The mathematical constant π (written as "pi" when the Greek letter is not available) is commonly used in mathematics and physics. In Euclidean plane geometry, π may be defined as either the ratio of a circle's circumference to its diameter, or as the area of a circle of radius 1 (the unit circle). Most modern textbooks define π analytically using trigonometric functions, for example as the smallest positive x for which sin(x) = 0, or as twice the smallest positive x for which cos(x) = 0. All these definitions are equivalent.
Pi is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number.
The numerical value of π approximated to 64 decimal places is:
 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 5923
More digits of π are also available. See pi to 1,000 places , 10,000 places , 100,000 places , and 1,000,000 places . This page is also good for up to 200 million places: pi search page
Properties
π is an irrational number: that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In fact, the number is transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational (equivalently, integer) coefficients of which π is a root.
An important consequence of the transcendence of π is the fact that it is not constructible. This means that it is impossible to express π using only a finite number of integers, fractions and their square roots. This result establishes the impossibility of squaring the circle: it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The reason is that the coordinates of all points that can be constructed with ruler and compass are constructible numbers.
While the original Greek letter π was phonetically equivalent to the English letter p, it has now evolved to be pronounced like the word pie in most circles.
Formulas involving π
Geometry
π appears in many formulas in geometry involving circles and spheres.
Geometrical shape 
Formula 
Circumference of circle of radius r and diameter d


Area of circle of radius r


Area of ellipse with semiaxes a and b


Volume of sphere of radius r


Surface area of sphere of radius r


Volume of cylinder of height h and radius r


Surface area of cylinder of height h and radius r 

Volume of cone of height h and radius r


Surface area of cone of height h and radius r 

Also, the angle measurement 180° (in degrees) is equal to π radians.
Analysis
Many formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and socalled special functions.
 This commonly cited infinite series is usually written as above, but is more technically expressed as:
 and generally, ζ(2n) is a rational multiple of π^{2n} for positive integer n
 Area of one quarter of the unit circle:
Continued fractions
π has many continued fractions representations, including:
(Other representations are available at The Wolfram Functions Site.)
Number theory
Some results from number theory:
Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,..., N}, and then take the limit as N approaches infinity.
Dynamical systems / ergodic theory
In dynamical systems theory (see also ergodic theory), for almost every realvalued x_{0} in the interval [0,1],
where the x_{i} are iterates of the Logistic map for r = 4.
Physics
In physics, appearance of π in formulas is usually only a matter of convention and normalization. For example, by using the reduced Planck's constant one can avoid writing π explicitly in many formulas of quantum mechanics. In fact, the reduced version is the more fundamental, and presence of factor 1/2π in formulas using h can be considered an artifact of the conventional definition of Planck's constant.
Probability and statistics
In probability and statistics, there are many distributions whose formulas contain π, including:
Note that since , for any pdf f(x), the above formulas can be used to produce other integral formulas for π.
An interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using:
History
The symbol "π" for Archimedes' constant was first introduced in 1706 by William Jones when he published A New Introduction to Mathematics, although the same symbol had been used earlier to indicate the circumference of a circle. The notation became standard after it was adopted by Leonhard Euler. In either case, π is the first letter of περιμετρος (perimetros), meaning 'measure around' in Greek.
Here is a brief chronology of π:
Date 
Person 
Value of π
(world records in bold)

20th century BC

Babylonians

25/8 = 3.125 
20th century BC

Egyptian Rhind Mathematical Papyrus

(16/9)² = 3.160493... 
12th century BC

Chinese

3 
mid 6th century BC

1 Kings 7:23

3 
434 BC

Anaxagoras tried to square the circle with straightedge and compass


3rd century BC

Archimedes

223/71 < π < 22/7
(3.140845... < π < 3.142857...)
211875/67441 = 3.14163... 
20 BC

Vitruvius

25/8 = 3.125 
130

Chang Hong 
√10 = 3.162277... 
150

Ptolemy

377/120 = 3.141666... 
250

Wang Fau 
142/45 = 3.155555... 
263

Liu Hui

3.14159 
480

Zu Chongzhi

3.1415926 < π < 3.1415927 
499

Aryabhatta

62832/20000 = 3.1416 
598

Brahmagupta

√10 = 3.162277... 
800

Al Khwarizmi

3.1416 
12th Century

Bhaskara

3.14156 
1220

Fibonacci

3.141818 
1400

Madhava

3.14159265359 
All records from 1424 are given as the number of correct decimal places (dps). 
1424

Jamshid Masud Al Kashi

16 dps 
1573

Valenthus Otho 
6 dps 
1593

François Viète

9 dps 
1593

Adriaen van Roomen 
15 dps 
1596

Ludolph van Ceulen

20 dps 
1615

Ludolph van Ceulen

32 dps 
1621

Willebrord Snell (Snellius), a pupil of Van Ceulen

35 dps 
1665

Isaac Newton

16 dps 
1699

Abraham Sharp

71 dps 
1700

Seki Kowa

10 dps 
1706

John Machin

100 dps 
1706

William Jones introduced the Greek letter π


1730

Kamata 
25 dps 
1719

De Lagny calculated 127 decimal places, but not all were correct 
112 dps 
1723

Takebe 
41 dps 
1734

Leonhard Euler adopted the Greek letter π and assured its popularity


1739

Matsunaga

50 dps 
1761

Johann Heinrich Lambert proved that π is irrational


1775

Euler pointed out the possibility that π might be transcendental


1789

Jurij Vega calculated 140 decimal places, but not all are correct

137 dps 
1794

AdrienMarie Legendre showed that π² (and hence π) is irrational, and mentioned the possibility that π might be transcendental.


1841

Rutherford calculated 208 decimal places, but not all were correct 
152 dps 
1844

Zacharias Dase and Strassnitzky 
200 dps 
1847

Thomas Clausen 
248 dps 
1853

Lehmann 
261 dps 
1853

Rutherford 
440 dps 
1853

William Shanks 
527 dps 
1855

Richter 
500 dps 
1874

William Shanks took 15 years to calculate 707 decimal places, but not all were correct (the error was found by D. F. Ferguson in 1946)

527 dps 
1882

Lindemann proved that π is transcendental (the LindemannWeierstrass theorem)


1946

D. F. Ferguson used a desk calculator 
620 dps 
1947

710 dps 
1947

808 dps 
All records from 1949 onwards were calculated with electronic computers. 
1949

J. W. Wrench, Jr, and L. R. Smith were the first to use an electronic computer (the Eniac) to calculate π

2,037 dps 
1953

Mahler showed that π is not a Liouville number


1955

J. W. Wrench, Jr, and L. R. Smith 
3,089 dps 
1961

100,000 dps 
1966

250,000 dps 
1967

500,000 dps 
1974

1,000,000 dps 
1992

2,180,000,000 dps 
1995

Yasumasa Kanada

> 6,000,000,000 dps 
1997

Kanada and Takahashi 
> 51,500,000,000 dps 
1999

Kanada and Takahashi 
> 206,000,000,000 dps 
2002

Kanada and team 
> 1,240,000,000,000 dps 
2003

Kanada and team 
> 1,241,100,000,000 dps 
Numerical approximations of π
Due to the transcendental nature of π, there are no nice closed expressions for π. Therefore numerical calculations must use approximations to the number. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π.
An Egyptian scribe named Ahmes is the source of the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes states that he copied a Middle Kingdom papyrus—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160.
The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in 263 and suggested that 3.14 was a good approximation.
The Indian mathematician and astronomer Aryabhata gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixtytwo thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with radius 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.
The Chinese mathematician and astronomer Zu Chongzhi computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the 5th century.
The Iranian mathematician and astronomer, Ghyath addin Jamshid Kashani, 13501439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digit as:
 2 π = 6.2831853071795865
The German mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tombstone.
The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today.
None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as Machin's:
together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with
Formulas of this kind are known as Machinlike formulas.
Extremely long decimal expansions of π are typically computed with the GaussLegendre algorithm and Borwein's algorithm; the SalaminBrent algorithm which was invented in 1976 has also been used in the past.
The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machinlike formulas were used for this:

K. Takano (1982).

F. C. W. Störmer (1896).
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records.
In 1996 David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series:
This formula permits one to easily compute the k^{th} binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64bits around the quadrillionth bit of π (which turns out to be 0).
Other formulas that have been used to compute estimates of π include:

Newton.

Ramanujan.

David Chudnovsky and Gregory Chudnovsky.

Euler.
On computers running Microsoft Windows OS, the program PiFast can be used to quickly calculate a large amount of digits. The largest number of digits of π calculated on a home computer, 25,000,000,000, was calculated with PiFast in 17 days.
Open questions
The most pressing open question about π is whether it is a normal number  whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of π.
Bailey and Crandall showed in 2000 that the existence of the above mentioned BaileyBorweinPlouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.
It is also unknown whether π and e are algebraically independent, i.e. whether there is a polynomial relation between π and e with rational coefficients.
John Harrison, (16931776) (of Longitude fame), devised a meantone temperament musical tuning system derived from π. This Lucy Tuning system (due to the unique mathematical properties of π), can map all musical intervals, harmony and harmonics. This suggests that musical harmonics beat, and that using π could provide a more precise model for the analysis of both musical and other harmonics in vibrating systems.
The nature of π
In nonEuclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulae in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. The reason it occurs so often in physics is simply because it's convenient in many physical models. For example Coulomb's law , here 4πr^{2} is just the surface area of sphere of radius r, which is a convenient way of describing the inverse square relationship of the force at a distance r from a point source, it would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient, if Planck charge is used it can be written as and thus eliminate the need for π.
π culture
There is an entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, which is known as piphilology. See Pi mnemonics for examples.
March 14 (3/14) marks Pi Day which is celebrated by many lovers of π. On July 22, Pi Approximation Day is celebrated (22/7 is a popular approximation of π).
Furthermore, many talk of "pi o clock" [fifteen seconds past fourteen minutes past 3 (3:14:15) is slightly less than pi o clock; 3:08:30 would be closest to π hours past noon or midnight in whole seconds].
Another example of mathhumor is this approximation of π: Take the number "1234", transpose the first two digits and the last two digits, so the number becomes "2143". Divide that number by "twotwo" (22, so 2143/22 = 97.40909...). Take the twosquaredth root (4th root) of this number. The final outcome is remarkably close to π: 3.14159265.
Related articles
External links
Digit resources
Calculation
General
Mnemonics
Last updated: 10192005 23:29:19