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The number pi (symbolized with the greek letter "π"), also called Archimedes' Constant, expresses the ratio of a circle's circumference to its diameter in Euclidean geometry. Alternatively, one can define π as the area of a circle of radius 1, or as the smallest positive number x for which sin(x) = 0. The numerical value of π is approximately

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959 5082953311 ...

Formulas from Euclidean geometry involving π

Circumference of circle of radius r: C = 2 π r
Area of circle of radius r: A = π r
Volume of sphere of radius r: V = (4/3) π r
Surface area of sphere of radius r: A = 4 π r
Angles: 180 degrees = π radians

Formulas from analysis involving π

1/1 + 1/2 + 1/3 + 1/4 + ... = π / 6 (Euler)
1/1 - 1/3 + 1/5 - 1/7 + 1/9 - ... = π / 4 (Leibniz' formula)
2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * ... = π / 2 (Wallis product)
     ∞   -x
    ∫   e    dx  =  π
   -∞
n! ~ (2 π n) (n/e) (Stirling's formula)
e + 1 = 0 ("The most remarkable formula in the world")

Formulas from number theory involving π

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

Formulas from physics involving π

Δx Δph / (4π) (Heisenberg's uncertainty principle)
Rik - 1/2 gik R + Λ gik = 8 π G/c Tik (Einstein's field equation of general relativity)

Irrationality, Transcendence & Squaring the Circle:

The number π is not a rational number. That is, you cannot write it as the ratio of two natural numbers. This was proved in 1761 by Johann Heinrich Lambert. In fact, the number is transcendental, as was proved by 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.

Approximations

So there are no nice closed expressions for π. Therefore we have to use approximations to the number. These approximations were once useful to the applied sciences; the more recent approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers.

For example Ludolph van Ceulen (c1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tomb stone.

Slovene mathematician Jurij Vega 1789 calculated the first 140 decimal places for π and he held the world record for over 50 years at that time.

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:

4 arctan(1/5) - arctan(1/239) = π/4

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) · (-239 + i) = -114244-114244 i.

The first one million digits of π and /π are available from Project Gutenberg. The current record (August 2001) stands at 206,000,000,000 digits, which were computed in September 1999 using the Gauss-Legendre algorithm and Borwein's algorithm.

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

         ∞    1   /    4         2         1         1   
   π  =  ∑  ----  |  -----  -  -----  -  -----  -  ----- |
        k=0  16     8k+1      8k+4      8k+5      8k+6  /

This formula permits one to easily compute the n-th binary or hexadecimal digit of π, without having to compute the first n-1 digits. http://www.nersc.gov/~dhbailey/ is Bailey's website and contains the derivation as well as implementations in various programming languages.

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.

Bailey and Crandal 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.

PiPhilology

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 famous example of a mnemonic for π is from Isaac Asimov:

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

In this example, the number of letters in each word represents successive digits of π: 3.14159265358979. There are piphilologists who have written poems which encode over 100 digits.

Celebrations

March 14 marks Pi Day which is celebrated by many lovers of Pi.


See also:


External links