Editing Pi (section)

=== 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>