Pi

The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,

(1) (2)

has decimal expansion given by

(3)

(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .

A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon), Archimedes obtained

(4)

(Wells 1986, p. 49; Shanks 1993, p. 140; Borwein et al. 2004, pp. 1-3).

is known to be irrational (Lambert 1761; Legendre 1794; Hermite 1873; Nagell 1951; Niven 1956; Struik 1969; Königsberger 1990; Schröder 1993; Stevens 1999; Borwein and Bailey 2003, pp. 139-140). In 1794, Legendre also proved that is irrational (Wells 1986, p. 76). is also transcendental (Lindemann 1882). An immediate consequence of Lindemann's proof of the transcendence of also proved that the geometric problem of antiquity known as circle squaring is impossible. A simplified, but still difficult, version of Lindemann's proof is given by Klein (1955).

It is also known that is not a Liouville number (Mahler 1953), but it is not known if is normal to any base (Stoneham 1970). The following table summarizes progress in computing upper bounds on the irrationality measure for . It is likely that the exponent can be reduced to , where is an infinitesimally small number (Borwein et al. 1989).

upper bound reference 20 Mahler (1953), Le Lionnais (1983, p. 50) 14.65 Chudnovsky and Chudnovsky (1984) 8.0161 Hata (1992) 7.606308 Salikhov (2008) 7.10320534 Zeilberger and Zudilin (2020)

It is not known if , , or are irrational. However, it is known that they cannot satisfy any polynomial equation of degree with integer coefficients of average size (Bailey 1988ab, Borwein et al. 1989).

J. H. Conway has shown that there is a sequence of fewer than 40 fractions , , ... with the property that if you start with and repeatedly multiply by the first of the that gives an integer result until a power of 2 (say, ) occurs, then is the th decimal digit of .

crops up in all sorts of unexpected places in mathematics besides circles and spheres. For example, it occurs in the normalization of the normal distribution, in the distribution of primes, in the construction of numbers which are very close to integers (the Ramanujan constant), and in the probability that a pin dropped on a set of parallel lines intersects a line (Buffon's needle problem). Pi also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river (Stølum 1996, Singh 1997).

The Bible contains two references (I Kings 7:23 and Chronicles 4:2) which give a value of 3 for (Wells 1986, p. 48). It should be mentioned, however, that both instances refer to a value obtained from physical measurements and, as such, are probably well within the bounds of experimental uncertainty. I Kings 7:23 states, "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits in height thereof; and a line thirty cubits did compass it round about." This implies . The Babylonians gave an estimate of as , while the Egyptians gave in the Rhind papyrus and 22/7 elsewhere. The Chinese geometers, however, did best of all, rigorously deriving to 6 decimal places.

appeared in Alfred Hitchcock's insipid and poorly acted 1966 film Torn Curtain, including in one particularly strange but memorable scene where Paul Newman (Professor Michael Armstrong) draws a symbol in the dirt with his foot at the door of a farmhouse. In this film, the symbol is the pass-sign of an underground East German network that smuggles fugitives to the West.

The 1998 film Pi is a dark, strange, and hyperkinetic movie about a mathematician who is slowly going insane searching for a pattern to the Stock Market. Both a Hasidic cabalistic sect and a Wall Street firm learn of his investigation and attempt to seduce him. Unfortunately, the film has essentially nothing to do with real mathematics. 314159, the first six digits of , is the combination to Ellie's office safe in the novel Contact by Carl Sagan.

On Sept. 15, 2005, Google offered exactly 14159265 shares of Class A stock, which is the same as the first eight digits or after the decimal point (Markoff 2005).

The formula for the volume of a cylinder leads to the mathematical joke: "What is the volume of a pizza of thickness and radius ?" Answer: pi z z a. This result is sometimes known as the second pizza theorem.

The 2005 album Aerial features a song called "Pi" in which the first digits of are interspersed (unfortunately incorrectly) with lyrics.

There are many, many formulas for pi, from the simple to the very complicated.

Ramanujan (1913-1914) and Olds (1963) give geometric constructions for 355/113. Gardner (1966, pp. 92-93) gives a geometric construction for . Dixon (1991) gives constructions for and . Constructions for approximations of are approximations to circle squaring (which is itself impossible).