View source for Pi

{{Short description|Number, approximately 3.14}}
{{Use Oxford spelling|date=July 2020}}
The number '''{{pi}}''' ({{IPAc-en|p|aɪ}}; spelled out as "'''pi'''") is a [[Mathematical constant|mathematical constant]] that is the [[Ratio|ratio]] of a [[Circle|circle]]'s [[Circumference|circumference]] to its [[Diameter|diameter]], approximately equal to 3.14159. The number {{pi}} appears in many formulae across [[Mathematics|mathematics]] and [[Physics:Physics|physics]]. It is an [[Irrational number|irrational number]], meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as <math>\tfrac{22}{7}</math> are commonly [[Approximations of π|used to approximate it]]. Consequently, its [[Decimal representation|decimal representation]] never ends, nor [[Repeating decimal|enters a permanently repeating pattern]]. It is a [[Transcendental number|transcendental number]], meaning that it cannot be a solution of an [[Equation|equation]] involving only finite sums, products, powers, and integers. The transcendence of {{pi}} implies that it is impossible to solve the ancient challenge of [[Squaring the circle|squaring the circle]] with a compass and straightedge. The decimal digits of {{pi}} appear to be [[Random sequence|randomly distributed]],{{efn|In particular, {{pi}} is conjectured to be a [[Normal number|normal number]], which implies a specific kind of statistical randomness on its digits in all bases.}} but no proof of this conjecture has been found.

For thousands of years, mathematicians have attempted to extend their understanding of {{pi}}, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of {{pi}} for practical computations. Around 250{{Nbsp}}BC, the Greek mathematician [[Biography:Archimedes|Archimedes]] created an algorithm to approximate {{pi}} with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated {{pi}} to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for {{pi}}, based on [[Series (mathematics)|infinite series]], was discovered a millennium later.{{sfn|Andrews|Askey|Roy|1999|p=59}}<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}}
</ref> The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician [[Biography:William Jones (mathematician)|William Jones]] in 1706.<ref name=jones>{{cite book |last=Jones |first=William |author-link=William Jones (mathematician) |year=1706 |title=Synopsis Palmariorum Matheseos |place=London |publisher=J. Wale |url=https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ |pages=[https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/ 243], [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ 263] |quote-page=263 |quote=There are various other ways of finding the ''Lengths'', or ''Areas'' of particular ''Curve Lines'' or ''Planes'', which may very much facilitate the Practice; as for instance, in the ''Circle'', the Diameter is to Circumference as 1 to <br><math>
\overline{\tfrac{16}5 - \tfrac4{239}}
- \tfrac13\overline{\tfrac{16}{5^3} - \tfrac4{239^3}}
+ \tfrac15\overline{\tfrac{16}{5^5} - \tfrac4{239^5}}
-,\, \&c. =</math><br>{{math|1=3.14159, &''c.'' = ''π''}}. This ''Series'' (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. ''John Machin''; and by means thereof, ''[[Biography:Ludolph van Ceulen|Van Ceulen]]''{{'}}s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
}}
<p>Reprinted in {{cite book |last=Smith |first=David Eugene |year=1929 |title=A Source Book in Mathematics |publisher=McGraw–Hill |chapter=William Jones: The First Use of {{mvar|π}} for the Circle Ratio |chapter-url=https://archive.org/details/sourcebookinmath1929smit/page/346/ |pages=346–347 }}</p></ref>

The invention of [[Calculus|calculus]] soon led to the calculation of hundreds of digits of {{pi}}, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and [[Computer science|computer scientists]] have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of {{pi}} to many trillions of digits.<ref>{{cite web |url=http://www.pi2e.ch/ |title=π<sup>e</sup> trillion digits of π |archive-url=https://web.archive.org/web/20161206063441/http://www.pi2e.ch/ |website=pi2e.ch |archive-date=6 December 2016 |url-status=live}} <!-- – the exact number of digits increases periodically – it should not be included in this article by citing only a primary reference source. --></ref><ref>{{Cite web |last=Haruka Iwao |first=Emma |url=https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud |title=Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud |website=[[Google Cloud Platform]] |date=14 March 2019 |access-date=12 April 2019 |archive-url=https://web.archive.org/web/20191019023120/https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud |archive-date=19 October 2019 |url-status=live}}</ref> These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records.{{sfn|Arndt|Haenel|2006|p=17}}<ref>{{cite journal |last1=Bailey |first1=David H. |last2=Plouffe |first2=Simon M. |last3=Borwein |first3=Peter B. |last4=Borwein |first4=Jonathan M. |title=The quest for PI |journal=[[The Mathematical Intelligencer]] |volume=19 |issue=1 |year=1997 |pages=50–56 |issn=0343-6993 |doi=10.1007/BF03024340 |citeseerx=10.1.1.138.7085|s2cid=14318695 }}</ref> The extensive computations involved have also been used to test [[Supercomputer|supercomputer]]s as well as stress testing consumer computer hardware. 

Because its definition relates to the circle, {{pi}} is found in many formulae in [[Trigonometry|trigonometry]] and [[Geometry|geometry]], especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as [[Astronomy:Cosmology|cosmology]], [[Fractal|fractal]]s, [[Physics:Thermodynamics|thermodynamics]], [[Physics:Mechanics|mechanics]], and [[Physics:Electromagnetism|electromagnetism]]. It also appears in areas having little to do with geometry, such as [[Number theory|number theory]] and [[Statistics|statistics]], and in modern [[Mathematical analysis|mathematical analysis]] can be defined without any reference to geometry. The ubiquity of {{pi}} makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to {{pi}} have been published, and record-setting calculations of the digits of {{pi}} often result in news headlines.


== Fundamentals ==

=== Name ===
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter {{pi}}, sometimes spelled out as ''pi.''<ref name=firstPi/> In English, {{pi}} is pronounced as "pie" ({{IPAc-en|p|aɪ}} {{respell|PY}}).<ref>{{cite web |url=http://dictionary.reference.com/browse/pi?s=t |title=pi |publisher=Dictionary.reference.com |date=2 March 1993 |access-date=18 June 2012|url-status=live |archive-url=https://web.archive.org/web/20140728121603/http://dictionary.reference.com/browse/pi?s=t |archive-date=28 July 2014}}</ref> In mathematical use, the lowercase letter {{pi}} is distinguished from its capitalized and enlarged counterpart {{math|Π}}, which denotes a [[Multiplication#Product of a sequence|product of a sequence]], analogous to how {{math|Σ}} denotes [[Summation|summation]].

The choice of the symbol {{pi}} is discussed in the section ''Adoption of the symbol {{pi}}''.

=== Definition ===
[[File:Pi eq C over d.svg|alt=A diagram of a circle, with the width labelled as diameter, and the perimeter labelled as circumference|thumb|right|The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called {{pi}}.]]
{{pi}} is commonly defined as the [[Ratio|ratio]] of a [[Circle|circle]]'s [[Circumference|circumference]] {{math|''C''}} to its [[Diameter|diameter]] {{math|''d''}}:{{sfn|Arndt|Haenel|2006|p=8}}

<math display="block"> \pi = \frac{C}{d}</math>

The ratio <math display="inline">\frac{C}{d}</math> 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 display="inline">\frac{C}{d}</math>. 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|curve (non-Euclidean) geometry]], these new circles will no longer satisfy the formula <math display="inline">\pi=\frac{C}{d}</math>.{{sfn|Arndt|Haenel|2006|p=8}}

Here, the circumference of a circle is the [[Arc length|arc length]] around the [[Perimeter|perimeter]] of the circle, a quantity which can be formally defined independently of geometry using [[Limit (mathematics)|limits]]—a concept in [[Calculus|calculus]].<ref>{{cite book |first=Tom |last=Apostol |title=Calculus|volume=1 |publisher=Wiley |edition=2nd |year=1967|page= 102|quote=From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length}}</ref>  For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation <math display="inline">x^2+y^2=1</math>, as the [[Integral|integral]]:{{sfn|Remmert|2012|p=129}}

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

An integral such as this was adopted as the definition of {{pi}} by [[Biography:Karl Weierstrass|Karl Weierstrass]], who defined it directly as an integral in 1841.{{efn|The precise integral that Weierstrass used was <math>\pi=\int_{-\infty}^\infty\frac{dx}{1+x^2}.</math> {{harvnb|Remmert|2012|p=148}} }}

Integration is no longer commonly used in a first analytical definition because, as {{harvnb|Remmert|2012}} explains, [[Differential 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>{{cite book |first=Richard |last=Baltzer |title=Die Elemente der Mathematik |language=de |trans-title=The Elements of Mathematics |year=1870 |page=195 |url=https://archive.org/details/dieelementederm02baltgoog |publisher=Hirzel |url-status=live |archive-url=https://web.archive.org/web/20160914204826/https://archive.org/details/dieelementederm02baltgoog |archive-date=14 September 2016}}</ref> and popularized by [[Biography:Edmund Landau|Edmund Landau]],<ref>{{cite book |first=Edmund |last=Landau |title=Einführung in die Differentialrechnung und Integralrechnung |language=de |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.{{sfn|Arndt|Haenel|2006|p=8}}{{sfn|Remmert|2012|p=129}}<ref name="Rudin 1976">{{cite book |last=Rudin |first=Walter |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi|url-access=registration |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8|page=183}}</ref> {{pi}} is also the smallest positive number at which the [[Sine|sine]] function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a [[Power series|power series]],<ref>{{cite book |last=Rudin |first=Walter |title=Real and complex analysis |publisher=McGraw-Hill |year=1986|page= 2}}</ref> or as the solution of a [[Differential equation|differential equation]].<ref name="Rudin 1976" />

In a similar spirit, {{pi}} can be defined 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 display=block>\{\dots,-2\pi i, 0, 2\pi i, 4\pi i,\dots\} = \{2\pi ki\mid k\in\mathbb Z\}</math>

and there is a unique positive real number {{pi}} with this property.{{sfn|Remmert|2012|p=129}}<ref>{{cite book |first=Lars |last=Ahlfors |title=Complex analysis |publisher=McGraw-Hill |year=1966 |page=46}}</ref>

A variation on the same idea, making use of sophisticated mathematical concepts of [[Topology|topology]] and [[Algebra|algebra]], is the following theorem:<ref>{{cite book |last=Bourbaki |first=Nicolas |title=Topologie generale |publisher=Springer |year=1981|at=§VIII.2}}</ref> there is a unique ([[Up to|up to]] [[Automorphism|automorphism]]) [[Continuous function|continuous]] [[Isomorphism|isomorphism]] from the [[Group (mathematics)|group]] '''R'''/'''Z''' of real numbers under addition [[Quotient group|modulo]] integers (the [[Circle group|circle group]]), onto the multiplicative group of complex numbers of [[Absolute value|absolute value]] one. The number {{pi}} is then defined as half the magnitude of the derivative of this homomorphism.<ref name="Nicolas Bourbaki">{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Fonctions d'une variable réelle |language=fr |publisher=Springer |year=1979|at= §II.3}}</ref>

=== Irrationality and normality ===
{{pi}} is an [[Irrational number|irrational number]], meaning that it cannot be written as the [[Rational number|ratio of two integers]]. Fractions such as {{math|{{sfrac|22|7}}}} and {{math|{{sfrac|355|113}}}} are commonly used to approximate {{pi}}, but no common fraction (ratio of whole numbers) can be its exact value.{{sfn|Arndt|Haenel|2006|p=5}} Because {{pi}} is irrational, it has an infinite number of digits in its [[Decimal representation|decimal representation]], and 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 ''[[Philosophy:Reductio ad absurdum|reductio ad absurdum]]'' technique. The degree to which {{pi}} can be approximated by [[Rational number|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 Surveys |volume=53 |issue=3 |pages=570–572 |doi=10.1070/RM2008v063n03ABEH004543 |bibcode=2008RuMaS..63..570S|s2cid=250798202 }}</ref>

The digits of {{pi}} have no apparent pattern and have passed tests for [[Statistical randomness|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. The conjecture that {{pi}} is [[Normal number|normal]] has not been proven or disproven.{{sfn|Arndt|Haenel|2006|pp=22–23}}

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 frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.{{sfn|Arndt|Haenel|2006|pp=22, 28–30}} Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the [[Infinite monkey theorem|infinite monkey theorem]]. Thus, because the sequence of {{pi}}'s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a [[Physics:Six nines in pi|sequence of six consecutive 9s]] that begins at the 762nd decimal place of the decimal representation of {{pi}}.{{sfn|Arndt|Haenel|2006|p=3}} This is also called the "Feynman point" in [[Philosophy:Mathematical folklore|mathematical folklore]], after [[Biography:Richard Feynman|Richard Feynman]], although no connection to Feynman is known.

=== Transcendence ===
{{See also|Lindemann–Weierstrass theorem}}[[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|transcendental number]], [[Squaring the circle|squaring the circle]] is not possible in a finite number of steps using the classical tools of compass and straightedge.|left]]
In addition to being irrational, {{pi}} is also a [[Transcendental number|transcendental number]], which means that it is not the solution of any non-constant polynomial equation with [[Rational number|rational]] coefficients, such as <math display="inline">\frac{x^5}{120}-\frac{x^3}{6}+x=0</math>.{{sfn|Arndt|Haenel|2006|p=6}}{{efn|The polynomial shown is the first few terms of the [[Taylor series]] expansion of the [[Sine|sine]] function.}}

The transcendence of {{pi}} has two important consequences: First, {{pi}} cannot be expressed using any finite combination of rational numbers and square roots or [[Nth root|''n''-th roots]] (such as <math>\sqrt[3]{31}</math> or <math>\sqrt{10}</math>). Second, since no transcendental number can be [[Constructible number|constructed]] with 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 exactly 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 [[History:Classical antiquity|classical antiquity]].<ref>{{harvnb|Eymard|Lafon|2004|p=129}}</ref> Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.<ref>{{cite book |last=Beckmann |first=Peter |title=History of Pi |publisher=St. Martin's Press |year=1989 |orig-year=1974 |isbn=978-0-88029-418-8 |page=37}}</ref><ref>{{cite book |last1=Schlager |first1=Neil |last2=Lauer |first2=Josh |title=Science and Its Times: Understanding the Social Significance of Scientific Discovery |publisher=Gale Group |year=2001 |isbn=978-0-7876-3933-4|url-access=registration |url=https://archive.org/details/scienceitstimesu0000unse|access-date=19 December 2019|archive-url=https://web.archive.org/web/20191213112426/https://archive.org/details/scienceitstimesu0000unse|archive-date=13 December 2019|url-status=live}}, p. 185.</ref>

=== Continued fractions ===
As an irrational number, {{pi}} cannot be represented as a common fraction. But every number, including {{pi}}, can be represented by an infinite<!--rationals have infinitely many 0's in the CF representation--> series of nested fractions, called a [[Continued fraction|continued fraction]]:

<math display=block>
\pi = 3+\textstyle \cfrac{1}{7+\textstyle \cfrac{1}{15+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{292+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\ddots}}}}}}}
</math>

Truncating the continued fraction at any point yields a rational approximation for {{pi}}; the first four of these are {{math|3}}, {{math|{{sfrac|22|7}}}}, {{math|{{sfrac|333|106}}}}, and {{math|{{sfrac|355|113}}}}. These numbers are among the best-known and most 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|2004|p=78}}</ref> Because {{pi}} is transcendental, it is by definition not [[Algebraic number|algebraic]] and so cannot be a quadratic irrational. Therefore, {{pi}} cannot have a [[Periodic continued fraction|periodic continued fraction]]. Although the simple continued fraction for {{pi}} (shown above) also does not exhibit any other obvious pattern,{{sfn|Arndt|Haenel|2006|p=33}}<ref name=mollin>{{cite journal|last=Mollin|first=R. A.|issue=3|journal=Nieuw Archief voor Wiskunde|mr=1743850|pages=383–405|title=Continued fraction gems|volume=17|year=1999}}</ref> several [[Generalized continued fraction|generalized continued fraction]]s 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>

<math display=block>
\begin{align}
\pi &= 3+ \cfrac
  {1^2}{6+ \cfrac
    {3^2}{6+ \cfrac
      {5^2}{6+ \cfrac
        {7^2}{6+ \ddots}}}} 
= \cfrac
  {4}{1+ \cfrac
    {1^2}{2+ \cfrac
      {3^2}{2+ \cfrac
        {5^2}{2+ \ddots}}}} 
= \cfrac
  {4}{1+ \cfrac
    {1^2}{3+ \cfrac
      {2^2}{5+ \cfrac
        {3^2}{7+ \ddots}}}}
\end{align}
</math>

The middle of these is due to the mid-17th century mathematician William Brouncker, see §&nbsp;Brouncker's formula.

=== Approximate value and digits ===
Some [[Approximations of π|approximations of ''pi'']] include:
* '''Integers''': 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}}, {{sfrac|104348|33215}}, and {{sfrac|245850922|78256779}}.<ref name="Eymard 1999 78" /> (List is selected terms from {{OEIS2C|id=A063674}} and {{OEIS2C|id=A063673}}.)
* '''Digits''': The first 50 decimal digits are {{gaps|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510...}}{{sfn|Arndt|Haenel|2006|p=240}} (see {{OEIS2C|id=A000796}})

'''Digits in other number systems'''
* The first 48 [[Binary number#Representing real numbers|binary]] ([[Radix|base]] 2) digits (called [[Bit|bit]]s) are {{gaps|11.0010|0100|0011|1111|0110|1010|1000|1000|1000|0101|1010|0011...}} (see {{OEIS2C|id=A004601}})
* The first 38 digits in [[Ternary numeral system|ternary]] (base 3) are {{gaps|10.010|211|0122|220|102|110|021|111|102|212|222|201...}} (see {{OEIS2C|id=A004602}})
* The first 20 digits in [[Hexadecimal|hexadecimal]] (base 16) are  {{gaps|3.243F|6A88|85A3|08D3|1319...}}{{sfn|Arndt|Haenel|2006|p=242}} (see {{OEIS2C|id=A062964}})
* The first five [[Sexagesimal|sexagesimal]] (base 60) digits are 3;8,29,44,0,47<ref>{{cite journal |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 |year=1978|s2cid=126383231 }} [[Biography:Ptolemy|Ptolemy]] used a three-sexagesimal-digit approximation, and [[Biography:Jamshīd al-Kāshī|Jamshīd al-Kāshī]] expanded this to nine digits; see {{cite book |last=Aaboe |first=Asger |year=1964 |title=Episodes from the Early History of Mathematics |series=New Mathematical Library |volume=13 |publisher=Random House |location=New York |page=125 |url=https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA125 |url-status=live |archive-url=https://web.archive.org/web/20161129205051/https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA125 |archive-date=29 November 2016 |df=dmy-all |isbn=978-0-88385-613-0}}</ref> (see {{OEIS2C|id=A060707}})

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

Any [[Complex number|complex number]], say {{Mvar|z}}, can be expressed using a pair of [[Real number|real number]]s. In the [[Polar coordinate system#Complex numbers|polar coordinate system]], one number ([[Radius|radius]] or {{Mvar|r}}) is used to represent {{Mvar|z}}'s distance from the [[Origin (mathematics)|origin]] of the [[Complex plane|complex plane]], and the other (angle or {{Mvar|φ}}) the counter-clockwise [[Physics:Rotation|rotation]] from the positive real line:{{sfn|Abramson|2014|loc=[https://openstax.org/books/precalculus/pages/8-5-polar-form-of-complex-numbers Section 8.5: Polar form of complex numbers]}}

<math display=block>z = r\cdot(\cos\varphi + i\sin\varphi),</math>

where {{Mvar|i}} is the [[Imaginary unit|imaginary unit]] satisfying <math>i^2=-1</math>. The frequent appearance of {{pi}} in [[Complex analysis|complex analysis]] can be related to the behaviour of the [[Exponential function|exponential function]] of a complex variable, described by [[Euler's formula]]:<ref name="EF">{{harvnb|Bronshteĭn|Semendiaev|1971|p=592}}</ref>

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

where [[E (mathematical constant)|the constant {{math|''e''}}]] is the base of the [[Natural logarithm|natural logarithm]]. This formula establishes a correspondence between imaginary powers of {{math|''e''}} and points on the [[Unit circle|unit circle]] centred at the origin of the complex plane. Setting <math>\phi=\pi</math> in Euler's formula results in [[Euler's identity]], celebrated in mathematics due to it containing five important mathematical constants:<ref name="EF" /><ref>{{cite book|last=Maor|first=Eli|title=E: The Story of a Number|publisher=Princeton University Press|year=2009|page=160|isbn=978-0-691-14134-3}}</ref>

<math display=block>e^{i \pi} + 1 = 0.</math>

There are {{math|''n''}} different [[Complex number|complex number]]s {{Mvar|z}} satisfying <math>z^n=1</math>, and these are called the "{{math|''n''}}-th [[Root of unity|roots of unity]]"{{sfn|Andrews|Askey|Roy|1999|p=14}} and are given by the formula:

<math display=block>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math>

==History== <!-- History of pi links here -->
{{See also|Chronology of computation of π|l1=Chronology of computation of {{pi}}}}

=== Antiquity ===
The best-known approximations to {{pi}} dating [[History: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.

The earliest written approximations of {{pi}} are found in Babylon and Egypt, 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;3.125.{{sfn|Arndt|Haenel|2006|p=167}} 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 <math display="inline">\bigl(\frac{16}{9}\bigr)^2\approx3.16</math>.<ref name=mollin/>{{sfn|Arndt|Haenel|2006|p=167}} Although some [[Unsolved:Pyramidology|pyramidologists]] have theorized that the Great Pyramid of Giza was built with proportions related to {{pi}}, this theory is not widely accepted by scholars.<ref>{{Cite book |pages=67–77, 165–166 |title=The Shape of the Great Pyramid |first=Roger |last=Herz-Fischler |publisher=Wilfrid Laurier University Press |year=2000 |isbn=978-0-88920-324-2 |url=https://books.google.com/books?id=066T3YLuhA0C&pg=67 |access-date=5 June 2013 |url-status=live |archive-url=https://web.archive.org/web/20161129205154/https://books.google.com/books?id=066T3YLuhA0C&pg=67 |archive-date=29 November 2016}}</ref>
In the [[Religion:Shulba Sutras|Shulba Sutras]] of Indian mathematics, dating to an oral tradition from the first or second millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.<ref>{{cite book|page=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA27 27]|title=Mathematics in India|title-link=Mathematics in India (book)|first=Kim|last=Plofker|date= 2009|publisher=Princeton University Press|isbn=978-0691120676}}</ref>

=== Polygon approximation era ===
[[File:Archimedes pi.svg|upright=1.59|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.|left]]
[[File:Domenico-Fetti Archimedes 1620.jpg|alt=A painting of a man studying|left|thumb|Archimedes developed the polygonal approach to approximating {{pi}}.]]
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 [[Biography:Archimedes|Archimedes]].{{sfn|Arndt|Haenel|2006|p=170}} This polygonal algorithm dominated for over 1,000 years, and as a result {{pi}} is sometimes referred to as Archimedes's constant.{{sfn|Arndt|Haenel|2006|pp=175, 205}} 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 name=life-of-pi>{{cite book|last=Borwein|first=Jonathan M.|editor1-last=Sidoli|editor1-first=Nathan|editor2-last=Van Brummelen|editor2-first=Glen|contribution=The life of {{pi}}: from Archimedes to ENIAC and beyond|doi=10.1007/978-3-642-36736-6_24|location=Heidelberg|mr=3203895|pages=531–561|publisher=Springer|title=From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren|year=2014}}</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}}}}.{{sfn|Arndt|Haenel|2006|p=171}} Around 150&nbsp;AD, Greek-Roman scientist [[Biography:Ptolemy|Ptolemy]], in his ''[[Astronomy:Almagest|Almagest]]'', gave a value for {{pi}} of 3.1416, which he may have obtained from Archimedes or from [[Biography:Apollonius of Perga|Apollonius of Perga]].{{sfn|Arndt|Haenel|2006|p=176}}{{sfn|Boyer|Merzbach|1991|p=168}}<!--may be suspect--> 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>

In ancient China, values for {{pi}} included 3.1547 (around 1&nbsp;AD), <math>\sqrt{10}</math> (100&nbsp;AD, approximately 3.1623), and {{math|{{sfrac|142|45}}}} (3rd century, approximately 3.1556).{{sfn|Arndt|Haenel|2006|pp=176–177}} Around 265&nbsp;AD, the Wei Kingdom mathematician [[Biography:Liu Hui|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>{{sfn|Arndt|Haenel|2006|p=177}} 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 [[Biography:Zu Chongzhi|Zu Chongzhi]], around 480&nbsp;AD, calculated that <math>3.1415926<\pi<3.1415927</math> and suggested the approximations <math display="inline">\pi \approx \frac{355}{113} = 3.14159292035...</math> and <math display="inline">\pi \approx \frac{22}{7} = 3.142857142857...</math>, which he termed the ''[[Milü]]'' (<nowiki>''</nowiki>close ratio") and ''Yuelü'' ("approximate ratio"), respectively, 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 remained the most accurate approximation of {{pi}} available for the next 800 years.{{sfn|Arndt|Haenel|2006|p=178}}

The Indian astronomer Aryabhata used a value of 3.1416 in his ''Āryabhaṭīya'' (499&nbsp;AD).{{sfn|Arndt|Haenel|2006|p=179}} [[Biography:Fibonacci|Fibonacci]] in {{Circa|1220}} computed 3.1418 using a polygonal method, independent of Archimedes.{{sfn|Arndt|Haenel|2006|p=180}} Italian author Dante apparently employed the value <math display="inline">3+\frac{\sqrt{2}}{10} \approx 3.14142</math>.{{sfn|Arndt|Haenel|2006|p=180}}

The Persian astronomer [[Biography:Jamshīd al-Kāshī|Jamshīd al-Kāshī]] produced nine [[Sexagesimal|sexagesimal]] digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with <math display="inline">3\times 2^{28}</math> 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 |doi=10.35834/mjms/1312233136|doi-access=free}}</ref><ref>{{cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=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 |access-date=11 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20110412192025/http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html |archive-date=12 April 2011}}</ref> which stood as the world record for about 180 years.{{sfn|Arndt|Haenel|2006|p=182}} French mathematician [[Biography:François Viète|François Viète]] in 1579 achieved nine digits with a polygon of <math display="inline">3\times 2^{17}</math> sides.{{sfn|Arndt|Haenel|2006|p=182}} Flemish mathematician [[Biography:Adriaan van Roomen|Adriaan van Roomen]] arrived at 15 decimal places in 1593.{{sfn|Arndt|Haenel|2006|p=182}} In 1596, Dutch mathematician [[Biography:Ludolph van Ceulen|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).{{sfn|Arndt|Haenel|2006|pp=182–183}} Dutch scientist [[Biography:Willebrord Snellius|Willebrord Snellius]] reached 34 digits in 1621,{{sfn|Arndt|Haenel|2006|p=183}} and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10<sup>40</sup> sides.<ref>{{cite book |first=Christophorus |last=Grienbergerus |language=la |year=1630 |title=Elementa Trigonometrica |url=http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-url=https://web.archive.org/web/20140201234124/http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-date=1 February 2014}} His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < {{pi}} < 3.14159 26535 89793 23846 26433 83279 50288 4199.</ref> [[Biography:Christiaan Huygens|Christiaan Huygens]] was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to [[Richardson extrapolation]].<ref>{{cite book|last=Brezinski|first=C.|contribution=Some pioneers of extrapolation methods|date=2009|url=https://www.worldscientific.com/doi/10.1142/9789812836267_0001|title=The Birth of Numerical Analysis|pages=1–22|publisher=World Scientific|doi=10.1142/9789812836267_0001|isbn=978-981-283-625-0|editor1-first=Adhemar|editor1-last=Bultheel|editor1-link=Adhemar Bultheel|editor2-first=Ronald|editor2-last=Cools}}</ref><ref>{{Cite journal|last=Yoder|first=Joella G.|date=1996|title=Following in the footsteps of geometry: The mathematical world of Christiaan Huygens|journal=De Zeventiende Eeuw|volume=12|pages=83–93|url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php|via=[[Digital Library for Dutch Literature]]}}</ref>

=== 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. Infinite series allowed mathematicians to compute {{pi}} with much greater precision than [[Biography:Archimedes|Archimedes]] and others who used geometrical techniques.<ref name="Ais">{{harvnb|Arndt|Haenel|2006|pp=185–191}}</ref> Although infinite series were exploited for {{pi}} most notably by European mathematicians such as [[Biography:James Gregory (mathematician)|James Gregory]] and [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]], the approach also appeared in the [[Astronomy:Kerala school of astronomy and mathematics|Kerala school]] sometime in the 14th or 15th century.<ref name="Roypp"/>{{sfn|Arndt|Haenel|2006|pp=185–186}} Around 1500&nbsp;AD, a written description of an infinite series that could be used to compute {{pi}} was laid out in [[Social:Sanskrit|Sanskrit]] verse in ''[[Unsolved:Tantrasamgraha|Tantrasamgraha]]'' by Nilakantha Somayaji.<ref name="Roypp">{{cite journal |last=Roy |first=Ranjan |year=1990 |title=The Discovery of the Series Formula for {{mvar|π}} by Leibniz, Gregory and Nilakantha |journal=Mathematics Magazine |volume=63 |number=5 |pages=291–306 |url=https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |doi=10.1080/0025570X.1990.11977541 }}</ref> The series are presented without proof, but proofs are presented in a later work, ''[[Astronomy:Yuktibhāṣā|Yuktibhāṣā]]'', from around 1530&nbsp;AD. Several infinite series are described, including series for sine (which Nilakantha attributes to [[Biography:Madhava of Sangamagrama|Madhava of Sangamagrama]]), cosine, and arctangent which are now sometimes referred to as [[Madhava series]]. The series for arctangent is sometimes called [[Gregory's series]] or the Gregory–Leibniz series.<ref name="Roypp" /> Madhava used infinite series to estimate {{pi}} to 11 digits around 1400.<ref>{{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/books?id=c-xT0KNJp0cC&pg=PA264 |page=264}}<!-- This ISBN is for the third edition from 2011! --></ref>

In 1593, [[Biography:François Viète|François Viète]] published what is now known as [[Viète's formula]], an [[Infinite product|infinite product]] (rather than an infinite sum, which is more typically used in {{pi}} calculations):{{sfn|Arndt|Haenel|2006|p=187}}<ref>{{OEIS2C|id=A060294}}</ref><ref>{{cite book|url=https://books.google.com/books?id=7_BCAAAAcAAJ|title=Variorum de rebus mathematicis responsorum|volume=VIII|first=Franciscus|last=Vieta|year=1593}}</ref>

<math display=block> \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots</math>

In 1655, [[Biography:John Wallis|John Wallis]] published what is now known as [[Wallis product]], also an infinite product:{{sfn|Arndt|Haenel|2006|p=187}}

<math display=block>
\frac{\pi}{2} = \Big(\frac{2}{1} \cdot \frac{2}{3}\Big) \cdot \Big(\frac{4}{3} \cdot \frac{4}{5}\Big) \cdot \Big(\frac{6}{5} \cdot \frac{6}{7}\Big) \cdot \Big(\frac{8}{7} \cdot \frac{8}{9}\Big) \cdots
</math>

[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright|alt=A formal portrait of a man, with long hair|[[Biography:Isaac Newton|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" />]]
In the 1660s, the English scientist [[Biography:Isaac Newton|Isaac Newton]] and German mathematician [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]] discovered [[Calculus|calculus]], which led to the development of many infinite series for approximating {{pi}}. Newton himself used an arcsine series to compute a 15-digit approximation of {{pi}} in 1665 or 1666, 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 1671, [[Biography:James Gregory (mathematician)|James Gregory]], and independently, Leibniz in 1673, discovered the [[Taylor series]] expansion for arctangent:<ref name=Roypp/><ref>{{cite journal |last=Horvath |first=Miklos |title=On the Leibnizian quadrature of the circle. |journal=Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) |volume=4 |year=1983 |pages=75–83 |url=http://ac.inf.elte.hu/Vol_004_1983/075.pdf }}</ref><ref name="LS" />

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

This series, sometimes called the [[Gregory's series|Gregory–Leibniz series]], equals <math display="inline">\frac{\pi}{4}</math> when evaluated with <math>z=1</math>.<ref name="LS">{{harvnb|Eymard|Lafon|2004|pp=53–54}}</ref> But for <math>z=1</math>, [[Leibniz formula for π#Convergence|it converges impractically slowly]] (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.<ref>{{cite journal |last=Cooker |first=M.J. |year=2011 |title=Fast formulas for slowly convergent alternating series |journal=Mathematical Gazette |volume=95 |number=533 |pages=218–226 |doi=10.1017/S0025557200002928 |s2cid=123392772 |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |access-date=23 February 2023 |archive-date=4 May 2019 |archive-url=https://web.archive.org/web/20190504091131/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |url-status=bot: unknown }}</ref>

In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for <math display="inline">z=\frac{1}{\sqrt{3}}</math> to compute {{pi}} to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.{{sfn|Arndt|Haenel|2006|p=189}}

In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:<ref name=jones/><ref name=tweddle>{{cite journal |last=Tweddle |first=Ian |year=1991 |title=John Machin and Robert Simson on Inverse-tangent Series for {{mvar|π}} |journal= Archive for History of Exact Sciences |volume=42 |number=1 |pages=1–14 |doi= 10.1007/BF00384331 |jstor=41133896 |s2cid=121087222 }}</ref>{{sfn|Arndt|Haenel|2006|pp=192–193}}

<math display=block> \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>{{cite journal |last=Lehmer |first=D. H. |year=1938 |title=On Arccotangent Relations for {{mvar|π}} |journal=American Mathematical Monthly |volume=45 |number=10 |pages=657-664 Published by: Mathematical Association of America |jstor=2302434 |doi=10.1080/00029890.1938.11990873 |url=https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf }}</ref><ref name="A72n4" />

Isaac Newton [[Series acceleration|accelerated the convergence]] of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):<ref>{{cite book |last=Roy |first=Ranjan |year=2021 |orig-year=1st ed. 2011 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |pages=215–216, 219–220}}
<p>{{cite book |last=Newton |first=Isaac |year=1971 |editor-last=Whiteside |editor-first=Derek Thomas |title=The Mathematical Papers of Isaac Newton |volume=4, 1674–1684 |publisher=Cambridge University Press |pages=526–653 }}</p></ref>

:<math>
\arctan x
= \frac{x}{1 + x^2} + \frac23\frac{x^3}{(1 + x^2)^2}
+ \frac{2\cdot 4}{3 \cdot 5}\frac{x^5}{(1 + x^2)^3} + \cdots
</math>

[[Biography:Leonhard Euler|Leonhard Euler]] popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including <math display=inline>\tfrac\pi4 = 5\arctan\tfrac17 + 2\arctan\tfrac{3}{77},</math> with which he computed 20 digits of {{pi}} in one hour.<ref>
{{cite web |last=Sandifer |first=Ed |year=2009 |title=Estimating π |website=How Euler Did It |url=http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf }} Reprinted in {{cite book |last=Sandifer |first=Ed |display-authors=0 |year=2014 |title=How Euler Did Even More |pages=109–118 |publisher=Mathematical Association of America}}
<p>{{cite book |last=Euler |first=Leonhard |year=1755 |title=[[Institutiones calculi differentialis|Institutiones Calculi Differentialis]] |chapter=§2.2.30 |page=318 |publisher=Academiae Imperialis Scientiarium Petropolitanae |lang=la |chapter-url=https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/318 |id=[https://scholarlycommons.pacific.edu/euler-works/212/ E 212]}}</p>
<p>{{cite journal |last=Euler |first=Leonhard |year=1798 |orig-year=written 1779 |title=Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae |journal=Nova Acta Academiae Scientiarum Petropolitinae |volume=11 |pages=133–149, 167–168 |url=https://archive.org/details/novaactaacademia11petr/page/133 |id=[https://scholarlycommons.pacific.edu/euler-works/705/ E 705] }}</p>
<p>{{cite journal |last=Chien-Lih |first=Hwang |year=2004 |title=88.38 Some Observations on the Method of Arctangents for the Calculation of {{mvar|π}} |journal=Mathematical Gazette |volume=88 |number=512 |pages=270–278 |doi=10.1017/S0025557200175060 |s2cid=123532808 }}</p>
<p>{{cite journal |last=Chien-Lih |first=Hwang |year=2005 |title=89.67 An elementary derivation of Euler's series for the arctangent function |journal=Mathematical Gazette |volume=89 |number=516 |pages=469–470 |doi=10.1017/S0025557200178404 |s2cid=123395287 }}</p>
</ref>

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.{{sfn|Arndt|Haenel|2006|pp=192–196, 205}}

In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of German mathematician [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]].<ref name="A194">{{harvnb|Arndt|Haenel|2006|pp=194–196}}</ref>

In 1853, British mathematician William Shanks calculated {{pi}} to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.<ref name="hayes-2014">{{cite magazine |last=Hayes |first=Brian |url=https://www.americanscientist.org/article/pencil-paper-and-pi |title=Pencil, Paper, and Pi |volume=102 |issue=5 |page=342 |magazine=American Scientist |date=September 2014 |access-date=22 January 2022 |doi=10.1511/2014.110.342}}</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 |last1=Borwein |first1=J.M. |last2=Borwein |first2=P.B. |title=Ramanujan and Pi |year=1988 |journal=Scientific American |volume=256 |issue=2 |pages=112–117 |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]]:{{sfn|Arndt|Haenel|2006|pp=69–72}}

<math display=block>
\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 |last1=Borwein |first1=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 |jstor=2324715|hdl=1959.13/1043679 |hdl-access=free }}</ref>

An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:{{sfn|Arndt|Haenel|2006|loc = Formula 16.10, p. 223}}<ref>{{cite book |last=Wells |first=David |page=35 |title=The Penguin Dictionary of Curious and Interesting Numbers |edition=revised |publisher=Penguin |year=1997 |isbn=978-0-14-026149-3}}</ref>

<math display=block>
\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:

{|class="wikitable" style="text-align: center; margin: auto;"
|-
! 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 ...
|}

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. 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|prime number]]s that later contributed to the development and study of the [[Riemann zeta function]]:<ref name="Posamentier">{{harvnb|Posamentier|Lehmann|2004|p=284}}</ref>

<math display=block> \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 1768 proved that {{pi}} is [[Irrational number|irrational]], meaning it is not equal to the quotient of any two integers.{{sfn|Arndt|Haenel|2006|p=5}} [[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 [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] proved in 1794 that {{pi}}<sup>2</sup> is also irrational. In 1882, German mathematician [[Biography:Ferdinand von Lindemann|Ferdinand von Lindemann]] proved that {{pi}} is [[Transcendental number|transcendental]],<ref>{{cite journal | last=Lindemann | first=F. |  year=1882 | title=Über die Ludolph'sche Zahl | journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin | volume=2 | pages=679–682 | url=https://archive.org/details/sitzungsberichte1882deutsch/page/679 }}</ref> confirming a conjecture made by both [[Biography:Adrien-Marie Legendre|Legendre]] and Euler.{{sfn|Arndt|Haenel|2006|p=196}}<ref>Hardy and Wright 1938 and 2000: 177 footnote §&nbsp;11.13–14 references Lindemann's proof as appearing at ''Math. Ann''. 20 (1882), 213–225.</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 §&nbsp;11.13–14. The proofs that e and π are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full citations.</ref>

=== Adoption of the symbol {{pi}} ===
{{Multiple image
| image1            = William Jones, the Mathematician.jpg
| caption1          = The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[Biography:William Jones (mathematician)|William Jones]] in 1706
| caption2          = [[Biography:Leonhard Euler|Leonhard Euler]] popularized the use of the Greek letter π in works he published in 1736 and 1748.
| total_width       = 300
| image2            = Leonhard Euler.jpg
| align             = left
}}
In the earliest usages, the Greek letter {{Pi}} was used to denote the [[Semiperimeter|semiperimeter]] (''semiperipheria'' in Latin) of a circle<ref name=firstPi>{{Cite book |url=https://books.google.com/books?id=KTgPAAAAQAAJ&pg=PP3 |title=Theorematum in libris Archimedis de sphaera et cylindro declarario |last=Oughtred |first=William |date=1652 |publisher=Excudebat L. Lichfield, Veneunt apud T. Robinson |language=la |quote={{math|''δ''.''π''}} :: semidiameter. semiperipheria}}</ref> and was combined in ratios with {{mvar|δ}} (for [[Diameter|diameter]] or semidiameter) or {{mvar|ρ}} (for [[Radius|radius]]) to form circle constants.<ref name="Cajori-2007">{{Cite book |url=https://books.google.com/books?id=bT5suOONXlgC&pg=PA9 |title=A History of Mathematical Notations: Vol. II |last=Cajori |first=Florian |date=2007 |publisher=Cosimo, Inc. |isbn=978-1-60206-714-1 |pages=8–13 |language=en |quote=the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented {{math|3.14159...}} by {{math|''δ'':''π''}}, as did Oughtred more than a century earlier}}</ref><ref>{{wikicite |ref={{harvid|Schepler|1950}} |reference = Schepler, H.C. (1950) "The Chronology of Pi" ''Mathematics Magazine''. '''23'''.<br/>Part 1. Jan/Feb. (3): 165–170. doi:[https://doi.org/10.2307/3029284 10.2307/3029284].<br>Part 2. Mar/Apr. (4): 216-228. doi:[https://doi.org/10.2307/3029832 10.2307/3029832].<br>Part 3. May/Jun. (5): 279-283. doi:[https://doi.org/10.2307/3029000 10.2307/3029000].}}<br>See p. 220: William Oughtred used the letter {{pi}} to represent the periphery (that is, the circumference) of a circle.</ref><ref name="Smith-1958">{{Cite book |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA312 |title=History of Mathematics |last=Smith |first=David E. |date=1958 |publisher=Courier Corporation |isbn=978-0-486-20430-7 |page=312 |language=en}}</ref><ref>{{Cite journal|last=Archibald|first=R.C.|date=1921|title=Historical Notes on the Relation {{math|1=''e''<sup>−(''π''/2)</sup> = ''i''<sup>''i''</sup>}}|jstor=2972388|journal=The American Mathematical Monthly|volume=28|issue=3|pages=116–121|doi=10.2307/2972388|quote=It is noticeable that these letters are ''never'' used separately, that is, {{pi}} is ''not'' used for 'Semiperipheria'}}</ref> (Before then, mathematicians sometimes used letters such as {{mvar|c}} or {{mvar|p}} instead.{{sfn|Arndt|Haenel|2006|p=166}}) The first recorded use is Oughtred's {{nobr|"<math>\delta . \pi</math>"}}, to express the ratio of periphery and diameter in the 1647 and later editions of {{lang|la|Clavis Mathematicae|italic=yes}}.<ref>See, for example, {{cite book |url=https://archive.org/details/bub_gb_ddMxgr27tNkC |title=Clavis Mathematicæ |last=Oughtred |first=William |date=1648 |publisher=Thomas Harper |location=London |page=[https://archive.org/details/bub_gb_ddMxgr27tNkC/page/n220 69] |language=la|trans-title=The key to mathematics}} (English translation: {{Cite book |url=https://books.google.com/books?id=S50yAQAAMAAJ&pg=PA99 |title=Key of the Mathematics |last=Oughtred |first=William |date=1694 |publisher=J. Salusbury |language=en}})</ref>{{sfn|Arndt|Haenel|2006|p=166}} Barrow likewise used {{nobr|"<math display=inline>\frac \pi \delta</math>"}} to represent the constant {{math|3.14...}},<ref>{{Cite book|chapter-url=https://archive.org/stream/mathematicalwor00whewgoog#page/n405/mode/1up |title=The mathematical works of Isaac Barrow |last=Barrow |first=Isaac |date=1860 |publisher=Cambridge University press |others=Harvard University |editor-last=Whewell|editor-first=William |pages=381 |language=la |chapter=Lecture XXIV}}</ref> while Gregory instead used {{nobr|"<math display=inline>\frac \pi \rho</math>"}} to represent {{math|6.28...&thinsp;}}.<ref>{{Cite journal |last=Gregorius |first=David |date=1695 |title=Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae |jstor=102382 |journal=Philosophical Transactions |language=la |volume=19 |issue=231 |pages=637–652 |doi=10.1098/rstl.1695.0114 |bibcode=1695RSPT...19..637G|doi-access=free|url=https://archive.org/download/crossref-pre-1909-scholarly-works/10.1098%252Frstl.1684.0084.zip/10.1098%252Frstl.1695.0114.pdf}}</ref><ref name="Smith-1958" />

The earliest known use of the Greek letter {{pi}} alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[Biography:William Jones (mathematician)|William Jones]] in his 1706 work ''{{lang|la|Synopsis Palmariorum Matheseos|italic=unset}}; or, a New Introduction to the Mathematics''.<ref name=jones/>{{sfn|Arndt|Haenel|2006|p=165|ps=: A facsimile of Jones' text is in {{harvnb|Berggren|Borwein|Borwein|1997|pp=108–109}}.}} The Greek letter appears on p. 243 in the phrase "<math display=inline>\tfrac12</math> Periphery ({{pi}})", calculated for a circle with radius one. However, Jones writes that his equations for {{pi}} are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.{{sfn|Arndt|Haenel|2006|p=166}} Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.<ref name="Cajori-2007" /><ref>{{Cite book |url=https://books.google.com/books?id=NmYVAAAAQAAJ&pg=PA282 |title=Cursus Mathematicus |last=Segner |first=Joannes Andreas |date=1756 |publisher=Halae Magdeburgicae |page=282 |language=la|access-date=15 October 2017|archive-url=https://web.archive.org/web/20171015150340/https://books.google.co.uk/books?id=NmYVAAAAQAAJ&pg=PA282|archive-date=15 October 2017|url-status=live}}</ref>

Euler started using the single-letter form beginning with his 1727 ''Essay Explaining the Properties of Air'', though he used {{math|1=''π'' = 6.28...}}, the ratio of periphery to radius, in this and some later writing.<ref>{{Cite journal |last=Euler |first=Leonhard |date=1727 |title=Tentamen explicationis phaenomenorum aeris|url=http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |journal=Commentarii Academiae Scientiarum Imperialis Petropolitana |language=la |volume=2 |page=351 |id=[http://eulerarchive.maa.org/pages/E007.html E007] |quote=Sumatur pro ratione radii ad peripheriem, {{math|I : π}} |access-date=15 October 2017|archive-url=https://web.archive.org/web/20160401072718/http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5|archive-date=1 April 2016|url-status=live}} [http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610172054/http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 |date=10 June 2016 }}: "{{mvar|π}} is taken for the ratio of the radius to the periphery [note that in this work, Euler's {{pi}} is double our {{pi}}.]"</ref><ref>{{Cite book |url=https://books.google.com/books?id=3C1iHFBXVEcC&pg=PA139 |title=Lettres inédites d'Euler à d'Alembert |last=Euler |first=Leonhard |series=Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche |year=1747 |editor-last=Henry|editor-first=Charles |volume=19 |publication-date=1886 |page=139 |language=fr |id=[http://eulerarchive.maa.org/pages/E858.html E858] |quote=Car, soit π la circonference d'un cercle, dout le rayon est {{math|{{=}} 1}}}} English translation in {{Cite journal |last=Cajori |first=Florian |date=1913 |title=History of the Exponential and Logarithmic Concepts |jstor=2973441 |journal=The American Mathematical Monthly |volume=20 |issue=3 |pages=75–84 |doi=10.2307/2973441 |quote=Letting {{pi}} be the circumference (!) of a circle of unit radius}}</ref>   Euler first used {{nowrap|1={{pi}} = 3.14...}} in his 1736 work ''[[Mechanica]]'',<ref>{{Cite book|last=Euler|first=Leonhard|title=Mechanica sive motus scientia analytice exposita. (cum tabulis)|date=1736|publisher=Academiae scientiarum Petropoli|volume=1|page=113|language=la|chapter=Ch. 3 Prop. 34 Cor. 1|id=[http://eulerarchive.maa.org/pages/E015.html E015]|quote=Denotet {{math|1 : ''π''}} rationem diametri ad peripheriam|chapter-url=https://books.google.com/books?id=jgdTAAAAcAAJ&pg=PA113}} [http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610183753/http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26|date=10 June 2016}} : "Let {{math|1 : ''π''}} denote the ratio of the diameter to the circumference"</ref> and continued in his widely read 1748 work {{lang|la|[[Introductio in analysin infinitorum]]|italic=yes}} (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 {{math|1}}").<ref>{{Cite book |url=http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |title=Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio |last=Euler |first=Leonhard (1707–1783) |date=1922 |publisher=B.G. Teubneri |location=Lipsae |pages=133–134 |language=la |id=[http://eulerarchive.maa.org/pages/E101.html E101]|access-date=15 October 2017|archive-url=https://web.archive.org/web/20171016022758/http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155|archive-date=16 October 2017|url-status=live}}</ref> Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world,{{sfn|Arndt|Haenel|2006|p=166}} though the definition still varied between {{math|3.14...}} and {{math|6.28...}} as late as 1761.<ref>{{Cite book |url=https://books.google.com/books?id=P-hEAAAAcAAJ&pg=PA374 |title=Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm |last=Segner |first=Johann Andreas von |date=1761 |publisher=Renger |page=374 |language=la |quote=Si autem {{pi}} notet peripheriam circuli, cuius diameter eſt {{math|2}}}}</ref>

== Modern quest for more digits ==

=== Computer era and iterative algorithms ===
{{quote box|quote=
The [[Gauss–Legendre algorithm|Gauss–Legendre iterative algorithm]]:<br />Initialize
<math display=block>\textstyle a_0 = 1, \quad b_0 = \frac{1}{\sqrt 2}, \quad t_0 = \frac{1}{4}, \quad p_0 = 1.</math>
Iterate
<math display=block>\textstyle a_{n+1} = \frac{a_n+b_n}{2}, \quad \quad b_{n+1} = \sqrt{a_n b_n},</math>
<math display=block>\textstyle 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 display=block>\textstyle \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}}. Mathematicians [[Biography:John Wrench|John Wrench]] and Levi Smith reached 1,120 digits in 1949 using a desk calculator.{{sfn|Arndt|Haenel|2006|p=205}} Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and [[Biography:John von Neumann|John von Neumann]] that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the [[Engineering:ENIAC|ENIAC]] computer.{{sfn|Arndt|Haenel|2006|p=197}}<ref>{{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 |jstor=2002695}}</ref> The record, always relying on an arctan series, was broken repeatedly (3089 digits in 1955,<ref>{{cite journal|first1=J. C.|last1=Nicholson| first2=J. |last2=Jeenel|journal=Math. Tabl. Aids. Comp.|volume=9|number=52|year=1955|doi=10.2307/2002052|jstor=2002052|title=Some comments on a NORC Computation of π|pages=162–164}}</ref> 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973.{{sfn|Arndt|Haenel|2006|p=197}}

Two additional developments around 1980 once again accelerated the ability to compute {{pi}}. First, the discovery of new iterative algorithms 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.{{sfn|Arndt|Haenel|2006|pp=15–17}} Such algorithms are particularly important in modern {{pi}} computations because most of the computer's time is devoted to multiplication.{{sfn|Arndt|Haenel|2006|p=131}} They include the [[Karatsuba algorithm]], [[Toom–Cook multiplication]], and Fourier transform-based methods.{{sfn|Arndt|Haenel|2006|pp=132, 140}}

The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent.{{sfn|Arndt|Haenel|2006|p=87}} 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 [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]], in what is now termed the arithmetic–geometric mean method (AGM method) or [[Gauss–Legendre algorithm]].{{sfn|Arndt|Haenel|2006|p=87}} 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, brothers 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)}}. For details of algorithms, see {{cite book |last1=Borwein |first1=Jonathan|last2=Borwein |first2=Peter|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> 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 |access-date=12 April 2012|url-status=live |archive-url=https://web.archive.org/web/20120415102439/http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/dhb-kanada.pdf |archive-date=15 April 2012}}</ref>

=== Motives for computing {{pi}} ===
[[File:Record pi approximations.svg|thumb|As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of {{pi}} increased dramatically. The vertical scale is [[Logarithmic scale|logarithmic]].|left|300x300px]]
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 [[Astronomy:Observable universe|observable universe]] with a precision of one atom. Accounting for additional digits needed to compensate for computational [[Round-off error|round-off error]]s, Arndt concludes that a few hundred digits would suffice for any scientific application. 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=https://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 |access-date=14 April 2012 |location=London |first=Steve |last=Connor|url-status=live |archive-url=https://web.archive.org/web/20120402220803/http://www.independent.co.uk/news/science/the-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html |archive-date=2 April 2012}}</ref> They also have practical benefits, such as testing [[Supercomputer|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}}.{{sfn|Arndt|Haenel|2006|p=18}}

=== Rapidly convergent series ===
[[File:Srinivasa Ramanujan - OPC - 2 (cleaned).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 Indian mathematician [[Biography:Srinivasa Ramanujan|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|modular equation]]s, is

<math display=block>
\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4\left(396^{4k}\right)}.
</math>

This series converges much more rapidly than most arctan series, including Machin's formula.<ref>{{harvnb|Arndt|Haenel|2006|p=104}}</ref> [[Biography:Bill Gosper|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 (Jonathan and Peter) and the Chudnovsky brothers.<ref>{{harvnb|Arndt|Haenel|2006|pp=110–111}}</ref> The [[Chudnovsky algorithm|Chudnovsky formula]] developed in 1987 is

<math display=block>
\frac{1}{\pi} = \frac{\sqrt{10005}}{4270934400} \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|2004|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, 10 trillion (10<sup>13</sup>) digits in 2011 by Alexander Yee and Shigeru Kondo,<ref name="NW">{{cite book|last1=Bailey|first1=David H.|last2=Borwein|first2=Jonathan M.|contribution=15.2 Computational records|contribution-url=https://books.google.com/books?id=K26zDAAAQBAJ&pg=PA469|doi=10.1007/978-3-319-32377-0|page=469|publisher=Springer International Publishing|title=Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation|year=2016|isbn=978-3-319-32375-6 }}</ref> and 100 trillion digits by Emma Haruka Iwao in 2022.<ref>{{Cite magazine |url=https://thenewstack.io/how-googles-emma-haruka-iwao-helped-set-a-new-record-for-pi/ |title=How Google's Emma Haruka Iwao Helped Set a New Record for Pi |date=11 June 2022|magazine=The New Stack|first=David|last=Cassel}}</ref> For similar formulae, see also the [[Ramanujan–Sato series]].

In 2006, mathematician Simon Plouffe used the PSLQ [[Integer relation algorithm|integer relation algorithm]]<ref>PSLQ means Partial Sum of Least Squares.</ref> to generate several new formulae for {{pi}}, conforming to the following template:

<math display=block>
\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 {{math|[[Gelfond's constant|''e''<sup>''&pi;''</sup>]]}} (Gelfond's constant), {{math|''k''}} is an [[Odd number|odd number]], and {{math|''a'', ''b'', ''c''}} are certain rational numbers that Plouffe computed.<ref>{{cite web |first=Simon |last=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 |access-date=10 April 2009|url-status=live |archive-url=https://web.archive.org/web/20120114101641/http://www.plouffe.fr/simon/inspired2.pdf |archive-date=14 January 2012}}</ref>

=== Monte Carlo methods ===
{{multiple image
| direction         = horizontal
| 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''
| image2            = Pi 30K.gif
| caption2          = Random dots are placed on a square and a circle inscribed inside.
| alt2              = Thousands of dots randomly covering a square and a circle inscribed in the square.
| align             = left
| total_width       = 225
}}
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>

<math display=block>\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>

[[File:Five random walks.png|thumb|Five random walks with 200 steps.  The sample mean of {{math|{{abs|''W''<sub>200</sub>}}}} is {{math|''μ'' {{=}} 56/5}}, and so {{math|2(200)''μ''<sup>−2</sup> ≈ 3.19}} is within {{math|0.05}} of&nbsp;{{pi}}.|left]]
Another way to calculate {{pi}} using probability is to start with a [[Random walk|random walk]], generated by a sequence of (fair) coin tosses: independent [[Random variable|random variable]]s {{math|''X<sub>k</sub>''}} such that {{math|''X<sub>k</sub>'' ∈ {{mset|−1,1}}}} with equal probabilities.  The associated random walk is

<math display=block>W_n = \sum_{k=1}^n X_k</math>

so that, for each {{mvar|n}}, {{math|''W<sub>n</sub>''}} is drawn from a shifted and scaled [[Binomial distribution|binomial distribution]].  As {{mvar|n}} varies, {{math|''W<sub>n</sub>''}} defines a (discrete) [[Stochastic process|stochastic process]].  Then {{pi}} can be calculated by<ref>{{cite journal |last=Grünbaum |first=B. |title=Projection Constants |journal=[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]] |volume=95 |issue=3 |pages=451–465 |year=1960 |doi=10.1090/s0002-9947-1960-0114110-9|doi-access=free}}</ref>

<math display=block>\pi = \lim_{n\to\infty} \frac{2n}{E[|W_n|]^2}.</math>

This Monte Carlo method is independent of any relation to circles, and is a consequence of the [[Central limit theorem|central limit theorem]], discussed below.

These Monte Carlo methods for approximating {{pi}} are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they 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>

=== Spigot algorithms ===
Two algorithms were discovered in 1995 that opened up new avenues of research into {{pi}}. They are called [[Spigot algorithm|spigot algorithm]]s because, like water dripping from a [[Engineering:Tap (valve)|spigot]], they produce single digits of {{pi}} that are not reused after they are calculated.{{sfn|Arndt|Haenel|2006|pp=77–84}}<ref name="Gibbons">{{cite journal|last=Gibbons|first=Jeremy|doi=10.2307/27641917|issue=4|journal=[[The American Mathematical Monthly]]|jstor=27641917|mr=2211758|pages=318–328|title=Unbounded spigot algorithms for the digits of pi|url=https://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf|volume=113|year=2006}}</ref> This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.{{sfn|Arndt|Haenel|2006|pp=77–84}}

Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.<ref name="Gibbons" />{{sfn|Arndt|Haenel|2006|p=77}}<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 |jstor=2975006}}</ref> Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.{{sfn|Arndt|Haenel|2006|p=77}}

Another spigot algorithm, the [[Bailey–Borwein–Plouffe formula|BBP]] digit extraction algorithm, was discovered in 1995 by Simon Plouffe:{{sfn|Arndt|Haenel|2006|pp=117, 126–128}}<ref name="bbpf">{{cite journal |last1=Bailey |first1=David H. |last2=Borwein |first2=Peter B. |last3=Plouffe |first3=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 |doi=10.1090/S0025-5718-97-00856-9 |url-status=live |archive-url=https://web.archive.org/web/20120722015837/http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/digits.pdf |archive-date=22 July 2012 |citeseerx=10.1.1.55.3762 |bibcode=1997MaCom..66..903B|s2cid=6109631 }}</ref>

<math display=block> \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|hexadecimal]] digit of {{pi}} without calculating all the preceding digits.{{sfn|Arndt|Haenel|2006|pp=117, 126–128}} Individual binary digits may be extracted from individual hexadecimal digits, and [[Octal|octal]] digits can be extracted from one or two hexadecimal digits. 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|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 |access-date=27 October 2007 |archive-url=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 --> |archive-date=12 September 2007}}</ref> In September 2010, a [[Company:Yahoo!|Yahoo!]] employee used the company's [[Software:Apache Hadoop|Hadoop]] application on one thousand computers over a 23-day period to compute 256 [[Bit|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 |last=Palmer |first=Jason |newspaper=BBC News |date=16 September 2010 |url=https://www.bbc.co.uk/news/technology-11313194 |access-date=26 March 2011 |url-status=live |archive-url=https://web.archive.org/web/20110317170643/http://www.bbc.co.uk/news/technology-11313194 |archive-date=17 March 2011}}</ref>

In 2022, Plouffe found a base-10 algorithm for calculating digits of {{pi}}.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Digit-Extraction Algorithm |url=https://mathworld.wolfram.com/ |access-date=2023-10-21 |website=mathworld.wolfram.com |language=en}}</ref>

== Role and characterizations in mathematics ==
Because {{pi}} is closely related to the circle, it is found in [[List of formulae involving π|many formulae]] from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, [[Fourier analysis]], and number theory, also include {{pi}} in some of their important formulae.

=== 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. The area of the [[Unit circle|unit circle]] is {{pi}}.]]

{{pi}} appears in formulae for areas and volumes of geometrical shapes based on circles, such as [[Ellipse|ellipse]]s, [[Sphere|sphere]]s, 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 circle with radius {{math|''r''}} is {{math|π''r''<sup>2</sup>}}.
* The area of an ellipse with semi-major axis {{math|''a''}} and semi-minor axis {{math|''b''}} is {{math|π''ab''}}.
* 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>}}.
Some of the formulae above are special cases of the volume of the ''n''-dimensional ball and the surface area of its boundary, the [[N-sphere|(''n''−1)-dimensional sphere]], given below.

Apart from circles, there are other [[Curve of constant width|curves of constant width]]. By [[Barbier's theorem]], every curve of constant width has perimeter {{pi}} times its width. The [[Reuleaux triangle]] (formed by the intersection of three circles with the sides of an [[Equilateral triangle|equilateral triangle]] as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular [[Smoothness|smooth]] and even [[Algebraic curve|algebraic curve]]s of constant width.<ref>{{cite book|last1=Martini|first1=Horst|last2=Montejano|first2=Luis|last3=Oliveros|first3=Déborah|doi=10.1007/978-3-030-03868-7|isbn=978-3-030-03866-3|mr=3930585|publisher=Birkhäuser|s2cid=127264210|title=Bodies of Constant Width: An Introduction to Convex Geometry with Applications|year=2019}}{{pb}}
See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, § 5.3.3, pp. 111–112.</ref>

[[Integral|Definite integrals]] that describe circumference, area, or volume of shapes generated by circles typically have values that involve {{pi}}. For example, an integral that specifies half the area of a circle of radius one is given by:<ref>{{cite book|last1=Herman|first1=Edwin|last2=Strang|first2=Gilbert|contribution=Section 5.5, Exercise 316|contribution-url=https://openstax.org/books/calculus-volume-1/pages/5-5-substitution|page=594|publisher=OpenStax|title=Calculus|volume=1|year=2016}}</ref>

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

In that integral, the function <math>\sqrt{1-x^2}</math> represents the height over the <math>x</math>-axis of a [[Semicircle|semicircle]] (the [[Square root|square root]] is a consequence of the [[Pythagorean theorem]]), and the integral computes the area below the semicircle.

=== Units of angle ===
[[File:Sine cosine one period.svg|thumb|upright=1.54|alt=Diagram showing graphs of functions|[[Sine|Sine]] and [[Cosine|cosine]] functions repeat with period 2{{pi}}.|left]]The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. {{pi}} plays an important role in angles measured in [[Radian|radian]]s, which are defined so that a complete circle spans an angle of 2{{pi}} radians. The angle measure of 180° is equal to {{pi}} radians, and 1° = {{pi}}/180 radians.{{sfn|Abramson|2014|loc=[https://openstax.org/books/precalculus/pages/5-1-angles Section 5.1: Angles]}}

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''}},<ref name="WCS" />

<math display=block> \sin\theta = \sin\left(\theta + 2\pi k \right) \text{ and } \cos\theta = \cos\left(\theta + 2\pi k \right).</math>

=== Eigenvalues ===
[[File:Harmonic partials on strings.svg|thumb|right|The [[Physics:Overtone|overtone]]s of a vibrating string are [[Eigenfunction|eigenfunction]]s of the second derivative, and form a harmonic progression.  The associated eigenvalues form the [[Arithmetic progression|arithmetic progression]] of integer multiples of {{pi}}.]]
Many of the appearances of {{pi}} in the formulae of mathematics and the sciences have to do with its close relationship with geometry.  However, {{pi}} also appears in many natural situations having apparently nothing to do with geometry.

In many applications, it plays a distinguished role as an eigenvalue.  For example, an idealized vibrating string can be modelled as the graph of a function {{math|''f''}} on the unit interval {{closed-closed|0, 1}}, with fixed ends {{math|1=''f''(0) = ''f''(1) = 0}}. The modes of vibration of the string are solutions of the [[Differential equation|differential equation]] <math>f''(x) + \lambda f(x) = 0</math>, or <math>f''(t) = -\lambda f(x)</math>.  Thus {{math|λ}} is an eigenvalue of the second derivative [[Differential operator|operator]] <math>f \mapsto f''</math>, and is constrained by [[Sturm–Liouville theory]] to take on only certain specific values. It must be positive, since the operator is negative definite, so it is convenient to write {{math|1=''λ'' = ''ν''<sup>2</sup>}}, where {{math|''ν'' > 0}} is called the [[Physics:Wavenumber|wavenumber]].  Then {{math|1=''f''(''x'') = sin(''π'' ''x'')}} satisfies the boundary conditions and the differential equation with {{math|1=''ν'' = ''π''}}.<ref>{{cite book |last1=Hilbert |first1=David |last2=Courant |first2=Richard |title=Methods of mathematical physics, volume 1 |pages=286–290 |year=1966 |publisher=Wiley}}</ref>

The value {{pi}} is, in fact, the ''least'' such value of the wavenumber, and is associated with the fundamental mode of vibration of the string.  One way to show this is by estimating the [[Physics:Energy|energy]], which satisfies [[Wirtinger's inequality for functions|Wirtinger's inequality]]:{{sfn|Dym|McKean|1972|page=47}} for a function <math>f : [0, 1] \to \Complex</math> with {{math|1=''f''(0) = ''f''(1) = 0}} and {{math|''f''}}, {{math|''f''{{′}}}} both square integrable, we have:

<math display=block>\pi^2\int_0^1|f(x)|^2\,dx\le \int_0^1|f'(x)|^2\,dx,</math>

with equality precisely when {{math|''f''}} is a multiple of {{math|sin(π ''x'')}}. Here {{pi}} appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the variational characterization of the eigenvalue. As a consequence, {{pi}} is the smallest [[Singular value|singular value]] of the derivative operator on the space of functions on {{closed-closed|0, 1}} vanishing at both endpoints (the [[Sobolev space]] <math>H^1_0[0,1]</math>).

=== Inequalities ===
[[File:Sir William Thompson illustration of Carthage.png|thumb|The [[History:Ancient Carthage|ancient city of Carthage]] was the solution to an isoperimetric problem, according to a legend recounted by Lord Kelvin:<ref>{{cite journal |first=William |last=Thompson |title=Isoperimetrical problems |year=1894 |journal=Nature Series: Popular Lectures and Addresses |volume=II |pages=571–592}}</ref> those lands bordering the sea that Queen Dido could enclose on all other sides within a single given oxhide, cut into strips.|left]]

The number {{pi}} serves appears in similar eigenvalue problems in higher-dimensional analysis.  As mentioned above, it can be characterized via its role as the best constant in the [[Isoperimetric inequality|isoperimetric inequality]]: the area {{mvar|A}} enclosed by a plane Jordan curve of perimeter {{mvar|P}} satisfies the inequality

<math display=block>4\pi A\le P^2,</math>

and equality is clearly achieved for the circle, since in that case {{math|1=''A'' = π''r''{{sup|2}}}} and {{math|1=''P'' = 2π''r''}}.<ref>{{cite book |first=Isaac |last=Chavel |title=Isoperimetric inequalities |publisher=Cambridge University Press |year=2001}}</ref>

Ultimately, as a consequence of the isoperimetric inequality, {{pi}} appears in the optimal constant for the critical [[Sobolev inequality]] in ''n'' dimensions, which thus characterizes the role of {{pi}} in many physical phenomena as well, for example those of classical [[Potential theory|potential theory]].<ref>{{cite journal |last=Talenti|first= Giorgio |title=Best constant in Sobolev inequality |journal=Annali di Matematica Pura ed Applicata |volume=110 |number=1 |pages=353–372 |issn=1618-1891 |doi=10.1007/BF02418013 |citeseerx=10.1.1.615.4193 |year=1976|s2cid=16923822 }}</ref><ref>{{cite arXiv |title=Best constants in Poincaré inequalities for convex domains |eprint=1110.2960 |author1=L. Esposito |author2=C. Nitsch |author3=C. Trombetti |year=2011 |class=math.AP}}</ref><ref>{{cite journal |title=Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions |first1=M.|last1=Del Pino |first2=J.|last2= Dolbeault |journal=Journal de Mathématiques Pures et Appliquées |year=2002 |volume=81 |issue=9 |pages=847–875 |doi=10.1016/s0021-7824(02)01266-7 |citeseerx=10.1.1.57.7077|s2cid=8409465 }}</ref>  In two dimensions, the critical Sobolev inequality is

<math display=block>2\pi\|f\|_2 \le \|\nabla f\|_1</math>

for ''f'' a smooth function with compact support in {{math|'''R'''<sup>2</sup>}}, <math>\nabla f</math> is the [[Gradient|gradient]] of ''f'', and <math>\|f\|_2</math> and <math>\|\nabla f\|_1</math> refer respectively to the [[Lp space|{{math|L<sup>2</sup>}} and {{math|L<sup>1</sup>}}-norm]].  The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.

Wirtinger's inequality also generalizes to higher-dimensional [[Poincaré inequality|Poincaré inequalities]] that provide best constants for the [[Dirichlet energy]] of an ''n''-dimensional membrane.  Specifically, {{pi}} is the greatest constant such that

<math display=block>
\pi \le \frac{\left (\int_G |\nabla u|^2\right)^{1/2}}{\left (\int_G|u|^2\right)^{1/2}}
</math>

for all [[Convex set|convex]] subsets {{math|''G''}} of {{math|'''R'''<sup>''n''</sup>}} of diameter 1, and square-integrable functions ''u'' on {{math|''G''}} of mean zero.<ref>{{cite journal |last1=Payne |first1=L.E. |last2=Weinberger |first2=H.F. |title=An optimal Poincaré inequality for convex domains |year=1960 |journal=Archive for Rational Mechanics and Analysis |volume=5 |issue=1 |issn=0003-9527 |pages=286–292 |doi=10.1007/BF00252910 |bibcode=1960ArRMA...5..286P|s2cid=121881343 }}</ref>  Just as Wirtinger's inequality is the [[Calculus of variations|variational]] form of the [[Dirichlet eigenvalue]] problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension.

=== Fourier transform and Heisenberg uncertainty principle ===
[[File:Animation of Heisenberg geodesic.gif|thumb|right|An animation of a geodesic in the Heisenberg group]]
The constant {{pi}} also appears as a critical spectral parameter in the [[Fourier transform]].  This is the [[Integral transform|integral transform]], that takes a complex-valued integrable function {{math|''f''}} on the real line to the function defined as:

<math display=block>\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi}\,dx.</math>

Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve {{pi}} ''somewhere''. The above is the most canonical definition, however, giving the unique unitary operator on {{math|''L''{{sup|2}}}} that is also an algebra homomorphism of {{math|''L''{{sup|1}}}} to {{math|''L''{{sup|∞}}}}.<ref>{{cite book |title=Harmonic analysis in phase space |first=Gerald|last= Folland |publisher=Princeton University Press |year=1989 |page=5|}}</ref>

The [[Physics:Heisenberg uncertainty principle|Heisenberg uncertainty principle]] also contains the number {{pi}}.  The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

<math display=block>
\left(\int_{-\infty}^\infty x^2|f(x)|^2\,dx\right)
\left(\int_{-\infty}^\infty \xi^2|\hat{f}(\xi)|^2\,d\xi\right)
 \ge
\left(\frac{1}{4\pi}\int_{-\infty}^\infty |f(x)|^2\,dx\right)^2.
</math>

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a [[Physics:Quantum mechanical|quantum mechanical]] system, is discussed below.  The appearance of {{pi}} in the formulae of Fourier analysis is ultimately a consequence of the [[Stone–von Neumann theorem]], asserting the uniqueness of the Schrödinger representation of the [[Heisenberg group]].<ref name=howe>{{cite journal |first=Roger |last=Howe |title=On the role of the Heisenberg group in harmonic analysis |journal=[[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] |volume=3 |pages=821–844 |number=2 |year=1980 |doi=10.1090/S0273-0979-1980-14825-9 |mr=578375|doi-access=free}}</ref>

=== Gaussian integrals ===
[[File:E^(-x^2).svg|thumb|A graph of the [[Gaussian function|Gaussian function]] {{math|1=''ƒ''(''x'') = ''e''{{sup|−''x''{{sup|2}}}}}}. The coloured region between the function and the ''x''-axis has area {{math|{{sqrt|π}}}}.|left]]

The fields of [[Probability|probability]] and [[Statistics|statistics]] frequently use the [[Normal distribution|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> The [[Gaussian function]], which is the [[Physics:Probability density function|probability density function]] of the normal distribution with [[Mean|mean]] {{math|μ}} and [[Standard deviation|standard deviation]] {{math|σ}}, naturally contains {{pi}}:<ref name="GaussProb">{{harvnb|Bronshteĭn|Semendiaev|1971|pp=106–107, 744, 748}}</ref>

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

The factor of <math>\tfrac{1}{\sqrt{2\pi}}</math> makes the area under the graph of {{math|''f''}}  equal to one, as is required for a probability distribution. This follows from a [[Integration by substitution|change of variables]] in the [[Gaussian integral]]:<ref name="GaussProb" />

<math display=block>\int_{-\infty}^\infty e^{-u^2} \, du=\sqrt{\pi}</math>

which says that the area under the basic bell curve in the figure is equal to the square root of {{pi}}.

The [[Central limit theorem|central limit theorem]] explains the central role of normal distributions, and thus of {{pi}}, in probability and statistics. This theorem is ultimately connected with the spectral characterization of {{pi}} as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.{{sfn|Dym|McKean|1972|loc=Section 2.7}} Equivalently, {{pi}} is the unique constant making the Gaussian normal distribution {{math|''e''{{sup|−π''x''{{sup|2}}}}}} equal to its own Fourier transform.<ref>{{cite book |first1=Elias|last1=Stein |first2=Guido|last2=Weiss |title=Fourier analysis on Euclidean spaces |year=1971 |publisher=Princeton University Press |page=6|author1-link=Elias Stein}}; Theorem 1.13.</ref> Indeed,  according to {{harvtxt|Howe|1980}}, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.<ref name=howe/>

=== Topology ===
[[File:Order-7 triangular tiling.svg|thumb|right|[[Uniformization theorem|Uniformization]] of the [[Klein quartic|Klein quartic]], a surface of [[Genus (mathematics)|genus]] three and Euler characteristic −4, as a quotient of the [[Poincaré disk model|hyperbolic plane]] by the [[Symmetry group|symmetry group]] [[PSL(2,7)|PSL(2,7)]] of the [[Fano plane|Fano plane]].  The hyperbolic area of a fundamental domain is {{math|8π}}, by Gauss–Bonnet.]]
The constant {{pi}} appears in the Gauss–Bonnet formula which relates the [[Differential geometry of surfaces|differential geometry of surfaces]] to their [[Topology|topology]].  Specifically, if a [[Compact space|compact]] surface {{math|Σ}} has Gauss curvature ''K'', then

<math display=block>\int_\Sigma K\,dA = 2\pi \chi(\Sigma)</math>

where {{math|''χ''(Σ)}} is the [[Euler characteristic]], which is an integer.<ref>{{cite book |title=A Comprehensive Introduction to Differential Geometry |volume=3 |first=Michael|last= Spivak |year=1999 |publisher=Publish or Perish Press|}}; Chapter 6.</ref>  An example is the surface area of a sphere ''S'' of curvature 1 (so that its [[Physics:Radius of curvature|radius of curvature]], which coincides with its radius, is also 1.)  The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two.  Thus we have

<math display=block>A(S) = \int_S 1\,dA = 2\pi\cdot 2 = 4\pi</math>

reproducing the formula for the surface area of a sphere of radius 1.

The constant appears in many other integral formulae in topology, in particular, those involving [[Characteristic class|characteristic class]]es via the [[Chern–Weil homomorphism]].<ref>{{cite book |last1=Kobayashi |first1=Shoshichi |last2=Nomizu |first2=Katsumi |title=Foundations of Differential Geometry |volume=2 |publisher=Wiley Interscience |year=1996 |edition=New |page=293|title-link=Foundations of Differential Geometry}}; Chapter XII ''Characteristic classes''</ref>

=== Cauchy's integral formula ===
[[File:Factorial05.jpg|thumb|right|Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles.  Here illustrated is the complex logarithm of the Gamma function.]]

One of the key tools in [[Complex analysis|complex analysis]] is [[Contour integration|contour integration]] of a function over a positively oriented (rectifiable) Jordan curve {{math|''γ''}}.  A form of [[Cauchy's integral formula]] states that if a point {{math|''z''<sub>0</sub>}} is interior to {{math|''γ''}}, then<ref>{{cite book |first=Lars|last= Ahlfors |title=Complex analysis |publisher=McGraw-Hill |year=1966 |page=115|}}</ref>

<math display=block>\oint_\gamma \frac{dz}{z-z_0} = 2\pi i.</math>

Although the curve {{math|''γ''}} is not a circle, and hence does not have any obvious connection to the constant {{pi}}, a standard proof of this result uses [[Morera's theorem]], which implies that the integral is invariant under [[Homotopy|homotopy]] of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates.  More generally, it is true that if a rectifiable closed curve {{math|γ}} does not contain {{math|''z''<sub>0</sub>}}, then the above integral is {{math|2π''i''}} times the [[Winding number|winding number]] of the curve.

The general form of Cauchy's integral formula establishes the relationship between the values of a complex analytic function {{math|''f''(''z'')}} on the Jordan curve {{math|''γ''}} and the value of {{math|''f''(''z'')}} at any interior point {{math|''z''<sub>0</sub>}} of {{math|γ}}:<ref>{{cite book|last=Joglekar|first=S. D.|title=Mathematical Physics|publisher=Universities Press|year=2005|page=166|isbn=978-81-7371-422-1}}</ref>

<math display=block>\oint_\gamma { f(z) \over z-z_0 }\,dz = 2\pi i f (z_{0})</math>

provided {{math|''f''(''z'')}} is analytic in the region enclosed by {{math|''γ''}} and extends continuously to {{math|''γ''}}.  Cauchy's integral formula is a special case of the [[Residue theorem|residue theorem]], that if {{math|''g''(''z'')}} is a [[Meromorphic function|meromorphic function]] the region enclosed by {{math|''γ''}} and is continuous in a neighbourhood of {{math|''γ''}}, then

<math display=block>\oint_\gamma g(z)\, dz =2\pi i \sum \operatorname{Res}( g, a_k ) </math>

where the sum is of the residues at the poles of {{math|''g''(''z'')}}.

=== Vector calculus and physics ===
The constant {{pi}} is ubiquitous in [[Vector calculus|vector calculus]] and [[Potential theory|potential theory]], for example in [[Physics:Coulomb's law|Coulomb's law]],<ref>{{cite book|first=H. M.|last=Schey|year=1996|title=Div, Grad, Curl, and All That: An Informal Text on Vector Calculus|publisher=W.W. Norton |isbn=0-393-96997-5}}</ref> Gauss' law, [[Physics:Maxwell's equations|Maxwell's equations]], and even the [[Physics:Einstein field equations|Einstein field equations]].<ref>{{cite book|last=Yeo|first=Adrian|title=The pleasures of pi, e and other interesting numbers|publisher=World Scientific Pub.|year=2006|page=21|isbn=978-981-270-078-0}}</ref><ref>{{cite book|last=Ehlers|first=Jürgen|title=Einstein's Field Equations and Their Physical Implications|publisher=Springer|year=2000|page=7|isbn=978-3-540-67073-5}}</ref>  Perhaps the simplest example of this is the two-dimensional [[Newtonian potential]], representing the potential of a point source at the origin, whose associated field has unit outward [[Physics:Flux|flux]] through any smooth and oriented closed surface enclosing the source:
<math display="block">\Phi(\mathbf x) = \frac{1}{2\pi}\log|\mathbf x|.</math>
The factor of <math>1/2\pi</math> is necessary to ensure that <math>\Phi</math> is the [[Fundamental solution|fundamental solution]] of the Poisson equation in <math>\mathbb R^2</math>:<ref>{{citation|first1=D.|last1=Gilbarg|first2=Neil|last2=Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}</ref>
<math display="block">\Delta\Phi = \delta</math>
where <math>\delta</math> is the [[Dirac delta function]].

In higher dimensions, factors of {{pi}} are present because of a normalization by the n-dimensional volume of the unit n sphere.  For example, in three dimensions, the Newtonian potential is:<ref>{{citation|first1=D.|last1=Gilbarg|first2=Neil|last2=Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}</ref>
<math display="block">\Phi(\mathbf x) = -\frac{1}{4\pi|\mathbf x|},</math>
which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.

=== Total curvature ===
{{excerpt|Total curvature}}

=== The gamma function and Stirling's approximation ===
[[File:Gamma plot points marked.svg|thumb|Plot of the gamma function on the real axis]]
The [[Factorial|factorial]] function <math>n!</math> is the product of all of the positive integers through {{math|''n''}}. The [[Gamma function|gamma function]] extends the concept of [[Factorial|factorial]] (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity <math>\Gamma(n)=(n-1)!</math>. When the gamma function is evaluated at half-integers, the result contains {{pi}}. For example, <math> \Gamma(1/2) = \sqrt{\pi} </math> and <math display="inline">\Gamma(5/2) = \frac {3 \sqrt{\pi}} {4}</math>.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=191–192}}</ref>

The gamma function is defined by its Weierstrass product development:<ref>{{cite book |title=The Gamma Function |first=Emil|last= Artin |publisher=Holt, Rinehart and Winston |year=1964 |series=Athena series; selected topics in mathematics |edition=1st|}}</ref>

<math display=block>\Gamma(z) = \frac{e^{-\gamma z}}{z}\prod_{n=1}^\infty \frac{e^{z/n}}{1+z/n}</math>

where {{math|γ}} is the [[Euler–Mascheroni constant]].  Evaluated at {{math|''z'' {{=}} 1/2}} and squared, the equation {{math|Γ(1/2)<sup>2</sup> {{=}} π}} reduces to the Wallis product formula.  The gamma function is also connected to the [[Riemann zeta function]] and identities for the [[Functional determinant|functional determinant]], in which the constant {{pi}} plays an important role.

The gamma function is used to calculate the volume {{math|''V''<sub>''n''</sub>(''r'')}} of the ''n''-dimensional ball of radius ''r'' in Euclidean ''n''-dimensional space, and the surface area {{math|''S''<sub>''n''−1</sub>(''r'')}} of its boundary, the [[N-sphere|(''n''−1)-dimensional sphere]]:<ref>{{cite book |first=Lawrence|last= Evans |title=Partial Differential Equations |publisher=AMS |year=1997 |page=615}}</ref>

<math display=block>V_n(r) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}r^n,</math>

<math display=block>S_{n-1}(r) = \frac{n\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}r^{n-1}.</math>

Further, it follows from the [[Functional equation|functional equation]] that

<math display=block>2\pi r = \frac{S_{n+1}(r)}{V_n(r)}.</math>

The gamma function can be used to create a simple approximation to the factorial function {{math|''n''!}} for large {{math|''n''}}: <math display="inline"> 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>  Equivalently,

<math display=block>\pi = \lim_{n\to\infty} \frac{e^{2n}n!^2}{2 n^{2n+1}}.</math>

As a geometrical application of Stirling's approximation, let {{math|Δ<sub>''n''</sub>}} denote the [[Simplex|standard simplex]] in ''n''-dimensional Euclidean space, and {{math|(''n''&nbsp;+&nbsp;1)Δ<sub>''n''</sub>}} denote the simplex having all of its sides scaled up by a factor of {{math|''n''&nbsp;+&nbsp;1}}.  Then

<math display=block>\operatorname{Vol}((n+1)\Delta_n) = \frac{(n+1)^n}{n!} \sim \frac{e^{n+1}}{\sqrt{2\pi n}}.</math>

[[Ehrhart's volume conjecture]] is that this is the (optimal) upper bound on the volume of a [[Convex body|convex body]] containing only one lattice point.<ref>{{cite journal |author1=Benjamin Nill |author2=Andreas Paffenholz |title=On the equality case in Erhart's volume conjecture |year=2014 |arxiv=1205.1270 |journal=Advances in Geometry |volume=14 |issue=4 |pages=579–586 |issn=1615-7168 |doi=10.1515/advgeom-2014-0001|s2cid=119125713 }}</ref>

=== Number theory and Riemann zeta function ===
[[File:Prüfer.png|thumb|right|Each prime has an associated [[Prüfer group|Prüfer group]], which are arithmetic localizations of the circle.  The [[L-function|L-function]]s of analytic number theory are also localized in each prime ''p''.]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|right|Solution of the Basel problem using the Weil conjecture: the value of {{math|''ζ''(2)}} is the [[Poincaré half-plane model|hyperbolic]] area of a fundamental domain of the [[Modular group|modular group]], times {{math|{{pi}}/2}}.]]
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 display=block>\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]]. [[Biography:Leonhard Euler|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|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 [[Biography:Ernesto Cesàro|Ernesto Cesàro]] in 1881. For a more rigorous proof than the intuitive and informal one given here, see {{cite book|last=Hardy|first=G. H.|title=An Introduction to the Theory of Numbers|publisher=Oxford University Press|year=2008|isbn=978-0-19-921986-5|at=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>{{cite book|last1=Ogilvy|first1=C. S.|last2=Anderson|first2=J. T.|title=Excursions in Number Theory|publisher=Dover Publications Inc.|year=1988|pages=29–35|isbn=0-486-25778-9}}</ref>

<math display=block>\begin{align}
\prod_p^\infty \left(1-\frac{1}{p^2}\right) &= \left( \prod_p^\infty \frac{1}{1-p^{-2}} \right)^{-1}\\[4pt]
&= \frac{1}{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots }\\[4pt]
&= \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 61\%.
\end{align}</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>

The solution to the Basel problem implies that the geometrically derived quantity {{pi}} is connected in a deep way to the distribution of prime numbers.  This is a special case of [[Weil's conjecture on Tamagawa numbers]], which asserts the equality of similar such infinite products of ''arithmetic'' quantities, localized at each prime ''p'', and a ''geometrical'' quantity: the reciprocal of the volume of a certain locally symmetric space.  In the case of the Basel problem, it is the [[Hyperbolic 3-manifold|hyperbolic 3-manifold]] {{math|[[SL2(R)|SL<sub>2</sub>('''R''')]]/[[Modular group|SL<sub>2</sub>('''Z''')]]}}.<ref>{{cite book |title=Algebraic Groups and Number Theory |first1=Vladimir|last1=Platonov |first2=Andrei |last2=Rapinchuk |publisher=Academic Press |year=1994 |pages=262–265}}</ref>

The zeta function also satisfies Riemann's functional equation, which involves {{pi}} as well as the gamma function:

<math display=block>\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s).</math>

Furthermore, the derivative of the zeta function satisfies

<math display=block>\exp(-\zeta'(0)) = \sqrt{2\pi}.</math>

A consequence is that {{pi}} can be obtained from the [[Functional determinant|functional determinant]] of the [[Physics:Harmonic oscillator|harmonic oscillator]].  This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.<ref>{{cite journal |last=Sondow|first= J. |title=Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series |journal=[[Organization:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] |volume=120 |issue=2 |pages=421–424 |year=1994 |doi=10.1090/s0002-9939-1994-1172954-7 |citeseerx=10.1.1.352.5774|s2cid= 122276856 }}</ref>  The calculation can be recast in [[Physics:Quantum mechanics|quantum mechanics]], specifically the [[Calculus of variations|variational approach]] to the [[Physics:Bohr model|spectrum of the hydrogen atom]].<ref>{{cite journal |doi=10.1063/1.4930800 |author1=T. Friedmann |author2=C.R. Hagen |title=Quantum mechanical derivation of the Wallis formula for pi |journal=Journal of Mathematical Physics |volume=56 |issue=11 |pages=112101 |year=2015 |arxiv=1510.07813 |bibcode=2015JMP....56k2101F|s2cid=119315853 }}</ref>

=== Fourier series ===
[[File:2-adic integers with dual colorings.svg|thumb|{{pi}} appears in characters of p-adic numbers (shown), which are elements of a [[Prüfer group|Prüfer group]]. [[Tate's thesis|Tate's thesis]] makes heavy use of this machinery.<ref>{{cite conference |last1=Tate |first1=John T. |editor1-first=J. W. S. |editor1-last=Cassels|editor2-first=A.|editor2-last=Fröhlich|title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) |publisher=Thompson, Washington, DC |isbn=978-0-9502734-2-6 |mr=0217026 |year=1950 |contribution=Fourier analysis in number fields, and Hecke's zeta-functions |pages=305–347}}</ref>|left]]
The constant {{pi}} also appears naturally in [[Fourier series]] of [[Periodic function|periodic function]]s.  Periodic functions are functions on the group {{math|'''T''' {{=}}'''R'''/'''Z'''}} of fractional parts of real numbers.  The Fourier decomposition shows that a complex-valued function {{math|''f''}} on {{math|'''T'''}} can be written as an infinite linear superposition of unitary characters of {{math|'''T'''}}.   That is, continuous [[Group homomorphism|group homomorphism]]s from {{math|'''T'''}} to the [[Circle group|circle group]] {{math|''U''(1)}} of unit modulus complex numbers.  It is a theorem that every character of {{math|'''T'''}} is one of the complex exponentials <math>e_n(x)=  e^{2\pi i n x}</math>.

There is a unique character on {{math|'''T'''}}, up to complex conjugation, that is a group isomorphism.  Using the [[Haar measure]] on the circle group, the constant {{pi}} is half the magnitude of the Radon–Nikodym derivative of this character.  The other characters have derivatives whose magnitudes are positive integral multiples of 2{{pi}}.<ref name="Nicolas Bourbaki" />  As a result, the constant {{pi}} is the unique number such that the group '''T''', equipped with its Haar measure, is Pontrjagin dual to the [[Lattice (group)|lattice]] of integral multiples of 2{{pi}}.{{sfn|Dym|McKean|1972|loc=Chapter 4}}  This is a version of the one-dimensional [[Poisson summation formula]].

=== Modular forms and theta functions ===
[[File:Lattice with tau.svg|thumb|right|Theta functions transform under the [[Lattice (group)|lattice]] of periods of an elliptic curve.]]

The constant {{pi}} is connected in a deep way with the theory of [[Modular form|modular form]]s and [[Theta function|theta function]]s.  For example, the [[Chudnovsky algorithm]] involves in an essential way the [[J-invariant|j-invariant]] of an [[Elliptic curve|elliptic curve]].

[[Modular form]]s are [[Holomorphic function|holomorphic function]]s in the upper half plane characterized by their transformation properties under the [[Modular group|modular group]] <math>\mathrm{SL}_2(\mathbb Z)</math> (or its various subgroups), a lattice in the group <math>\mathrm{SL}_2(\mathbb R)</math>.  An example is the Jacobi theta function

<math display=block>\theta(z,\tau) = \sum_{n=-\infty}^\infty e^{2\pi i nz + i\pi n^2\tau}</math>

which is a kind of modular form called a [[Jacobi form]].<ref name="Mumford 1983 1–117">{{cite book |first=David |last=Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=Birkhauser |location=Boston |isbn=978-3-7643-3109-2 |pages=1–117}}</ref>  This is sometimes written in terms of the [[Nome (mathematics)|nome]] <math>q=e^{\pi i \tau}</math>.

The constant {{pi}} is the unique constant making the Jacobi theta function an [[Automorphic form|automorphic form]], which means that it transforms in a specific way.  Certain identities hold for all automorphic forms.  An example is

<math display=block>\theta(z+\tau,\tau) = e^{-\pi i\tau -2\pi i z}\theta(z,\tau),</math>

which implies that {{math|θ}} transforms as a representation under the discrete [[Heisenberg group]].  General modular forms and other [[Theta function|theta function]]s also involve {{pi}}, once again because of the [[Stone–von Neumann theorem]].<ref name="Mumford 1983 1–117" />

=== Cauchy distribution and potential theory ===
[[File:Witch of Agnesi, construction.svg|thumb|The [[Witch of Agnesi|Witch of Agnesi]], named for Maria Agnesi (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.|left]]
[[File:2d random walk ag adatom ag111.gif|left|thumb|The Cauchy distribution governs the passage of [[Brownian motion|Brownian particles]] through a membrane.]]
The [[Cauchy distribution]]

<math display=block>g(x)=\frac{1}{\pi}\cdot\frac{1}{x^2+1}</math>

is a [[Physics:Probability density function|probability density function]].  The total probability is equal to one, owing to the integral:

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

The Shannon entropy of the Cauchy distribution is equal to {{math|ln(4π)}}, which also involves {{pi}}.

The Cauchy distribution plays an important role in [[Potential theory|potential theory]] because it is the simplest [[Physics:Furstenberg boundary|Furstenberg measure]], the classical [[Poisson kernel]] associated with a [[Brownian motion]] in a half-plane.<ref>{{cite book |first1=Sidney|last1= Port |first2=Charles|last2= Stone |title=Brownian motion and classical potential theory |publisher=Academic Press |year=1978 |page=29}}</ref> Conjugate harmonic functions and so also the [[Hilbert transform]] are associated with the asymptotics of the Poisson kernel. The Hilbert transform ''H'' is the integral transform given by the [[Cauchy principal value]] of the [[Singular integral|singular integral]]

<math display=block>Hf(t) = \frac{1}{\pi}\int_{-\infty}^\infty \frac{f(x)\,dx}{x-t}.</math>

The constant {{pi}} is the unique (positive) normalizing factor such that ''H'' defines a [[Linear complex structure|linear complex structure]] on the Hilbert space of square-integrable real-valued functions on the real line.<ref>{{cite book |last=Titchmarsh |first=E. |title=Introduction to the Theory of Fourier Integrals |isbn=978-0-8284-0324-5 |year=1948 |edition=2nd|publication-date=1986 |publisher=Clarendon Press |location=Oxford University}}</ref>  The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space {{math|L<sup>2</sup>('''R''')}}: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.<ref>{{cite book |first=Elias |last=Stein |author-link=Elias Stein |title=Singular Integrals and Differentiability Properties of Functions |publisher=Princeton University Press |year=1970}}; Chapter II.</ref>  The constant {{pi}} is the unique normalizing factor that makes this transformation unitary.

=== In the Mandelbrot set===
[[File:Mandel zoom 00 mandelbrot set.jpg|alt=An complex black shape on a blue background.|thumb|The [[Mandelbrot set|Mandelbrot set]] can be used to approximate {{pi}}.]]

An occurrence of {{pi}} in the [[Fractal|fractal]] called the [[Mandelbrot set]] was discovered by 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 |archive-url=https://web.archive.org/web/20111027155739/http://home.comcast.net/~davejanelle/mandel.pdf |archive-date=27 October 2011 |access-date=14 April 2012 |doi=10.1142/S0218348X01000828  |url-status=dead }}</ref> He examined the behaviour of the Mandelbrot set near the "neck" at {{math|(−0.75, 0)}}. When the number of iterations until divergence for the point {{math|(−0.75, ''ε'')}} is multiplied by {{mvar|ε}}, the result approaches {{pi}} as {{mvar|ε}} approaches zero. The point {{math|(0.25 + ''ε'', 0)}} 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 {{mvar|ε}} tends to {{pi}}.<ref name="KA" /><ref>{{cite book|last=Peitgen|first=Heinz-Otto|title=Chaos and fractals: new frontiers of science|publisher=Springer|year=2004|pages=801–803|isbn=978-0-387-20229-7}}</ref>

=== Projective geometry ===
Let {{math|''V''}} be the set of all twice differentiable real functions <math>f:\mathbb R\to\mathbb R</math> that satisfy the [[Ordinary differential equation|ordinary differential equation]] <math>f''(x)+f(x)=0</math>.  Then {{math|''V''}} is a two-dimensional real [[Vector space|vector space]], with two parameters corresponding to a pair of initial conditions for the differential equation.  For any <math>t\in\mathbb R</math>, let <math>e_t:V\to\mathbb R</math> be the evaluation functional, which associates to each <math>f\in V</math> the value <math>e_t(f)=f(t)</math> of the function {{math|''f''}} at the real point {{math|''t''}}.  Then, for each ''t'', the kernel of <math>e_t</math> is a one-dimensional linear subspace of {{math|''V''}}.  Hence <math>t\mapsto\ker e_t</math> defines a function from <math>\mathbb R\to\mathbb P(V)</math> from the real line to the [[Real projective line|real projective line]].   This function is periodic, and the quantity {{pi}} can be characterized as the period of this map.<ref>{{cite book |title=Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups |series=Cambridge Tracts in Mathematics |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83186-4 |first1=V.|last1=Ovsienko|first2=S.|last2=Tabachnikov|contribution=Section 1.3}}</ref> This is notable in that the constant {{pi}}, rather than 2{{pi}}, appears naturally in this context.

== Outside mathematics ==

=== Describing physical phenomena ===
Although not a [[Physics:Physical constant|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|spherical coordinate system]]s. A simple formula from the field of [[Physics:Classical mechanics|classical mechanics]] gives the approximate period {{math|''T''}} of a simple [[Physics:Pendulum|pendulum]] of length {{math|''L''}}, swinging with a small amplitude ({{math|''g''}} is the [[Physics:Gravity of Earth|earth's gravitational acceleration]]):<ref>{{cite book|last1=Halliday|first1=David|last2=Resnick|first2=Robert|last3=Walker|first3=Jearl|title=Fundamentals of Physics|edition=5th|publisher=John Wiley & Sons|year=1997|page=381|isbn=0-471-14854-7}}</ref>

<math display=block>T \approx 2\pi \sqrt\frac{L}{g}.</math>

One of the key formulae of [[Physics:Quantum mechanics|quantum mechanics]] is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δ{{math|''x''}}) and [[Physics:Momentum|momentum]] (Δ{{math|''p''}}) cannot both be arbitrarily small at the same time (where {{math|''h''}} is the [[Physics:Planck constant|Planck constant]]):<ref>{{cite book|title=College Physics 2e|contribution=29.7 Probability: The Heisenberg Uncertainty Principle|contribution-url=https://openstax.org/books/college-physics-2e/pages/29-7-probability-the-heisenberg-uncertainty-principle|publisher=OpenStax|first1=Paul Peter|last1=Urone|first2=Roger|last2=Hinrichs|year=2022}}</ref>

<math display=block> \Delta x\, \Delta p \ge \frac{h}{4\pi}.</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 [[Physics:Fine-structure constant|fine-structure constant]] {{math|''α''}} is<ref>{{cite book |last1=Itzykson |first1=C. |last2=Zuber |first2=J.-B. |author-link2=Jean-Bernard Zuber |title=Quantum Field Theory |date=1980 |publisher=Dover Publications |location=Mineola, NY |isbn=978-0-486-44568-7 |edition=2005 |url=https://books.google.com/books?id=4MwsAwAAQBAJ |lccn=2005053026 |oclc=61200849}}</ref>
<math display=block>\frac{1}{\tau} = 2\frac{\pi^2 - 9}{9\pi}m_\text{e}\alpha^{6},</math>
where {{math|''m''<sub>e</sub>}} is the mass of the electron.

{{pi}} is present in some structural engineering formulae, such as the [[Physics:Buckling|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>{{cite book|last=Low|first=Peter|title=Classical Theory of Structures Based on the Differential Equation|publisher=Cambridge University Press|year=1971|pages=116–118|isbn=978-0-521-08089-7}}</ref>

<math display=block>F =\frac{\pi^2EI}{L^2}.</math>

The field of [[Physics:Fluid dynamics|fluid dynamics]] contains {{pi}} in [[Physics:Stokes' law|Stokes' law]], which approximates the frictional force {{math|''F''}} exerted on small, [[Sphere|spherical]] objects of radius {{math|''R''}}, moving with velocity {{math|''v''}} in a [[Physics:Fluid|fluid]] with [[Physics:Dynamic viscosity|dynamic viscosity]] {{math|''η''}}:<ref>{{cite book|last=Batchelor|first=G. K.|title=An Introduction to Fluid Dynamics|publisher=Cambridge University Press|year=1967|page=233|isbn=0-521-66396-2}}</ref>

<math display=block>F =6\pi\eta  Rv.</math>

In electromagnetics, the [[Physics:Vacuum permeability|vacuum permeability]] constant ''μ''<sub>0</sub> appears in [[Physics:Maxwell's equations|Maxwell's equations]], which describe the properties of [[Physics:Electric field|electric]] and [[Physics:Magnetic field|magnetic]] fields and [[Physics:Electromagnetic radiation|electromagnetic radiation]]. Before 20 May 2019, it was defined as exactly

<math display=block>\mu_0 = 4 \pi \times 10^{-7}\text{ H/m} \approx 1.2566370614 \ldots \times 10 ^{-6} \text{ N/A}^2. </math>

Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the [[Sinuosity|sinuosity]] of a [[Earth:Meander|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 |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|s2cid=19219185 }}</ref><ref>{{harvnb|Posamentier|Lehmann|2004|pp=140–141}}</ref>

=== Memorizing digits ===
Piphilology is the practice of memorizing large numbers of digits of {{pi}},<ref name="A445">{{harvnb|Arndt|Haenel|2006|pp=44–45}}</ref> and world-records are kept by the ''Guinness World Records''.  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"] {{webarchive|url=https://web.archive.org/web/20160214205333/http://www.guinnessworldrecords.com/world-records/most-pi-places-memorised |date=14 February 2016 }}, 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 |access-date=27 October 2007|url-status=live |archive-url=https://web.archive.org/web/20130818004142/http://www.japantimes.co.jp/life/2006/12/17/life/how-can-anyone-remember-100000-numbers/ |archive-date=18 August 2013}}</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. Such memorization aids are called [[Biology:Mnemonic|mnemonic]]s. An early example of a mnemonic for pi, originally devised by English scientist 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''.<ref>{{cite book|last=Danesi|first=Marcel|chapter=Chapter 4: Pi in Popular Culture|date=January 2021|doi=10.1163/9789004433397|page=[https://books.google.com/books?id=tAsOEAAAQBAJ&pg=PA97 97]|publisher=Brill|title=Pi ({{pi}}) in Nature, Art, and Culture|isbn=9789004433373 |s2cid=224869535 }}</ref> Poems for memorizing {{pi}} have been composed in several languages in addition to English.<ref name="A445" />  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 |issue=5 |pages=361–372 |doi=10.1080/13554790902776896 |pmid=19585350 |pmc=4323087}}</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 |url=http://www.cadaeic.net/comments.htm |title=Cadaeic Cadenza Notes & Commentary |access-date=29 July 2009|url-status=live |archive-url=https://web.archive.org/web/20090118060210/http://cadaeic.net/comments.htm |archive-date=18 January 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 10,000 decimals |publisher=Vinculum Press |isbn=978-0-9630097-1-5 |author2=Diana Keith |date=17 February 2010}}</ref>

=== In popular culture ===
[[File:Pi pie2.jpg|thumb|right|alt=Pi Pie at Delft University|A pi pie. Many pies are circular, and "pie" and {{pi}} are homophones, making pie a frequent subject of pi puns.]]

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 Givenchy π perfume, Pi (film), and Pi Day as examples. See: {{cite book |title=Keys to Infinity |first=Clifford A. |last=Pickover |publisher=Wiley & Sons |year=1995 |isbn=978-0-471-11857-2 |page=[https://archive.org/details/keystoinfinity00clif/page/59 59] |url=https://archive.org/details/keystoinfinity00clif/page/59}} {{cite book |title=Mathematical Treks: From Surreal Numbers to Magic Circles |series=MAA spectrum |first=Ivars |last=Peterson |author-link=Ivars Peterson |publisher=Mathematical Association of America |year=2002 |isbn=978-0-88385-537-9 |page=17 |url=https://books.google.com/books?id=4gWSAraVhtAC&pg=PA17|url-status=live |archive-url=https://web.archive.org/web/20161129190818/https://books.google.com/books?id=4gWSAraVhtAC&pg=PA17 |archive-date=29 November 2016}}</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 1873 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 [[Biography:Carl Sagan|Carl Sagan]]'s 1985 novel ''Contact'' it is suggested that the creator of the universe buried a message deep within the digits of {{pi}}. This part of the story was omitted from the film adaptation of the novel.<ref>{{harvnb|Arndt|Haenel|2006|p=14}}</ref><ref>{{cite book|first1=Burkard |last1=Polster |first2=Marty |last2=Ross |title=Math Goes to the Movies |year=2012 |isbn=978-1-421-40484-4 |publisher=Johns Hopkins University Press |pages=56–57}}</ref> The digits of {{pi}} have also been incorporated into the lyrics of the song "Pi" from the 2005 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)|url-status=live |archive-url=https://web.archive.org/web/20061015122229/http://gaffa.org/reaching/rev_aer_UK5.html |archive-date=15 October 2006}}</ref> In the 1967 ''Star Trek'' episode "Wolf in the Fold", an out-of-control computer is contained by being instructed to "Compute to the last digit the value of {{pi}}".<ref name=life-of-pi/>

In the United States, Pi Day falls on 14&nbsp;March (written 3/14 in the US style), and is popular among students.<ref name=life-of-pi/> {{pi}} and its digital representation are often used by self-described "math [[Social:Geek|geek]]s" for inside jokes among mathematically and technologically minded groups. A college cheer variously attributed to the [[Organization:Massachusetts Institute of Technology|Massachusetts Institute of Technology]] or the [[Organization:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] includes "3.14159".<ref>{{cite journal|last=Rubillo|first=James M.|date=January 1989|issue=1|journal=The Mathematics Teacher|jstor=27966082|page=10|title=Disintegrate 'em|volume=82}}</ref><ref>{{cite book|last=Petroski|first=Henry|isbn=978-1-139-50530-7|page=47|publisher=Cambridge University Press|title=Title An Engineer's Alphabet: Gleanings from the Softer Side of a Profession|url=https://books.google.com/books?id=oVXzxvS3MLUC&pg=PA47|year=2011}}</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 news |url=https://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 |newspaper=USAToday.com |date=14 March 2015 |access-date=14 March 2015 |url-status=live |archive-url=https://web.archive.org/web/20150315005038/http://www.usatoday.com/story/news/nation-now/2015/03/14/pi-day-kids-videos/24753169/ |archive-date=15 March 2015}}</ref><ref>{{Cite journal |url=http://probability.ca/jeff/writing/PiInstant.html |title=Pi Instant |last=Rosenthal |first=Jeffrey S. |date=February 2015 |page=22 |journal=Math Horizons|volume=22 |issue=3 |doi=10.4169/mathhorizons.22.3.22 |s2cid=218542599 }}</ref> In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi Approximation Day", as 22/7&nbsp;=&nbsp;3.142857.<ref>{{cite web |last1=Griffin |first1=Andrew |title=Pi Day: Why some mathematicians refuse to celebrate 14 March and won't observe the dessert-filled day |url=https://www.independent.co.uk/news/science/pi-day-march-14-maths-google-doodle-pie-baking-celebrate-30-anniversary-a8254036.html |website=The Independent |access-date=2 February 2019 |archive-url=https://web.archive.org/web/20190424151944/https://www.independent.co.uk/news/science/pi-day-march-14-maths-google-doodle-pie-baking-celebrate-30-anniversary-a8254036.html |archive-date=24 April 2019 |url-status=live}}</ref>

{{anchor|tau}} <!-- (was) used in the redirect Tau (number) -->
Some have proposed replacing {{pi}} by {{math|1=''τ'' = 2''π''}},<ref>{{cite book|last1=Freiberger|first1=Marianne|last2=Thomas|first2=Rachel|contribution=Tau – the new {{pi}}|contribution-url=https://books.google.com/books?id=IbR-BAAAQBAJ&pg=PT133|isbn=978-1-62365-411-5|page=159|publisher=Quercus|title=Numericon: A Journey through the Hidden Lives of Numbers|year=2015}}</ref> arguing that {{mvar|τ}}, as the number of radians in one [[Physics:Turn (angle)|turn]] or the ratio of a circle's circumference to its radius, is more natural than {{pi}} and simplifies many formulae.<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 |s2cid=126179022 |url=http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf|url-status=live |archive-url=https://web.archive.org/web/20130928095819/http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf |archive-date=28 September 2013}}</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|s2cid=120965049 |url=http://www.math.utah.edu/~palais/pi.pdf|url-status=live|archive-url=https://web.archive.org/web/20120622070009/http://www.math.utah.edu/~palais/pi.pdf|archive-date=22 June 2012}}</ref> 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=30 June 2011|url-status=dead |archive-url=https://web.archive.org/web/20130713084345/http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp |archive-date=13 July 2013}}</ref> but since 2010 this has led to people celebrating Two Pi Day or Tau Day on June 28.<ref>{{Cite web |title=Forget Pi Day. We should be celebrating Tau Day {{!}} Science News |url=https://www.sciencenews.org/blog/science-the-public/forget-pi-day-we-should-be-celebrating-tau-day |access-date=2023-05-02 |language=en-US}}</ref> 

In 1897, an amateur mathematician attempted to persuade the 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 mathematical constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, and thus it did not become a law.<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>

=== In computer culture ===
In contemporary [[Social:Internet culture|internet culture]], individuals and organizations frequently pay homage to the number {{pi}}. For instance, the [[Computer scientist|computer scientist]] [[Biography:Donald Knuth|Donald Knuth]] let the version numbers of his program [[Software:TeX|TeX]] approach {{pi}}. The versions are 3, 3.1, 3.14, and so forth.<ref>{{cite journal |url=http://www.ntg.nl/maps/05/34.pdf |title=The Future of TeX and Metafont |first=Donald |last=Knuth |journal=TeX Mag |volume=5 |issue=1 |page=145 |date=3 October 1990 |access-date=17 February 2017|url-status=live |archive-url=https://web.archive.org/web/20160413230304/http://www.ntg.nl/maps/05/34.pdf |archive-date=13 April 2016}}</ref> ''{{mvar|τ}}'' has been added to several [[Programming language|programming language]]s as a predefined constant.<ref>{{cite web| url = https://www.python.org/dev/peps/pep-0628/| title = PEP 628 – Add math.tau}}</ref><ref>{{cite web |url=https://docs.rs/tau/latest/tau/ |title=Crate tau |access-date=2022-12-06}}</ref>

== See also ==
* [[Approximations of π|Approximations of {{pi}}]]
* [[Chronology of computation of π|Chronology of computation of {{pi}}]]
* [[List of mathematical constants]]

== References ==
=== Explanatory notes ===
{{notelist}}

=== Citations ===
{{reflist}}

=== General and cited sources ===
{{refbegin|30em|indent=yes}}
* {{cite book|last=Abramson|first=Jay|title=Precalculus|publisher=OpenStax|year=2014|url=https://openstax.org/details/books/precalculus}}
* {{cite book |last1=Andrews |first1=George E. |last2=Askey |first2=Richard |last3=Roy |first3=Ranjan |title=Special Functions |url=https://books.google.com/books?id=kGshpCa3eYwC&pg=PA59 |year=1999 |publisher=University Press |location=Cambridge |isbn=978-0-521-78988-2}}
* {{cite book |last1=Arndt |first1=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/books?id=QwwcmweJCDQC |access-date=5 June 2013}} English translation by Catriona and David Lischka.
* {{cite book |last1=Berggren |first1=Lennart |last2=Borwein |first2=Jonathan|last3=Borwein |first3=Peter|title=Pi: a Source Book |publisher=Springer-Verlag |year=1997 |isbn=978-0-387-20571-7 }}
* {{cite book |last1=Boyer |first1=Carl B. |last2=Merzbach |first2=Uta C.|year=1991 |title=A History of Mathematics |url=https://archive.org/details/historyofmathema00boye|url-access=registration |edition=2 |publisher=Wiley |isbn=978-0-471-54397-8 }}<!-- 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 p. 202 of 1991 edition; p. 228 of 1968 edition. Google snippet has a hit for 3.1456 on p. 168 for 1991, but does not show the number. -->
* {{cite book |last1=Bronshteĭn |first1=Ilia |last2=Semendiaev |first2=K.A. |title=A Guide Book to Mathematics |publisher=[[Company:Verlag Harri Deutsch|Verlag Harri Deutsch]] |year=1971 |isbn=978-3-87144-095-3 |title-link=A Guide Book to Mathematics}}
* {{cite book |first1=H. |last1=Dym |first2=H. P. |last2=McKean |title=Fourier series and integrals |publisher=Academic Press |year=1972 }}
* {{wikicite |ref={{harvid|Eymard|Lafon|2004}} |reference={{cite book |ref=none |last1=Eymard |first1=Pierre |last2=Lafon |first2=Jean Pierre |title=The Number {{pi}} |publisher=American Mathematical Society |year=2004 |isbn=978-0-8218-3246-2 |translator-last=Wilson |translator-first=Stephen }} English translation of {{cite book |ref=none |display-authors=0 |last1=Eymard |first1=Pierre |last2=Lafon |first2=Jean Pierre |title=Autour du nombre {{pi}} |language=fr |publisher=Hermann |year=1999 }} }}
* {{cite book |last1=Posamentier |first1=Alfred S. |last2=Lehmann |first2=Ingmar |title={{pi}}: A Biography of the World's Most Mysterious Number |url=https://archive.org/details/pi00alfr_0|url-access=registration |publisher=Prometheus Books |year=2004 |isbn=978-1-59102-200-8 }}
* {{cite book |last=Remmert |first=Reinhold |editor1=Heinz-Dieter Ebbinghaus |editor2=Hans Hermes |editor3=Friedrich Hirzebruch |editor4=Max Koecher |editor5=Klaus Mainzer |editor6=Jürgen Neukirch |editor7=Alexander Prestel |editor8=Reinhold Remmert |title=Numbers|chapter-url=https://books.google.com/books?id=Z53SBwAAQBAJ&pg=PA123 |chapter=Ch. 5 What is π? |date=2012 |publisher=Springer |isbn=978-1-4612-1005-4}}
{{refend}}

== Further reading ==
{{refbegin|indent=yes}}
* {{cite book |last=Blatner |first=David |title=The Joy of {{pi}} |publisher=Walker & Company |year=1999 |isbn=978-0-8027-7562-7 }}
* {{cite book |last=Delahaye |first=Jean-Paul |title=Le fascinant nombre {{pi}} |location=Paris |publisher=Bibliothèque Pour la Science |date=1997 |isbn=2-902918-25-9}}
{{refend}}

== External links ==
* {{mathworld|title=Pi|urlname=Pi}}
* 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 analysed ''[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).
* [http://pisearch.org/pi {{pi}} Search Engine] 2 billion searchable digits of {{pi}}, {{mvar|e}} and {{radic|2}}
* [https://www.geogebra.org/m/kwty4hsz ''approximation von π by lattice points''] and [https://www.geogebra.org/m/bxfa364u ''approximation of π with rectangles and trapezoids''] (interactive illustrations)
{{Irrational number}}

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

{{Sourceattribution|Pi}}