There is an entire field of humorous yet serious study that involves the use of mnemonic devices 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. |
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. |
Formulas from Euclidean geometry involving π
Formulas from analysis involving π
∞ -x2 ∫ e dx = π1/2 -∞
Formulas from number theory involving π
Formulas from physics involving π
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:
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:
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.