 The title given to this article is incorrect due to technical limitations. The correct title is π.
 For other uses, see Pi (disambiguation).
The mathematical constant π (written as "pi" when the Greek letter is not available) is ubiquitous 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, e.g. 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 (sequence A000796 in OEIS) 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. Properties Pi 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 for pi 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 Pi appears in many formulas in geometry involving circles and spheres. Geometrical shape  Formula  Circumference of circle of radius r   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
 Stirling's approximation:
 Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"):
 Area of one quarter of the unit circle:
 A special case of Euler's formula
Continued fractions Pi has many continued fractions representations, including: (You can see other representations at The Wolfram Functions Site (http://functions.wolfram.com/Constants/Pi/10/).)
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 Formulas from physics.  Magnetic permeability of free space:
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 (http://www.nersc.gov/~dhbailey/) 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.
Open questions The most pressing open question about π is whether it is a normal number, i.e. 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.
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. For example, part of the school cheer of MIT is: "Cosine, secant, tangent, sine! 3 point 1 4 1 5 9!" See Pi mnemonics for more 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]. 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  Wikisource  Pi to 1,000 Places (http://sources.wikipedia.org/wiki/Pi_to_1,000_places)  10,000 Places (http://sources.wikipedia.org/wiki/Pi_to_10,000_places)  100,000 Places (http://sources.wikipedia.org/wiki/Pi_to_100,000_places)  1,000,000 Places (http://sources.wikipedia.org/wiki/Pi_to_1,000,000_places)
 Project Gutenberg EText containing a million digits of Pi (http://www.gutenberg.net/etext/50)
 Archives of Pi calculated to 1,000,000 or 10,000,000 places. (http://www.solidz.com/pi/)
 Search Pi for any sequence of digits (http://www.pisearch.de.vu/)
 Statistics about the first 1.2 trillion digits of Pi (http://www.supercomputing.org/pidecimal_current.html)
 A banner of approximately 220 million digits of pi (http://3.14.maxg.org/)
Calculation General Mnemonics  One of the more popular mnemonic devices for remembering pi (http://users.aol.com/s6sj7gt/mikerav.htm)
 Andreas P. Hatzipolakis: PiPhilology. A site with hundreds of examples of π mnemonics (http://www.cilea.it/~bottoni/wwwcilea/F90/piph.htm)
 Pi memorised as poetry (http://www.startfromhere.freeserve.co.uk/nudesci/abc/pi.htm)
