae for {{pi}}, remarkable for their elegance, mathematical depth, and rapid convergence.<ref>{{harvnb|Arndt|Haenel|2006|pp=103–104}}</ref> One of his formulae, based on [[modular equation]]s, is
:<math>\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4(396^{4k})}.</math>
This series converges much more rapidly than most arctan series, including Machin's formula.<ref>{{harvnb|Arndt|Haenel|2006|p=104}}</ref> [[Bill Gosper]] was the first to use it for advances in the calculation of {{pi}}, setting a record of 17 million digits in 1985.<ref>{{harvnb|Arndt|Haenel|2006|pp=104, 206}}</ref> Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the [[Chudnovsky brothers]].<ref>{{harvnb|Arndt|Haenel|2006|pp=110–111}}</ref> The [[Chudnovsky algorithm|Chudnovsky formula]] developed in 1987 is
:<math>\frac{1}{\pi} = \frac{12}{640320^{3/2}} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}.</math>
It produces about 14 digits of {{pi}} per term,<ref>{{harvnb|Eymard|Lafon|1999|p=254}}</ref> and has been used for several record-setting {{pi}} calculations, including the first to surpass 1 billion (10<sup>9</sup>) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×10<sup>12</sup>) digits by [[Fabrice Bellard]] in 2009, and 10 trillion (10<sup>13</sup>) digits in 2011 by Alexander Yee and Shigeru Kondo.<ref>{{harvnb|Arndt|Haenel|2006|pp=110–111, 206}}<br />Bellard, Fabrice, [http://bellard.org/pi/pi2700e9/pipcrecord.pdf "Computation of 2700 billion decimal digits of Pi using a Desktop Computer"], 11 Feb 2010.</ref><ref name="NW">[http://www.numberworld.org/misc_runs/pi-12t "Round 2... 10 Trillion Digits of Pi"], NumberWorld.org, 17 Oct 2011. Retrieved 30 May 2012.</ref>  For similar formulas, see also the [[Ramanujan–Sato series]].

In 2006, Canadian mathematician [[Simon Plouffe]] used the PSLQ [[integer relation algorithm]]<ref>PSLQ means Partial Sum of Least Squares.</ref> to generate several new formulas for {{pi}}, conforming to the following template:
:<math>\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n-1} + \frac{b}{q^{2n}-1} + \frac{c}{q^{4n}-1}\right),</math>
where {{math.|''q''}} is [[Gelfond's constant|{{math.|e}}<sup>{{pi}}</sup>]] (Gelfond's constant), {{math.|''k''}} is an [[odd number]], and {{math.|''a'', ''b'', ''c''}} are certain rational numbers that Plouffe computed.<ref>{{cite web|first=Simon|last=Plouffe|authorlink=Simon Plouffe|title=Identities inspired by Ramanujan's Notebooks (part 2)|date=April 2006|url=<!-- http://www.lacim.uqam.ca/~plouffe/inspired2.pdf -->http://plouffe.fr/simon/inspired2.pdf|accessdate=10 April 2009}}</ref>

===Spigot algorithms===
Two algorithms were discovered in 1995 that opened up new avenues of research into {{pi}}. They are called [[spigot algorithm]]s because, like water dripping from a [[Tap (valve)|spigot]], they produce single digits of {{pi}} that are not reused after they are calculated.<ref name="Arndtpp" /><ref name="Gibbons">Gibbons, Jeremy, [http://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf "Unbounded Spigot Algorithms for the Digits of Pi"], 2005. Gibbons produced an improved version of Wagon's algorithm.</ref> This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.<ref name="Arndtpp">{{harvnb|Arndt|Haenel|2006|pp=77–84}}</ref>

American mathematicians [[Stan Wagon]] and Stanley Rabinowitz produced a simple spigot algorithm in 1995.<ref name="Gibbons" /><ref name="Arndt_k">{{harvnb|Arndt|Haenel|2006|p=77}}</ref><ref>{{cite journal|first1=Stanley|last1=Rabinowitz|last2=Wagon|first2=Stan|date=March 1995|title=A spigot algorithm for the digits of Pi|journal=American Mathematical Monthly|volume=102|issue=3|pages=195–203|doi=10.2307/2975006|ref=harv}} A computer program has been created that implements Wagon's spigot algorithm in only 120 characters of software.</ref> Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.<ref name="Arndt_k" />

Another spigot algorithm, the [[Bailey–Borwein–Plouffe formula|BBP]] [[digit extraction algorithm]], was discovered in 1995 by Simon Plouffe:<ref name="Arndtpp_a">{{harvnb|Arndt|Haenel|2006|pp=117, 126–128}}</ref><ref name="bbpf">{{cite journal|author=[[David H. Bailey|Bailey, David H.]]; [[Peter Borwein|Borwein, Peter B.]]; and [[Simon Plouffe|Plouffe, Simon]]|date=April 1997|title=On the Rapid Computation of Various Polylogarithmic Constants|journal=Mathematics of Computation|volume=66| issue=218|pages=903–913|url=<!-- http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf -->http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/digits.pdf|format=PDF|doi=10.1090/S0025-5718-97-00856-9|ref=harv}}</ref>
:<math> \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)</math>
This formula, unlike others before it, can produce any individual [[hexadecimal]] digit of {{pi}} without calculating all the preceding digits.<ref name="Arndtpp_a" /> Individual binary digits may be extracted from individual hexadecimal digits, and [[octal]] digits can be extracted from one or two hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.<ref>{{harvnb|Arndt|Haenel|2006|p=128}}. Plouffe did create a decimal digit extraction algorithm, but it is slower than full, direct computation of all preceding digits.</ref> An important application of digit extraction algorithms is to validate new claims of record {{pi}} computations: After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several random hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct.<ref name="NW" />

Between 1998 and 2000, the [[distributed computing]] project [[PiHex]] used [[Bellard's formula]] (a modification of the BBP algorithm) to compute the quadrillionth (10<sup>15</sup>th) bit of {{pi}}, which turned out to be 0.<ref>{{harvnb|Arndt|Haenel|2006|p=20}}<br />Bellards formula in: {{cite web|url=http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html|title=A new formula to compute the n<sup>th</sup> binary digit of pi|first=Fabrice|last=Bellard|authorlink=Fabrice Bellard|accessdate=27 October 2007|archiveurl=https://web.archive.org/web/20070912084453/http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html <!-- http://www.lacim.uqam.ca/~plouffe/inspired2.pdf -->|archivedate=12 September 2007}} {{Webarchive|url=https://web.archive.org/web/20070912084453/http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html |date=12 September 2007 }}</ref> In September 2010, a [[Yahoo!]] employee used the company's [[Apache Hadoop|Hadoop]] application on one thousand computers over a 23-day period to compute 256 [[bit]]s of {{pi}} at the two-quadrillionth (2×10<sup>15</sup>th) bit, which also happens to be zero.<ref>{{cite news|title= Pi record smashed as team finds two-quadrillionth digit|author=Palmer, Jason|newspaper=BBC News|date=16 September 2010|url=http://www.bbc.co.uk/news/technology-11313194|accessdate=26 March 2011}}</ref>

==Use==
{{Main|List of formulae involving π|l1=List of formulae involving {{pi}}}}
Because {{pi}} is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Formulae from other branches of science also include {{pi}} in some of their important formulae, including sciences such as statistics, fractals, thermodynamics, mechanics, cosmology, number theory, and electromagnetism.

===Geometry and trigonometry===
[[File:Circle Area.svg|thumb|alt=A diagram of a circle with a square coving the circle's upper right quadrant.|right|The area of the circle equals {{pi}} times the shaded area.]]
{{pi}} appears in formulae for areas and volumes of geometrical shapes based on circles, such as [[ellipse]]s, [[sphere]]s, [[cone (geometry)|cones]], and [[torus|tori]]. Below are some of the more common formulae that involve {{pi}}.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=200, 209}}</ref>
* The circumference of a circle with radius {{math.|''r''}} is {{math.|2π''r''}}.
* The [[area of a disk|area of a circle]] with radius {{math.|''r''}} is {{math.|π''r''<sup>2</sup>}}.
* The volume of a sphere with radius {{math.|''r''}} is {{math.|{{sfrac|4|3}}π''r''<sup>3</sup>}}.
* The surface area of a sphere with radius {{math.|''r''}} is {{math.|4π''r''<sup>2</sup>}}.

The formulae above are special cases of the surface area {{math.|''S''<sub>''n''</sub>(''r'')}} and volume {{math.|''V''<sub>''n''</sub>(''r'')}} of an [[N-sphere|''n''-dimensional sphere]].

: <math>S_n(r) = \frac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n-1}</math>

: <math>V_n(r) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^n</math>

{{pi}} appears in [[integral|definite integrals]] that describe circumference, area, or volume of shapes generated by circles. For example, an integral that specifies half the area of a circle of radius one is given by:<ref name="udi">{{MathWorld|Semicircle|Semicircle}}</ref>

: <math>\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}.</math>

In that integral the function {{math.|{{sqrt|1&nbsp;−&nbsp;''x''<sup>2</sup>}}}} represents the top half of a circle (the [[square root]] is a consequence of the [[Pythagorean theorem]]), and the integral {{math.|{{intmath|int|−1|1}}}} computes the area between that half of a circle and the [[X axis|{{math.|''x''}} axis]].
[[File:Sine cosine one period.svg|thumb|340px|alt=Diagram showing graphs of functions|[[Sine]] and [[cosine]] functions repeat with period 2{{pi}}.]]
The [[trigonometric function]]s rely on angles, and mathematicians generally use radians as units of measurement. {{pi}} plays an important role in angles measured in [[radian]]s, which are defined so that a complete circle spans an angle of 2{{pi}} radians.<ref name="WR">{{harvnb|Ayers|1964|p=60}}</ref> The angle measure of 180° is equal to {{pi}} radians, and 1° = {{pi}}/180 radians.<ref name="WR" />

Common trigonometric functions have periods that are multiples of {{pi}}; for example, sine and cosine have period 2{{pi}},<ref name="WCS">{{harvnb|Bronshteĭn|Semendiaev|1971|pp=210–211}}</ref> so for any angle {{math.|''θ''}} and any integer {{math.|''k''}},

: <math> \sin\theta = \sin\left(\theta + 2\pi k \right) \text{ and } \cos\theta = \cos\left(\theta + 2\pi k \right).</math><ref name="WCS" />

====Monte Carlo methods====
{{multiple image
|direction=horizontal
|footer=[[Monte Carlo method]]s, based on random trials, can be used to approximate {{pi}}.
|width1=166
|image1=Buffon needle.svg
|caption1=[[Buffon's needle]]. Needles ''a'' and ''b'' are dropped randomly.
|alt1=Needles of length ''ℓ'' scattered on stripes with width ''t''
|width2=100
|image2=Pi 30K.gif|
|caption2=Random dots are placed on the quadrant of a square with a circle inscribed in it.
|alt2=Thousands of dots randomly covering a square and a circle inscribed in the square.
}}
[[Monte Carlo methods]], which evaluate the results of multiple random trials, can be used to create approximations of {{pi}}.<ref>{{harvnb|Arndt|Haenel|2006|p=39}}</ref> [[Buffon's needle]] is one such technique: If a needle of length {{math.|''ℓ''}} is dropped {{math.|''n''}} times on a surface on which parallel lines are drawn {{math.|''t''}} units apart, and if {{math.|''x''}} of those times it comes to rest crossing a line ({{math.|''x''}}&nbsp;>&nbsp;0), then one may approximate {{pi}} based on the counts:<ref name="bn">{{cite journal|last=Ramaley|first=J. F.|title=Buffon's Noodle Problem|jstor=2317945|journal=The American Mathematical Monthly|volume=76|issue=8|date=October 1969|pages=916–918|doi=10.2307/2317945|ref=harv}}</ref>

: <math>\pi \approx \frac{2n\ell}{xt}</math>

Another Monte Carlo method for computing {{pi}} is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal {{math.|π/4}}.<ref>{{harvnb|Arndt|Haenel|2006|pp=39–40}}<br />{{harvnb|Posamentier|Lehmann|2004|p=105}}</ref>

Monte Carlo methods for approximating {{pi}} are very slow compared to other methods, and are never used to approximate {{pi}} when speed or accuracy is desired.<ref>{{harvnb|Arndt|Haenel|2006|pp=43}}<br />{{harvnb|Posamentier|Lehmann|2004|pp=105–108}}</ref>

===Complex numbers and analysis===
[[File:Euler's formula.svg|thumb|alt=A diagram of a unit circle centered at the origin in the complex plane, including a ray from the center of the circle to its edge, with the triangle legs labeled with sine and cosine functions.|The association between imaginary powers of the number {{math.|''e''}} and [[Point (geometry)|points]] on the [[unit circle]] centered at the [[Origin (mathematics)|origin]] in the [[complex plane]] given by [[Euler's formula]].]]

Any [[complex number]], say {{math.|''z''}}, can be expressed using a pair of [[real number]]s. In the [[Polar coordinate system#Complex numbers|polar coordinate system]], one number (radius or ''r'') is used to represent {{math.|''z''}}'s distance from the [[Origin (mathematics)|origin]] of the [[complex plane]] and the other (angle or {{math.|φ}}) to represent a counter-clockwise [[rotation]] from the positive real line as follows:<ref>{{harvnb|Ayers|1964|p=100}}</ref>
:<math>z = r\cdot(\cos\varphi + i\sin\varphi),</math>
where {{math.|''i''}} is the [[imaginary unit]] satisfying {{math.|''i''<sup>2</sup>}} = −1. The frequent appearance of {{pi}} in [[complex analysis]] can be related to the behavior of the [[exponential function]] of a complex variable, described by [[Euler's formula]]:<ref name="EF">{{harvnb|Bronshteĭn|Semendiaev|1971|p=592}}</ref>

:<math>e^{i\varphi} = \cos \varphi + i\sin \varphi,</math>

where [[E (mathematical constant)|the constant {{math.|''e''}}]] is the base of the [[natural logarithm]]. This formula establishes a correspondence between imaginary powers of {{math.|''e''}} and points on the [[unit circle]] centered at the origin of the complex plane. Setting {{math.|''φ''}} = {{pi}} in Euler's formula results in [[Euler's identity]], celebrated by mathematicians because it contains the five most important mathematical constants:<ref name="EF" /><ref>Maor, Eli, ''E: The Story of a Number'', Princeton University Press, 2009, p 160, ISBN 978-0-691-14134-3 ("five most important" constants).</ref>
:<math>e^{i \pi} + 1 = 0.</math>

There are {{math.|''n''}} different [[complex number]]s {{math.|''z''}} satisfying {{math.|1=''z''<sup>''n''</sup> = 1}}, and these are called the "{{math.|''n''}}-th [[root of unity|roots of unity]]".<ref>{{MathWorld|RootofUnity|Roots of Unity}}</ref> They are given by this formula:
:<math>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math>

[[Cauchy's integral formula]] governs [[complex analytic function]]s and establishes an important relationship between integration and differentiation, including the fact that the values of a complex function within a closed boundary are entirely determined by the values on the boundary:<ref>{{mathworld|CauchyIntegralFormula|Cauchy Integral Formula}}</ref><ref>Joglekar, S. D., ''Mathematical Physics'', Universities Press, 2005, p 166, ISBN 978-81-7371-422-1.</ref>
:<math>f (z_{0}) = \frac{1}{ 2\pi i } \oint_\gamma { f(z) \over z-z_0 }\,dz</math>

[[File:Mandel zoom 00 mandelbrot set.jpg|alt=An complex black shape on a blue background.|thumb|{{pi}} can be computed from the [[Mandelbrot set]], by counting the number of iterations required before point (−0.75, ε) diverges.]]
An occurrence of {{pi}} in the [[Mandelbrot set]] [[fractal]] was discovered by American David Boll in 1991.<ref name="KA">{{cite journal|last1=Klebanoff|first1=Aaron|year=2001|title=Pi in the Mandelbrot set|journal=Fractals|volume=9|issue=4|pages=393–402|url=http://home.comcast.net/~davejanelle/mandel.pdf|archiveurl=https://web.archive.org/web/20111027155739/http://home.comcast.net/~davejanelle/mandel.pdf|archivedate=27 October 2011|accessdate=14 April 2012|doi=10.1142/S0218348X01000828|ref=harv|url-status=dead}}</ref> He examined the behavior of the Mandelbrot set near the "neck" at (−0.75, 0). If points with coordinates (−0.75, ε) are considered, as ε tends to zero, the number of iterations until divergence for the point multiplied by ε converges to {{pi}}. The point (0.25, ε) at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of ε tends to {{pi}}.<ref name="KA" /><ref>Peitgen, Heinz-Otto, ''Chaos and fractals: new frontiers of science'', Springer, 2004, pp. 801–803, ISBN 978-0-387-20229-7.</ref>

The [[gamma function]] extends the concept of [[factorial]] (normally defined only for non-negative integers) to all complex numbers, except the negative real integers. When the gamma function is evaluated at half-integers, the result contains {{pi}}; for example <math> \Gamma(1/2) = \sqrt{\pi} </math> and <math>\Gamma(5/2) = \frac {3 \sqrt{\pi}} {4} </math>.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=191–192}}</ref> The gamma function can be used to create a simple approximation to {{math.|''n''!}} for large {{math.|''n''}}: <math> n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> which is known as [[Stirling's approximation]].<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|p=190}}</ref>

===Number theory and Riemann zeta function===
The [[Riemann zeta function]] {{math.|''ζ''(''s'')}} is used in many areas of mathematics. When evaluated at {{math.|1=''s'' = 2}} it can be written as
:<math> \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots</math>
Finding a [[closed-form expression|simple solution]] for this infinite series was a famous problem in mathematics called the [[Basel problem]]. [[Leonhard Euler]] solved it in 1735 when he showed it was equal to {{math.|π<sup>2</sup>/6}}.<ref name="Posamentier" /> Euler's result leads to the [[number theory]] result that the probability of two random numbers being [[relatively prime]] (that is, having no shared factors) is equal to {{math.|6/π<sup>2</sup>}}.<ref>{{harvnb|Arndt|Haenel|2006|pp=41–43}}</ref><ref>This theorem was proved by [[Ernesto Cesàro]] in 1881. For a more rigorous proof than the intuitive and informal one given here, see Hardy, G. H., ''An Introduction to the Theory of Numbers'', Oxford University Press, 2008, ISBN 978-0-19-921986-5, theorem 332.</ref> This probability is based on the observation that the probability that any number is [[divisible]] by a prime {{math.|''p''}} is {{math.|1/''p''}} (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is {{math.|1/''p''<sup>2</sup>}}, and the probability that at least one of them is not is {{math.|1&nbsp;−&nbsp;1/''p''<sup>2</sup>}}. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:<ref>[[C. Stanley Ogilvy|Ogilvy, C. S.]]; Anderson, J. T., ''Excursions in Number Theory'', Dover Publications Inc., 1988, pp. 29–35, ISBN 0-486-25778-9.</ref>
: <math>\prod_p^\infty \left(1-\frac{1}{p^2}\right) = \left( \prod_p^\infty \frac{1}{1-p^{-2}} \right)^{-1} = \frac{1}{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots } = \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 61\% </math>
This probability can be used in conjunction with a [[random number generator]] to approximate {{pi}} using a Monte Carlo approach.<ref>{{harvnb|Arndt|Haenel|2006|p=43}}</ref>

===Probability and statistics===
[[File:E^(-x^2).svg|thumb|right|A graph of the [[Gaussian function]] <br /> ''ƒ''(''x'') =&nbsp;''e''<sup>−''x''<sup>2</sup></sup>. The colored region between the function and the ''x''-axis has area <math> \sqrt{\pi} </math>.]]

The fields of [[probability]] and [[statistics]] frequently use the [[normal distribution]] as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.<ref>Feller, W. ''An Introduction to Probability Theory and Its Applications, Vol. 1'', Wiley, 1968, pp 174–190.</ref> {{pi}} is found in the [[Gaussian function]] (which is the [[probability density function]] of the normal distribution) with [[mean]] {{math.|μ}} and [[standard deviation]] {{math.|σ}}:<ref name="GaussProb">{{harvnb|Bronshteĭn|Semendiaev|1971|pp=106–107, 744, 748}}</ref>

:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>

The area under the graph of the normal distribution curve is given by the [[Gaussian integral]]:<ref name="GaussProb" />

:<math>\int_{-\infty}^\infty e^{-x^2} \, dx=\sqrt{\pi},</math>

while the related integral for the [[Cauchy distribution]] is

:<math>\int_{-\infty }^{\infty } \frac{1}{x^2+1} \, dx = \pi.</math>

==Outside mathematics==

===Describing physical phenomena===
Although not a [[physical constant]], {{pi}} appears routinely in equations describing fundamental principles of the universe, often because of {{pi}}'s relationship to the circle and to [[spherical coordinate system]]s. A simple formula from the field of [[classical mechanics]] gives the approximate period {{math.|''T''}} of a simple [[pendulum]] of length {{math.|''L''}}, swinging with a small amplitude ({{math.|''g''}} is the [[Gravity of Earth|earth's gravitational acceleration]]):<ref>Halliday, David; Resnick, Robert; Walker, Jearl, ''Fundamentals of Physics, 5th Ed.'', John Wiley & Sons, 1997, p 381, ISBN 0-471-14854-7.</ref>
:<math>T \approx 2\pi \sqrt\frac{L}{g}.</math>
One of the key formulae of [[quantum mechanics]] is [[Heisenberg's uncertainty principle]], which shows that the uncertainty in the measurement of a particle's position (Δ{{math.|''x''}}) and [[momentum]] (Δ{{math.|''p''}}) cannot both be arbitrarily small at the same time (where {{math.|''h''}} is [[Planck's constant]]):<ref>{{cite web|first=James M|last=Imamura|url=http://zebu.uoregon.edu/~imamura/208/jan27/hup.html|title=Heisenberg Uncertainty Principle|publisher=[[University of Oregon]]|date=17 August 2005|accessdate=9 September 2007|archiveurl=https://web.archive.org/web/20071012060715/http://zebu.uoregon.edu/~imamura/208/jan27/hup.html <!-- http://www.lacim.uqam.ca/~plouffe/inspired2.pdf -->|archivedate=12 October 2007}} {{Webarchive|url=https://web.archive.org/web/20071012060715/http://zebu.uoregon.edu/~imamura/208/jan27/hup.html |date=12 October 2007 }}</ref>
:<math> \Delta x\, \Delta p \ge \frac{h}{4\pi}.</math>

In the domain of [[cosmology]], {{pi}} appears in [[Einstein field equations|Einstein's field equation]], a fundamental formula which forms the basis of the [[general theory of relativity]] and describes the [[fundamental interaction]] of [[gravitation]] as a result of [[spacetime]] being [[curvature|curved]] by [[matter]] and [[energy]]:<ref>Yeo, Adrian, ''The pleasures of pi, e and other interesting numbers'', World Scientific Pub., 2006, p 21, ISBN 978-981-270-078-0.<br>Ehlers, Jürgen, ''Einstein's Field Equations and Their Physical Implications'', Springer, 2000, p 7, ISBN 978-3-540-67073-5.</ref>
:<math> R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu},</math>
where <math>R_{\mu\nu}</math> is the [[Ricci curvature tensor]], {{math.|''R''}} is the [[scalar curvature]], <math>g_{\mu\nu}</math> is the [[metric tensor (general relativity)|metric tensor]], {{math.|&Lambda;}} is the [[cosmological constant]], {{math.|''G''}} is [[gravitational constant|Newton's gravitational constant]], {{math.|''c''}} is the [[speed of light]] in vacuum, and <math>T_{\mu\nu}</math> is the [[stress–energy tensor]].

[[Coulomb's law]], from the discipline of [[electromagnetism]], describes the [[electric field]] between two [[electric charge]]s ({{math.|''q''<sub>1</sub>}} and {{math.|''q''<sub>2</sub>}}) separated by distance {{math.|''r''}} (with {{math.|''ε''<sub>0</sub>}} representing the [[vacuum permittivity]] of free space):<ref>
{{cite web|first=C. Rod|last=Nave|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c3|title=Coulomb's Constant|work=[[HyperPhysics]]|publisher=[[Georgia State University]]|date=28 June 2005|accessdate=9 November 2007}}</ref>
:<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}.</math>

The fact that {{pi}} is approximately equal to 3 plays a role in the relatively long lifetime of [[orthopositronium]]. The inverse lifetime to lowest order in the [[fine-structure constant]] {{math.|''α''}} is<ref>C. Itzykson, J-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.</ref>
:<math>\frac{1}{\tau} = 2\frac{\pi^2 - 9}{9\pi}m\alpha^{6},</math>
where {{math.|''m''}} is the mass of the electron.

{{pi}} is present in some structural engineering formulae, such as the [[buckling]] formula derived by Euler, which gives the maximum axial load {{math.|''F''}} that a long, slender column of length {{math.|''L''}}, [[modulus of elasticity]] {{math.|''E''}}, and [[area moment of inertia]] {{math.|''I''}} can carry without buckling:<ref>Low, Peter, ''Classical Theory of Structures Based on the Differential Equation'', CUP Archive, 1971, pp 116–118, ISBN 978-0-521-08089-7.</ref>
:<math>F =\frac{\pi^2EI}{L^2}.</math>

The field of [[fluid dynamics]] contains {{pi}} in [[Stokes' law]], which approximates the [[drag force|frictional force]] {{math.|''F''}} exerted on small, [[sphere|spherical]] objects of radius {{math.|''R''}}, moving with velocity {{math.|''v''}} in a [[fluid]] with [[dynamic viscosity]] {{math.|''η''}}:<ref>Batchelor, G. K., ''An Introduction to Fluid Dynamics'', Cambridge University Press, 1967, p 233, ISBN 0-521-66396-2.</ref>
:<math>F =6 \, \pi \, \eta \, R \, v .</math>

The [[Fourier transform]], defined below, is a mathematical operation that expresses time as a [[function (mathematics)|function]] of [[frequency]], known as its [[frequency spectrum]]. It has many applications in [[physics]] and [[engineering]], particularly in [[signal processing]].<ref>Bracewell, R. N., ''The Fourier Transform and Its Applications'', McGraw-Hill, 2000, ISBN 0-07-116043-4.</ref>
:<math>\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx</math>

Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the [[sinuosity]] of a [[meander]]ing river approaches {{pi}}. The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river's bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and "short-circuit", creating an [[ox-bow lake]] in the process. The balance between these two opposing factors leads to an average ratio of {{pi}} between the actual length and the direct distance between source and mouth.<ref>{{cite journal|journal=[[Science (journal)|Science]]|volume=271|issue=5256|date=22 March 1996|pages=1710–1713|doi=10.1126/science.271.5256.1710|title=River Meandering as a Self-Organization Process|author=Hans-Henrik Stølum|bibcode=1996Sci...271.1710S|ref=harv}}</ref><ref>{{harvnb|Posamentier|Lehmann|2004|pp=140–141}}</ref>

The Wallis formula for pi can be obtained directly from the [[Calculus of variations|variational approach]] to the [[Bohr model|spectrum of the hydrogen atom]] in spaces of arbitrary dimensions greater than one, including the physical three dimensions.<ref>{{cite journal|doi=10.1063/1.4930800|author=T. Friedmann ; C.R. Hagen|title=Quantum mechanical derivation of the Wallis formula for pi|journal=Journal of Mathmatical Physics|volume=56|issue=11|year=2015}}</ref>

===Memorizing digits===
{{Main|Piphilology}}
Many persons have memorized large numbers of digits of {{pi}}, a practice called [[piphilology]].<ref name="A445">{{harvnb|Arndt|Haenel|2006|pp=44–45}}</ref> One common technique is to memorize a story or poem in which the word lengths represent the digits of {{pi}}: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. An early example of a memorization aid, originally devised by English scientist [[James Hopwood Jeans|James Jeans]], is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."<ref name="A445" /> When a poem is used, it is sometimes referred to as a ''piem''. Poems for memorizing {{pi}} have been composed in several languages in addition to English.<ref name="A445" />

The record for memorizing digits of {{pi}}, certified by ''[[Guinness World Records]]'', is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015.<ref>[http://www.guinnessworldrecords.com/world-records/most-pi-places-memorised "Most Pi Places Memorized"], Guinness World Records.</ref> In 2006, [[Akira Haraguchi]], a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.<ref name="japantimes">{{cite news|first=Tomoko|last=Otake|url=<!-- http://www.lacim.uqam.ca/~plouffe/inspired2.pdf -->http://www.japantimes.co.jp/life/2006/12/17/life/how-can-anyone-remember-100000-numbers/|title=How can anyone remember 100,000 numbers?|work=[[The Japan Times]]|date=17 December 2006|accessdate=27 October 2007}}</ref> Record-setting {{pi}} memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the [[method of loci]].<ref>{{cite journal|last1=Raz|first1=A.|last2=Packard|first2=M. G.|year=2009|title=A slice of pi: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist|journal=Neurocase|volume=15|pages=361–372|ref=harv|doi=10.1080/13554790902776896|pmid=19585350}}</ref>

A few authors have used the digits of {{pi}} to establish a new form of [[constrained writing]], where the word lengths are required to represent the digits of {{pi}}. The ''[[Cadaeic Cadenza]]'' contains the first 3835 digits of {{pi}} in this manner,<ref>{{cite web|first=Mike|last=Keith|authorlink=Mike Keith (mathematician)|url=http://www.cadaeic.net/comments.htm|title=Cadaeic Cadenza Notes & Commentary|accessdate=29 July 2009}}</ref> and the full-length book ''Not a Wake'' contains 10,000 words, each representing one digit of {{pi}}.<ref name=KeithNAW>{{cite book|last=Keith|first=Michael|title=Not A Wake: A dream embodying (pi)'s digits fully for 10000 decimals|publisher=Vinculum Press|isbn=978-0963009715|author2=Diana Keith|date=February 17, 2010}}</ref>

===In popular culture===
[[File:Pi pie2.jpg|thumb|right|alt=Pi Pie at Delft University|A pi pie. The circular shape of [[pie]] makes it a frequent subject of pi [[pun]]s.|bottom]]
Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, {{pi}} has been represented in popular culture more than other mathematical constructs.<ref>For instance, Pickover calls π "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the [[Parfums Givenchy|Givenchy]] π perfume, [[Pi (film)]], and [[Pi Day]] as examples. See {{citation|title=Keys to Infinity|first=Clifford A.|last=Pickover|authorlink=Clifford A. Pickover|publisher=Wiley & Sons|year=1995|isbn=9780471118572|page=59}}; {{citation|title=Mathematical Treks: From Surreal Numbers to Magic Circles|series=MAA spectrum|first=Ivars|last=Peterson|authorlink=Ivars Peterson|publisher=Mathematical Association of America|year=2002|isbn=9780883855379|page=17|url=https://books.google.com/books?id=4gWSAraVhtAC&pg=PA17}}</ref>

In the 2008 [[Open University]] and [[BBC]] documentary co-production, [[The Story of Maths]], aired in October 2008 on [[BBC Four]], British mathematician [[Marcus du Sautoy]] shows a [[Information graphics|visualization]] of the - historically first exact - [[Madhava of Sangamagrama#The value of π (pi)|formula for calculating {{pi}}]] when visiting India and exploring its contributions to trigonometry.<ref name="BBC_2008">[http://vimeo.com/74859005 BBC documentary "The Story of Maths", second part], showing a visualization of the historically first exact formula, starting at 35 min and 20 sec into the second part of the documentary.</ref>

In the [[Palais de la Découverte]] (a science museum in Paris) there is a circular room known as the ''pi room''. On its wall are inscribed 707 digits of {{pi}}. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1853 calculation by English mathematician [[William Shanks]], which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.<ref>{{harvnb|Posamentier|Lehmann|2004|p=118}}<br />{{harvnb|Arndt|Haenel|2006|p=50}}</ref>

In [[Carl Sagan]]'s novel ''[[Contact (novel)|Contact]]'' it is suggested that the creator of the universe buried a message deep within the digits of {{pi}}.<ref>{{harvnb|Arndt|Haenel|2006|p=14}}. This part of the story was omitted from the [[Contact (1997 US film)|film]] adaptation of the novel.</ref> The digits of {{pi}} have also been incorporated into the lyrics of the song "Pi" from the album ''[[Aerial (album)|Aerial]]'' by [[Kate Bush]],<ref>{{cite journal|title=Review of Aerial|first=Andy|last=Gill|journal=[[The Independent]]|date=4 November 2005|url=http://gaffa.org/reaching/rev_aer_UK5.html|quote=the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)|ref=harv}}</ref> and a song by [[Hard 'n Phirm]].<ref>{{cite web|quote=There was one song about Pi. Nothing like hearing people harmonizing over 200 digits.|title=PARTY CRASHER: Laughing With Hard 'N Phirm|publisher=SanDiego.com|first=Josh|last=Board|date=1 December 2010|url=http://local.sandiego.com/crasher/party-crasher-laughing-with-hard-n-phirm|ref=harv}}{{dead link|date=March 2016}}</ref>

Many schools in the United States observe [[Pi Day]] on 14&nbsp;March (written 3/14 in the US style).<ref>[http://www.piday.org/2008/2008-pi-day-activities-for-teachers/ Pi Day activities].</ref> {{pi}} and its digital representation are often used by self-described "math [[geek]]s" for [[inside joke]]s among mathematically and technologically minded groups. Several college [[cheering|cheers]] at the [[Massachusetts Institute of Technology]] include "3.14159".<ref name="MIT">[http://web.mit.edu/cheer/2004-2005SpecificWebPages/GeneralInformation/cheers.html MIT cheers]. Retrieved 12 April 2012.</ref> Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi.<ref>{{cite web|url=http://www.usatoday.com/story/news/nation-now/2015/03/14/pi-day-kids-videos/24753169/ |title=Happy Pi Day! Watch these stunning videos of kids reciting 3.14 |publisher=USAToday.com |date=2015-03-14 |accessdate=2015-03-14}}</ref> Pi Day in 2016 is also significant because the date including the year - 3/14/16 - is the value of pi rounded to 4 decimal places: 3.1416

During the 2011 auction for [[Nortel]]'s portfolio of valuable technology patents, [[Google]] made a series of unusually specific bids based on mathematical and scientific constants, including {{pi}}.<ref name="FP_2011">{{cite news|agency=Reuters|url=http://business.financialpost.com/2011/07/05/googles-strage-bids-for-nortel-patents/|title=Google's strange bids for Nortel patents|newspaper=FinancialPost.com|date=2011-07-05|accessdate=16 August 2011}}</ref>

[[File:Tau uses.jpg|thumb|right|alt=Pi Pie at Delft University|Some formulas using the 2{{pi}} definition of {{tau}}.|bottom]]

{{anchor|tau}} <!-- used in the redirect [[Tau (number)]] -->
In 1958 [[Albert Eagle]] proposed replacing {{pi}} by {{tau}} = {{pi}}/2 to simplify formulas.<ref name=AE1958>{{cite book|last=Eagle|first=Albert|title=The Elliptic Functions as They Should be: An Account, with Applications, of the Functions in a New Canonical Form|year=1958|publisher=Galloway and Porter, Ltd.|page=ix}}</ref> However, no other authors are known to use {{tau}} in this way. Some people use a different value, {{tau}} = 6.283185... = 2{{pi}},<ref>Sequence {{OEIS2C|A019692}},</ref> arguing that {{tau}}, as the number of radians in one [[Turn (geometry)|turn]] or as the ratio of a circle's circumference to its radius rather than its diameter, is more natural than {{pi}} and simplifies many formulas.<ref>{{cite journal|last=Abbott|first=Stephen|title=My Conversion to Tauism|journal=Math Horizons|date=April 2012|volume=19|issue=4|page=34|doi=10.4169/mathhorizons.19.4.34|url=http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf|ref=harv}}</ref><ref>{{cite journal|last=Palais|first=Robert|title={{pi}} Is Wrong!|journal=The Mathematical Intelligencer|year=2001|volume=23|issue=3|pages=7–8|doi=10.1007/BF03026846|url=http://www.math.utah.edu/~palais/pi.pdf|ref=harv}}</ref> Celebrations of this number, because it approximately equals 6.28, by making 28&nbsp;June "Tau Day" and eating "twice the pie",<ref>{{Cite web |title=Tau Day: Why you should eat twice the pie – Light Years – CNN.com Blogs |url=http://lightyears.blogs.cnn.com/2012/06/28/tau-day-why-you-should-eat-twice-the-pie/ |access-date=20 March 2016 |archive-date=12 January 2013 |archive-url=https://web.archive.org/web/20130112211136/http://lightyears.blogs.cnn.com/2012/06/28/tau-day-why-you-should-eat-twice-the-pie/ }}</ref> have been reported in the media. However this use of {{math.|τ}} has not made its way into mainstream mathematics.<ref>{{cite journal|url=http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp|title=Life of pi in no danger – Experts cold-shoulder campaign to replace with tau|journal=Telegraph India|date=2011-06-30}}</ref>

In 1897, an amateur American mathematician attempted to persuade the [[Indiana General Assembly|Indiana legislature]] to pass the [[Indiana Pi Bill]], which described a method to [[Squaring the circle|square the circle]] and contained text that implied various incorrect values for {{pi}}, including 3.2. The bill is notorious as an attempt to establish a value of scientific constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate.<ref>{{harvnb|Arndt|Haenel|2006|pp=211–212}}<br />{{harvnb|Posamentier|Lehmann|2004|pp=36–37}}<br />{{cite journal|last1=Hallerberg|first1=Arthur|date=May 1977|title=Indiana's squared circle|journal=Mathematics Magazine|volume=50|issue=3|pages=136–140|jstor=2689499|doi=10.2307/2689499|ref=harv}}</ref>

==Notes==
;Footnotes
{{Reflist|20em}}

;References
{{refbegin|30em}}
* {{cite book|last=Arndt|first=Jörg|last2=Haenel|first2=Christoph|title=Pi Unleashed|publisher=Springer-Verlag|year=2006|isbn=978-3-540-66572-4 <!--isbn only volume 1-->|url=https://books.google.com/?id=QwwcmweJCDQC&printsec=frontcover#v=onepage&q&f=false|ref=harv|accessdate=2013-06-05}} English translation by Catriona and David Lischka.
* {{cite book|last=Ayers|first=Frank|title=Calculus|publisher=McGraw-Hill|year=1964|isbn=978-0-070-02653-7|ref=harv}}
* {{cite book|last=Berggren|first=Lennart|last2=Borwein|first2=Jonathan|author2-link=Jonathan Borwein|last3=Borwein|first3=Peter|author3-link=Peter Borwein|title=Pi: a Source Book|publisher=Springer-Verlag|year=1997|isbn=978-0-387-20571-7|ref=harv}}
* {{cite book|last=Beckmann|first=Peter|title=History of Pi|publisher=St. Martin's Press|year=1989|origyear=1974|isbn=978-0-88029-418-8|ref=harv}}
* {{cite book|last=Borwein|first=Jonathan|author1-link=|last2=Borwein|first2=Peter|author2-link=|title=Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity|publisher=Wiley|year=1987|isbn=978-0-471-31515-5|ref=harv}}
* {{cite book|last=Boyer|first=Carl B.|last2=Merzbach|first2=Uta C.|year=1991|title=A History of Mathematics|edition=2|publisher=Wiley|isbn=978-0-471-54397-8|ref=harv}}<!-- Year from ISBN. Original citatation was just to Boyer. Possible that edition is wrong and therefore page is wrong. Editions: Boyer 1968, Boyer/Merzbach 1989, Boyer/Merzbach 1991, Merzbach/Boyer 2010, Merzbach/Boyer 2011. Verify second: Hui and 3072-sided polygon is on cited page 202 of 1991 edition; page 228 of 1968 edition. Google snippet has a hit for 3.1456 on page 168 for 1991, but does not show the number. -->
* {{cite book|last=Bronshteĭn|first=Ilia|last2=Semendiaev|first2=K. A.|title=A Guide Book to Mathematics|publisher=H. Deutsch|year=1971|isbn= 978-3-871-44095-3|ref=harv}}
* {{cite book|last=Eymard|first=Pierre|last2=Lafon|first2=Jean Pierre|title=The Number Pi|publisher=American Mathematical Society|year=1999|isbn=978-0-8218-3246-2|ref=harv}}, English translation by Stephen Wilson.
* {{cite book|last=Joseph|first=George Gheverghese|title=The Crest of the Peacock: Non-European Roots of Mathematics|publisher=Princeton University Press|year=1991|isbn=978-0-691-13526-7|url=https://books.google.com/?id=c-xT0KNJp0cC&printsec=frontcover#v=onepage&q&f=false%7C|ref=harv|accessdate=2013-06-05}}<!-- This ISBN is for the third edition from 2011! -->
* {{cite book|last=Posamentier|first=Alfred S.|last2=Lehmann|first2=Ingmar|title=Pi: A Biography of the World's Most Mysterious Number|publisher=Prometheus Books|year=2004|isbn=978-1-59102-200-8|ref=harv}}
* {{cite journal|last=Reitwiesner|first=George|title=An ENIAC Determination of pi and e to 2000 Decimal Places|journal=Mathematical Tables and Other Aids to Computation|year=1950|volume=4|issue= 29|pages=11–15|doi=10.2307/2002695|ref=harv }}
* {{cite journal|last=Roy|first=Ranjan|title=The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha|journal=Mathematics Magazine|volume=63|issue=5|year=1990|pages=291–306|doi=10.2307/2690896|ref=harv }}
* {{cite journal|last=Schepler|first=H. C.|title=The Chronology of Pi|journal=Mathematics Magazine|publisher=Mathematical Association of America|year=1950|volume=23|issue=3|pages=165–170 (Jan/Feb), 216–228 (Mar/Apr), and 279–283 (May/Jun)|doi=10.2307/3029284|ref=harv }}. [<!-- http://www.jstor.org/stable/3029284 -->http://www.jstor.org/discover/10.2307/3029284?sid=21105295462031&uid=2&uid=4 issue 3 Jan/Feb], [http://www.jstor.org/discover/10.2307/3029832?sid=21105295462031&uid=2&uid=4 issue 4 Mar/Apr], [http://www.jstor.org/stable/3029000 issue 5 May/Jun]
{{refend}}

==Further reading==
{{refbegin|30em}}
* {{cite book|last=Blatner|first=David|title=The Joy of Pi|publisher=Walker & Company|year=1999|isbn=978-0-8027-7562-7|doi= }}
* {{cite journal | last1 = Borwein | first1 = Jonathan | authorlink = Jonathan Borwein | authorlink2 = Peter Borwein | last2 = Borwein | first2 = Peter | year = 1984 | title = The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions | url = | journal = SIAM Review | volume = 26 | issue = 3| pages = 351–365 | doi=10.1137/1026073}}
* {{cite journal | last1 = Borwein | first1 = Jonathan | authorlink3 = David H. Bailey | last2 = Borwein | first2 = Peter | last3 = Bailey | first3 = David H. | year = 1989 | title = Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi | url = | journal = The American Mathematical Monthly | volume = 96 | issue = 3| pages = 201–219 | doi=10.2307/2325206| jstor = 2325206 }}
* [[Chudnovsky brothers|Chudnovsky, David V.]] and [[Chudnovsky brothers|Chudnovsky, Gregory V.]], "Approximations and Complex Multiplication According to Ramanujan", in ''Ramanujan Revisited'' (G.E. Andrews et al. Eds), Academic Press, 1988, pp 375–396, 468–472
* Cox, David A., "The Arithmetic-Geometric Mean of Gauss", ''L' Ensignement Mathematique'', '''30'''(1984) 275–330
* [[Jean-Paul Delahaye|Delahaye, Jean-Paul]], "Le Fascinant Nombre Pi", Paris: Bibliothèque Pour la Science (1997) ISBN 2902918259
* {{cite journal | last1 = Engels | first1 = Hermann | year = 1977 | title = Quadrature of the Circle in Ancient Egypt | url = | journal = Historia Mathematica | volume = 4 | issue = 2| pages = 137–140 | doi=10.1016/0315-0860(77)90104-5}}
* [[Leonhard Euler|Euler, Leonhard]], "On the Use of the Discovered Fractions to Sum Infinite Series", in ''Introduction to Analysis of the Infinite. Book I'', translated from the Latin by J. D. Blanton, Springer-Verlag, 1964, pp 137–153
* Hardy, G. H. and Wright E. M., ''An Introduction to the Theory of Numbers'' first published 1938, fifth Edition 1979 with additions 2000, Clarendon Press, Oxford U.K.
* Heath, T. L., ''The Works of Archimedes'', Cambridge, 1897; reprinted in ''The Works of Archimedes with The Method of Archimedes'', Dover, 1953, pp 91–98
* [[Christiaan Huygens|Huygens, Christiaan]], "De Circuli Magnitudine Inventa", ''Christiani Hugenii Opera Varia I'', Leiden 1724, pp 384–388
* {{cite journal | last1 = Lay-Yong | first1 = Lam | authorlink = Lam Lay Yong | last2 = Tian-Se | first2 = Ang | year = 1986 | title = Circle Measurements in Ancient China | url = | journal = Historia Mathematica | volume = 13 | issue = 4| pages = 325–340 | doi=10.1016/0315-0860(86)90055-8}}
* {{cite journal | last1 = Lindemann | first1 = Ferdinand | authorlink = Ferdinand von Lindemann | year = 1882 | title = Ueber die Zahl pi | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0020&DMDID=DMDLOG_0031&L=1 | journal = Mathematische Annalen | volume = 20 | issue = 2 | pages = 213–225 | doi = 10.1007/bf01446522 | access-date = 20 March 2016 | archive-date = 22 January 2015 | archive-url = https://web.archive.org/web/20150122054610/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0020&DMDID=DMDLOG_0031&L=1 | url-status = dead }}
* {{cite journal | last1 = Matar | first1 = K. Mukunda | last2 = Rajagonal | first2 = C. | year = 1944 | title = On the Hindu Quadrature of the Circle" (Appendix by K. Balagangadharan) | url = | journal = Journal of the Bombay Branch of the Royal Asiatic Society | volume = 20 | issue = | pages = 77–82 }}
* [[Ivan M. Niven|Niven, Ivan]], "A Simple Proof that pi Is Irrational", ''Bulletin of the American Mathematical Society'', '''53''':7 (July 1947), 507
* [[Srinivasa Ramanujan|Ramanujan, Srinivasa]], "Modular Equations and Approximations to π", ''Quarterly Journal of Pure and Applied Mathematics'', '''XLV''', 1914, 350–372. Reprinted in G.H. Hardy, P.V. Seshu Aiyar, and B. M. Wilson (eds), ''Srinivasa Ramanujan: Collected Papers'', 1927 (reprinted 2000), pp 23–29
* [[William Shanks|Shanks, William]], ''Contributions to Mathematics {{sic|Comprising|hide=y}} Chiefly of the Rectification of the Circle to 607 Places of Decimals'', 1853, pp. i–xvi, 10
* {{cite journal | last1 = Shanks | first1 = Daniel | authorlink = Daniel Shanks | authorlink2 = John Wrench | last2 = Wrench | first2 = John William | year = 1962 | title = Calculation of pi to 100,000 Decimals | url = | journal = Mathematics of Computation | volume = 16 | issue = 77| pages = 76–99 | doi=10.1090/s0025-5718-1962-0136051-9}}
* Tropfke, Johannes, ''Geschichte Der Elementar-Mathematik in Systematischer Darstellung'' (''The history of elementary mathematics''), BiblioBazaar, 2009 (reprint), ISBN 978-1-113-08573-3
* [[François Viète|Viete, Francois]], ''Variorum de Rebus Mathematicis Reponsorum Liber VII. F. Viete, Opera Mathematica'' (reprint), Georg Olms Verlag, 1970, pp 398–401, 436–446
* [[Stan Wagon|Wagon, Stan]], "Is Pi Normal?", ''The Mathematical Intelligencer'', '''7''':3(1985) 65–67
* [[John Wallis|Wallis, John]], ''Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadratum, aliaque difficiliora Matheseos Problemata'', Oxford 1655–6. Reprinted in vol. 1 (pp 357–478) of ''Opera Mathematica'', Oxford 1693
* Zebrowski, Ernest, ''A History of the Circle: Mathematical Reasoning and the Physical Universe'', Rutgers University Press, 1999, ISBN 978-0-8135-2898-4
{{refend}}

==External links==
{{Commons category|Pi}}
* {{dmoz|Science/Math/Recreations/Specific_Numbers/Pi/Digits/|Digits of Pi}}
* [http://mathworld.wolfram.com/Pi.html "Pi"] at Wolfram Mathworld
* [http://www.wolframalpha.com/input/?i=Representations+of+Pi Representations of Pi] at [[Wolfram Alpha]]
* [http://www.subidiom.com/pi Pi Search Engine]: 2 billion searchable digits of {{pi}}, {{math.|{{sqrt|2}}}}, and {{math.|''e''}}
* {{cite web|last=Eaves|first=Laurence|title={{pi}} – Pi|url=http://www.sixtysymbols.com/videos/pi.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|authorlink=Laurence Eaves|year=2009}}
* {{cite web|last=Grime|first=Dr. James|title=Pi is Beautiful – Numberphile|url=https://www.youtube.com/watch?v=NPoj8lk9Fo4|work=Numberphile|publisher=[[Brady Haran]]|year=2014}}
{{featured article}}
* Demonstration by Lambert (1761) of irrationality of {{pi}}, [https://www.bibnum.education.fr/mathematiques/theorie-des-nombres/lambert-et-l-irrationalite-de-p-1761 online] {{Webarchive|url=https://web.archive.org/web/20141231045534/https://www.bibnum.education.fr/mathematiques/theorie-des-nombres/lambert-et-l-irrationalite-de-p-1761 |date=31 December 2014 }} and analyzed ''[https://www.bibnum.education.fr/sites/default/files/24-lambert-analysis.pdf BibNum] {{Webarchive|url=https://web.archive.org/web/20150402115151/https://www.bibnum.education.fr/sites/default/files/24-lambert-analysis.pdf |date=2 April 2015 }}'' (PDF).

{{Irrational number}}

{{Authority control}}

[[Category:Beta| ]]
[[Category:Complex analysis]]
[[Category:Mathematical series]]
[[Category:Real transcendental numbers]]