Difference between revisions of "Pi(number)"
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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 | ||
− | + | $$\pi=3.141592653589793\ldots.$$ | |
− | One frequently arrives at the number | + | 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 | + | which, however, converges very slowly. There are more rapidly-converging series suitable for calculating $\pi$. |
− | The possibility of a pure analytic definition of | + | 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 | + | 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 | + | 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 & 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 & 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) |
Pi(number). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pi(number)&oldid=11364