Editing Pi (section)

== Fundamentals ==

=== Name ===
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase [[Pi (letter)|Greek letter {{pi}}]], sometimes spelled out as ''pi.''{{r|firstPi}} In English, {{pi}} is [[English pronunciation of Greek letters|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]].

The choice of the symbol {{pi}} is discussed in the section [[#Adoption of the symbol π|''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]] of a [[circle]]'s [[circumference]] {{math|''C''}} to its [[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]] around the [[perimeter]] of the circle, a quantity which can be formally defined independently of geometry using [[limit (mathematics)|limits]]—a concept in [[calculus]].<ref>{{cite book |first=Tom |last=Apostol |author-link=Tom M. 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]]:{{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 proposed as a definition of {{pi}} by [[Karl Weierstrass]], who defined it directly as an integral in 1841.{{efn|The specific integral that Weierstrass used was<ref>{{harvnb|Remmert|2012|p=148}}.{{pb}} {{cite book |last=Weierstrass |first=Karl |author-link=Karl Weierstrass |chapter=Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt |trans-chapter=Representation of an analytical function of a complex variable, whose absolute value lies between two given limits |language=de |title=Mathematische Werke |volume=1 |publication-place=Berlin |publisher=Mayer & Müller |year=1841 |publication-date=1894 |pages=51–66 |chapter-url=https://archive.org/details/mathematischewer01weieuoft/page/51/}}</ref>
{{br}}
<math display=block>\pi=\int_{-\infty}^\infty\frac{dx}{1+x^2}.</math>}}

Integration is no longer commonly used in a first analytical definition because, as {{harvnb|Remmert|2012}} explains, [[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 [[Edmund Landau]],<ref>{{cite book |first=Edmund |last=Landau |author-link=Edmund 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]] 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]] 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]],<ref>{{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Real and complex analysis |publisher=McGraw-Hill |year=1986 |page=2}}</ref> or as the solution of a [[differential equation]].{{r|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 |author-link=Lars 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]] and [[algebra]], is the following theorem:<ref>{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Topologie generale |publisher=Springer |year=1981 |at=§VIII.2}}</ref> there is a unique ([[up to]] [[automorphism]]) [[continuous function|continuous]] [[isomorphism]] from the [[group (mathematics)|group]] '''R'''/'''Z''' of real numbers under addition [[quotient group|modulo]] integers (the [[circle group]]), onto the multiplicative group of [[complex numbers]] of [[absolute value]] one. The number {{pi}} is then defined as half the magnitude of the derivative of this homomorphism.<ref 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]], 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]], 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 are generally [[proofs by contradiction]] and require calculus. The degree to which {{pi}} can be approximated by [[rational number]]s (called the [[irrationality measure]]) is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of {{math|''e''}} 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 |issn=0036-0279}}</ref>

The digits of {{pi}} have no apparent pattern and have passed tests for [[statistical randomness]], including tests for [[normal number|normality]]; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. 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 test]]s, 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]]. 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 [[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 [[mathematical folklore]], after [[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]], [[squaring the circle]] is not possible in a finite number of steps using the classical tools of [[Compass-and-straightedge construction|compass and straightedge]].|left]]
In addition to being irrational, {{pi}} is also a [[transcendental number]], which means that it is not the [[solution (equation)|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]] function.}} This follows from the so-called [[Lindemann–Weierstrass theorem#Transcendence of e and π|Lindemann–Weierstrass theorem]], which also establishes the transcendence of [[E (mathematical constant)|the constant ''{{mvar|e}}'']].

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 construction|compass and straightedge]], it is not possible to "[[squaring the circle|square the circle]]". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.{{sfn|Posamentier|Lehmann|2004|p=25}} Squaring a circle was one of the important geometry problems of the [[classical antiquity]].{{sfn|Eymard|Lafon|2004|p=129}} 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=Petr |author-link=Petr Beckmann |title=History of Pi |publisher=St. Martin's Press |year=1989 |orig-year=1974 |isbn=978-0-88029-418-8 |page=37}} {{pb}} {{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>

An [[List of unsolved problems in mathematics|unsolved problem]] thus far is the question of whether or not the numbers ''{{mvar|π}}'' and ''{{mvar|e}}'' are [[Algebraic independence|algebraically independent]] ("relatively transcendental"). This would be resolved by [[Schanuel's conjecture]]<ref>{{Cite book |last1=Murty |first1=M. Ram |author-link1=M. Ram Murty |url=https://link.springer.com/book/10.1007/978-1-4939-0832-5 |title=Transcendental Numbers |last2=Rath |first2=Purusottam |date=2014 |publisher=Springer |language=en |doi=10.1007/978-1-4939-0832-5 |isbn=978-1-4939-0831-8}} {{pb}} {{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}</ref> – a currently unproven generalization of the Lindemann–Weierstrass theorem.<ref>{{mathworld|title=Lindemann-Weierstrass Theorem|id=Lindemann-WeierstrassTheorem|ref=none}}</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 [[simple 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.{{sfn|Eymard|Lafon|2004|p=78}} 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]]. Although the simple continued fraction for {{pi}} (with numerators all 1, 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 non-simple [[continued fraction]]s do, such as:<ref>{{cite journal |title=An Elegant Continued Fraction for π |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>{{efn|The middle of these is due to the mid-17th century mathematician [[William Brouncker, 2nd Viscount Brouncker|William Brouncker]], see [[William Brouncker, 2nd Viscount Brouncker#Brouncker's formula|§&nbsp;Brouncker's formula]].}}

<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>

=== 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}}.{{sfn|Eymard|Lafon|2004|p=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]]s) are {{gaps|11.0010|0100|0011|1111|0110|1010|1000|1000|1000|0101|1010|0011...}} (see {{OEIS2C|id=A004601}})
* The first 36 digits in [[ternary numeral system|ternary]] (base 3) are {{gaps|10.010|211|012|222|010|211|002|111|110|221|222|220...}} (see {{OEIS2C|id=A004602}})
* The first 20 digits in [[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]] (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. |author-link=Edward Stewart Kennedy |journal=Journal for the History of Astronomy |volume=9 |page=65 |bibcode=1978JHA.....9...65K |doi=10.1177/002182867800900106 |year=1978 |s2cid=126383231}} [[Ptolemy]] used a three-sexagesimal-digit approximation, and [[Jamshīd al-Kāshī]] expanded this to nine digits; see {{cite book |last=Aaboe |first=Asger |author-link=Asger Aaboe |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]] centred at the [[Origin (mathematics)|origin]] in the [[complex plane]] given by [[Euler's formula]]]]

Any [[complex number]], say {{Mvar|z}}, can be expressed using a pair of [[real number]]s. In the [[Polar coordinate system#Complex numbers|polar coordinate system]], one number ([[radius]] or {{Mvar|r}}) is used to represent {{Mvar|z}}'s distance from the [[Origin (mathematics)|origin]] of the [[complex plane]], and the other (angle or {{Mvar|φ}}) the counter-clockwise [[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]] satisfying <math>i^2=-1</math>. The frequent appearance of {{pi}} in [[complex analysis]] can be related to the behaviour of the [[exponential function]] of a complex variable, described by [[Euler's formula]]:{{sfn|Bronshteĭn|Semendiaev|1971|p=592}}

<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]]. This formula establishes a correspondence between imaginary powers of {{math|''e''}} and points on the [[unit circle]] centred at the origin of the complex plane. Setting <math>\varphi=\pi</math> in Euler's formula results in [[Euler's identity]], celebrated in mathematics due to it containing five important mathematical constants:{{sfn|Bronshteĭn|Semendiaev|1971|p=592}}<ref>{{cite book |last=Maor |first=Eli |author-link=Eli Maor |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 numbers {{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>