Mathematics

March 14 has been labelled pi-day. The mathematical constant pi, the ratio of the circumference of a circle to its diameter (also the ratio of the area of a filled-in disc to the square of its radius) plays a vital role in much of mathematics and physics.

Pi (more often written using the Greek letter ?) is an irrational number; it cannot be written as a fraction, i.e., as a ratio of two integers (whole numbers). The proof of this is attributed to Johann Lambert in 1761. Furthermore, ? is known to be transcendental, first proved in 1882 by Ferdinand von Lindemann; it is impossible to write it as the root of a polynomial with rational coefficients. This fact is connected with an older mathematical problem, that of whether it is possible to square the circle, that is, is it possible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The transcendental nature of ? means that ? is not constructible and, therefore, that it is impossible to square the circle.

Over the years, much effort has been put into computing numerical approximations to the value of ?.

The first 50 (of an infinite number of) digits in the decimal expansion are:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 ...

A commonly used approximation to pi was the rational number 22/7 (this is one of the convergents in a continued fraction expansion of pi). It is possible to prove, using only elementary calculus, that pi exceeds this approximation of 22/7.

Pi occurs in many places in mathematics, including analysis, as continued fraction expansions, in the theory of numbers, in dynamical systems and ergodic theory, in the equations of physics, in probability and statistics, and elsewhere.

It is interesting to note that there are still open problems involving the nature of pi, one of the most important of which is whether pi is a so-called "normal number".

A normal number is one in which every finite block of consecutive digits occurs in its expansion with a probability just the same as if the expansion had been produced randomly. Furthermore, for the number to be called "normal", this must be true no matter which number base we choose in which to write the expansion. At present, it is not known whether pi is normal. In fact, it is not even known whether each of the digits (0,?,9) occurs infinitely often in the decimal expansion of ?. (If pi were a normal number, then each of the digits 0, ..., 9 would occur one tenth of the time. Given that the decimal expansion of pi has an infinite number of digits, this would imply that they should each occur infinitely often).

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