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The ratio of the length of a circle to its diameter; it is an infinite non-periodic decimal number
 
The ratio of the length of a circle to its diameter; it is an infinite non-periodic decimal number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726502.png" /></td> </tr></table>
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$$\pi=3.141592653589793\ldots.$$
  
One frequently arrives at the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726503.png" /> as the limit of certain arithmetic sequences involving simple laws. An example is Leibniz' series
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One frequently arrives at the number $\pi$ as the limit of certain arithmetic sequences involving simple laws. An example is Leibniz' series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726504.png" /></td> </tr></table>
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$$\frac\pi4=1-\frac13+\frac15-\frac17+\frac19-\ldots,$$
  
which, however, converges very slowly. There are more rapidly-converging series suitable for calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726505.png" />.
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which, however, converges very slowly. There are more rapidly-converging series suitable for calculating $\pi$.
  
The possibility of a pure analytic definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726506.png" /> is of essential significance for geometry. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726507.png" /> also participates in certain formulas in non-Euclidean geometry, but not as the ratio of the length of a circle to its diameter (this ratio is not constant in non-Euclidean geometry). The arithmetic nature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726508.png" /> was finally elucidated in analysis, with a decisive part played by Euler's formula:
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The possibility of a pure analytic definition of $\pi$ is of essential significance for geometry. For example, $\pi$ also participates in certain formulas in non-Euclidean geometry, but not as the ratio of the length of a circle to its diameter (this ratio is not constant in non-Euclidean geometry). The arithmetic nature of $\pi$ was finally elucidated in analysis, with a decisive part played by Euler's formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p0726509.png" /></td> </tr></table>
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$$e^{\pi i}=-1.$$
  
At the end of the 18th century, J. Lambert and A. Legendre established that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265010.png" /> is an [[Irrational number|irrational number]], while in the 19th century, F. Lindemann showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265011.png" /> is a [[Transcendental number|transcendental number]].
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At the end of the 18th century, J. Lambert and A. Legendre established that $\pi$ is an [[Irrational number|irrational number]], while in the 19th century, F. Lindemann showed that $\pi$ is a [[Transcendental number|transcendental number]].
  
  
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A nice account of Lindemann's proof can be found in [[#References|[a3]]], Chapt. 6.
 
A nice account of Lindemann's proof can be found in [[#References|[a3]]], Chapt. 6.
  
The number of known digits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265012.png" /> has increased exponentially in recent times. At the moment (1990), the record seems to be half a billion digits (D.V. Chudnovsky and G.V. Chudnovsky). For an account of such computations see [[#References|[a1]]]. Up to the 1960's the standard way to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265013.png" /> was to use Machin's formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265014.png" /> and the power series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265015.png" />. Nowadays, some powerful formulas of Ramanujan are used. It is still not known how randomly the digits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265016.png" /> are distributed; in particular, whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072650/p07265017.png" /> is a [[Normal number|normal number]].
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The number of known digits of $\pi$ has increased exponentially in recent times. At the moment (1990), the record seems to be half a billion digits (D.V. Chudnovsky and G.V. Chudnovsky). For an account of such computations see [[#References|[a1]]]. Up to the 1960's the standard way to calculate $\pi$ was to use Machin's formula $\pi/4=4\arctan(1/5)-\arctan(1/239)$ and the power series of $\arctan(z)$. Nowadays, some powerful formulas of Ramanujan are used. It is still not known how randomly the digits of $\pi$ are distributed; in particular, whether $\pi$ is a [[Normal number|normal number]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.M. Borwein,  P.B. Borwein,  "Pi and the AGM" , Interscience  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Beckmann,  "A history of pi" , The Golem Press , Boulder (Co.)  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.M. Borwein,  P.B. Borwein,  "Pi and the AGM" , Interscience  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Beckmann,  "A history of pi" , The Golem Press , Boulder (Co.)  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>

Revision as of 16:51, 11 April 2014

The ratio of the length of a circle to its diameter; it is an infinite non-periodic decimal number

$$\pi=3.141592653589793\ldots.$$

One frequently arrives at the number $\pi$ as the limit of certain arithmetic sequences involving simple laws. An example is Leibniz' series

$$\frac\pi4=1-\frac13+\frac15-\frac17+\frac19-\ldots,$$

which, however, converges very slowly. There are more rapidly-converging series suitable for calculating $\pi$.

The possibility of a pure analytic definition of $\pi$ is of essential significance for geometry. For example, $\pi$ also participates in certain formulas in non-Euclidean geometry, but not as the ratio of the length of a circle to its diameter (this ratio is not constant in non-Euclidean geometry). The arithmetic nature of $\pi$ was finally elucidated in analysis, with a decisive part played by Euler's formula:

$$e^{\pi i}=-1.$$

At the end of the 18th century, J. Lambert and A. Legendre established that $\pi$ is an irrational number, while in the 19th century, F. Lindemann showed that $\pi$ is a transcendental number.


Comments

A nice account of Lindemann's proof can be found in [a3], Chapt. 6.

The number of known digits of $\pi$ has increased exponentially in recent times. At the moment (1990), the record seems to be half a billion digits (D.V. Chudnovsky and G.V. Chudnovsky). For an account of such computations see [a1]. Up to the 1960's the standard way to calculate $\pi$ was to use Machin's formula $\pi/4=4\arctan(1/5)-\arctan(1/239)$ and the power series of $\arctan(z)$. Nowadays, some powerful formulas of Ramanujan are used. It is still not known how randomly the digits of $\pi$ are distributed; in particular, whether $\pi$ is a normal number.

References

[a1] J.M. Borwein, P.B. Borwein, "Pi and the AGM" , Interscience (1987)
[a2] P. Beckmann, "A history of pi" , The Golem Press , Boulder (Co.) (1971)
[a3] I. Stewart, "Galois theory" , Chapman & Hall (1979)
How to Cite This Entry:
Pi(number). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pi(number)&oldid=31518
This article was adapted from an original article by Material from the article "Pi" in BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article