Pi

π was also the name of a motion picture. See (π (movie)).

The number pi (denoted with the lower-case Greek letter π) is a mathematical constant which occurs in many areas of mathematics and physics. It is also known as Archimedes' constant or Ludolph's number and is equal to the ratio of a circle's circumference to its diameter in Euclidean geometry. Alternatively, π can be defined as the area of a circle of radius 1, or as the smallest positive number x for which sin(x) = 0.

The value of π to the first sixty-four digits is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 592...

Properties

The number π is an irrational number: that is, it cannot be written as the ratio of two integers. This was proved in 1761 by Johann Heinrich Lambert. In fact, the number is transcendental, as was proved by Ferdinand Lindemann in 1882. This means that there is no polynomial with integer (or rational) coefficients of which π is a root. As a consequence, it is impossible to express π using only a finite number of integers, fractions and their 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 special algebraic numbers.

Formulae involving π

Geometry:

Circumference of circle or sphere of radius r: C = 2 π r

Area of circle of radius r: A = π r²

Area of ellipse with semiaxes a and b: A = π ab

Volume of sphere of radius r: V = (4/3) π r³

Surface area of sphere of radius r: A = 4 π r²

Angles: 180 degrees is equivalent to π radians

Analysis:

{\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}

(Leibniz' formula)

{\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {\pi }{2}}

(Wallis' product)

\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}

(Euler)

\zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}

\int _{-\infty }^{\infty }e^{-x^{2}}dx={\sqrt {\pi }}

\Gamma \left({1 \over 2}\right)={\sqrt {\pi }}

n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}

(Stirling's formula)

\sum _{k=0}^{n}\phi (k)\sim \pi n^{2}

e^{\pi i}+1=0\;

(Euler's identity, also called "The most remarkable formula in the world")

π has beautiful continued fractional representations:

{\frac {4}{\pi }}=1+{\frac {1}{3+{\frac {4}{5+{\frac {9}{7+{\frac {16}{9+{\frac {25}{11+{\frac {36}{13+...}}}}}}}}}}}}

(You can see 12 other representations at [1] )

Number theory:

The probability that two randomly chosen integers are relatively prime is 6/π².

The probability that a randomly chosen integer is square-free is 6/π².

The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.

Dynamical Systems / Ergodic Theory:

\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}

for almost every x₀ in [0, 1] where the x_i are iterates of the Logistic map for r=4.

Physics:

\Delta x\Delta p\geq {\frac {h}{4\pi }}

(Heisenberg's uncertainty principle)

R_{ik}-{g_{ik}R \over 2}+\Lambda g_{ik}={8\pi G \over c^{4}}T_{ik}

(Einstein's field equation of general relativity)

Statistics:

f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2} \over 2\sigma ^{2}}}

(The probability density function for the normal distribution.)

Computing the value of pi

Due to the transcendental nature of π there are no nice closed expressions for π. Therefore calculations have to 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. 355/113, with two each of the first 3 odd digits, is a simple and easily memorised fraction which is good for 7 significant figures.

An Egyptian scribe called Ahmes is the source of the oldest known text to give an approximate value for π. The Rhind Papyrus dates from the 17th century B.C.E. 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 (incorrect in the fourth decimal digit) in 263 C.E. and suggested that 3.14 was a good approximation.

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 137 were correct and held the world record for over 50 years at that time. He 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 like Machin's:

{\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}

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

(5+i)⁴ · (-239 + i) = -114244-114244 i.

Formulas of this kind are known as Machin-like formulas.

Extremely large numbers of digits of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent 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 64-node 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 PiHex Project have 1000000000000000, 1^15 bits of pi. (1^30 digits of PI) PiHex,PiHex Site

the following Machin-like formulas were used for this:

{\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}

(K. Takano, 1982)

{\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}

(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.

In 1996 David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series:

\pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)

This formula permits one to easily compute the n^th binary or hexadecimal digit of π, without having to compute the preceeding n-1 digits. Bailey's website contains the derivation as well as implementations in various programming languages.

Other formulas that have been used to compute pi include:

{\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k!)(1103+26390k)}{(k!)^{4}396^{4k}}}

(Ramanujan)

{\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}

(David Chudnovsky and Gregory Chudnovsky)

Open questions

The most pressing open question about π is whether it is normal, 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 should be true in any base, not just in base 10.

It isn't 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 Bailey-Borwein-Plouffe 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.

The nature of π

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π, and the ratio of a circle's circumference to its diameter may also differ from π. This doesn't change the value of π, but does affect many formulae in which it appears. So in particular, the shape of the universe does not affect the value of π at all: it is a mathematical constant, not a physical value.

Pi 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. This is obviously a play on Pi itself and the linguistic field of philology.

The most common mnemonic technique is to memorise a sentence in which the number of letters in each word in turn is equal to the corresponding digit of π. The most famous example of this is from Isaac Asimov (a noted tee-totaller):

How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!

If less accuracy is required an easy one is also:

How I wish I could recollect pi easily today!

Part of the school cheer of MIT is: "Cosine, secant, tangent, sine! 3 point 1 4 1 5 9!"

There are piphilologists who have written poems which encode 100s of digits. This is an example of constrained writing.

March 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). Incidentally, a good approximation to the duration of a year is π×10⁷ seconds.

Pi