Pi

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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 = 3.141 592 653 589 793 238 462 643 383 ...

Formulas from Euclidean geometry involving π

Circumference of circle of radius r: C = 2 π r
Area of circle of radius r: A = π r2
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 = π radians

Formulas from analysis involving π

1/12 + 1/22 + 1/32 + 1/42 + ... = π2 / 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)
     ∞   -x2
    ∫   e    dx  =  π1/2
   -∞
n! ~ (2 π n)1/2 (n/e)n (Stirling's formula)
eπ i + 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/π2.

Formulas from physics involving π

Δx Δph / (4π) (Heisenberg's uncertainty principle)
Rik - 1/2 gik R + Λ gik = 8 π G/c4 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.

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)4 · (-239 + i) = -114244-114244 i.

The first one million digits of π and 1/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:

http://www.nersc.gov/~dhbailey/dhb-form.gif

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.

Applications

Pi plays an important role in various parts of mathematics, not only geometry.

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.


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