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Alternative meanings: Pi (letter), π (movie), Pi meson

The mathematical constant π (written as "pi" when the Greek letter is not available) is ubiquitous in many areas of 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. 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 of these definitions are equivalent.

Pi is also known as Archimedes' constant (not to be confused with Archimedes' number), Ludolph's number.

The first sixty-four decimal digits of π (sequence A000796 in OEIS) are:

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

More digits of π may be found at the following Wikisource links:Wikisource - Pi to 1,000 Places | 10,000 Places | 100,000 Places | 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 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.

Formulas involving π

Geometry

Pi appears in many formulas in geometry involving circles and spheres.

Geometrical shape	Formula
Circumference of circle of radius r	${\displaystyle C=2\pi r\,\!}$
Area of circle of radius r	${\displaystyle A=\pi r^{2}\,\!}$
Area of ellipse with semiaxes a and b	${\displaystyle A=\pi ab\,\!}$
Volume of sphere of radius r	${\displaystyle V={\frac {4}{3}}\pi r^{3}\,\!}$
Surface area of sphere of radius r	${\displaystyle A=4\pi r^{2}\,\!}$
Volume of cylinder of height h and radius r	${\displaystyle V=\pi r^{2}h\,\!}$
Surface area of cylinder of height h and radius r	${\displaystyle A=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!}$
Volume of cone of height h and radius r	${\displaystyle V={\frac {1}{3}}\pi r^{2}h\,\!}$
Surface area of cone of height h and radius r	${\displaystyle A=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!}$

Also, the angle measurement 180° (in degrees) is equivalent to π radians.

Analysis

Many formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.

• Leibniz' formula:

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

• Wallis' product:

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

• An integral formula from calculus (see also Error function and Normal distribution):

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

• Basel problem, first solved by Euler (see also Riemann zeta function):

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

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

• Gamma function evaluated at 1/2:

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

• Stirling's approximation:

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

• Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"):

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

• Property of Euler's totient function (see also Farey sequence):

${\displaystyle \sum _{k=0}^{n}\phi (k)\sim 3n^{2}/\pi ^{2}}$

Continued fractions

Pi has many continued fractions representations, including:

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

(You can see other representations at The Wolfram Functions Site.)

Number theory

Some results from number theory:

• The probability that two randomly chosen integers are relatively prime is 6/π.
• The probability that a randomly chosen integer is square-free is 6/π.
• 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

In dynamical systems theory (see also ergodic theory), for almost every real-valued x₀ in the interval ,

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

where the x_i are iterates of the Logistic map for r = 4.

Physics

Formulas from physics.

• Heisenberg's uncertainty principle:

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

• Einstein's field equation of general relativity:

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

• Coulomb's law for the electric force:

${\displaystyle F={\frac {\left|q_{1}q_{2}\right|}{4\pi \epsilon _{0}r^{2}}}}$

Probability and Statistics

In probability and statistics, there are many distributions whose formulas contain π, including:

• probability density function (pdf) for the normal distribution with mean μ and standard deviation σ:

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

• pdf for the (standard) Cauchy distribution:

${\displaystyle f(x)={\frac {1}{\pi (1+x^{2})}}}$

Note that since ${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1}$, 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:

${\displaystyle \pi \approx {\frac {2nL}{xS}}}$

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:

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 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 BCE 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 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 5th century.

The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, 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:

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

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

${\displaystyle (5+i)^{4}\cdot (-239+i)=-114244-114244i.}$

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

Extremely long decimal expansions 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 following Machin-like formulas were used for this:

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

K. Takano (1982).

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

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 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:

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

This formula permits one to easily compute the k 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 64-bits around the quadrillionth bit of π (which turns out to be 0).

Other formulas that have been used to compute estimates of π include:

${\displaystyle {\frac {\pi }{2}}=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=1+{\frac {1}{3}}\left(1+{\frac {2}{5}}\left(1+{\frac {3}{7}}\left(1+{\frac {4}{9}}(1+...)\right)\right)\right)}$

Newton.

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

Ramanujan.

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

David Chudnovsky and Gregory Chudnovsky.

${\displaystyle {\pi }=20\arctan {\frac {1}{7}}-8\arctan {\frac {3}{79}}}$

Euler.

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 must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., 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.

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 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 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.

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. For example, part of the school cheer of MIT is: "Cosine, secant, tangent, sine! 3 point 1 4 1 5 9!" See Piphilology 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).

Related articles

• Greek letter Pi
• Calculus
• Geometry
• Trigonometric function
• Pi through experiment
• Proof that π is transcendental
• A simple proof that 22/7 exceeds Pi
• Feynman point
• Petr Beckmann, A History of Pi
• Pi (movie)

External links

• Wikisource - Pi to 1,000 Places | 10,000 Places | 100,000 Places | 1,000,000 Places
• Project Gutenberg E-Text containing a million digits of Pi
• Statistics about the first 1.2 trillion digits of Pi
• PiHex Project
• J J O'Connor and E F Robertson: A history of Pi. Mac Tutor project
• Andreas P. Hatzipolakis: PiPhilology. A site with hundreds of examples of π mnemonics
• From the Wolfram Mathematics site lots of formulae for π
• Finding the value of Pi
• PlanetMath: Pi
• The pi-hacks Yahoo! Group
• A banner of approximately 220 million digits of pi
• http://3.141592653589793238462643383279502884197169399375105820974944592.com - Pi to 1,000,000 Places
• A collection of Machin-type formulas for Pi
• A proof that Pi Is Irrational
• Calculating Pi: The open source project for calculating Pi.

• Real numbers
• Irrational numbers
• Transcendental numbers