Pi(number)

The ratio of the length of a circle to its diameter; it is an infinite non-periodic decimal number

$$\pi=3.141592653589793\dots.$$

One frequently arrives at the number $\pi$ as the limit of certain arithmetic sequences involving simple laws. An example is Leibniz' series

$$\frac\pi4=1-\frac13+\frac15-\frac17+\frac19-\dotsb,$$

which, however, converges very slowly. There are more rapidly-converging series suitable for calculating $\pi$.

The possibility of a pure analytic definition of $\pi$ is of essential significance for geometry. For example, $\pi$ also participates in certain formulas in non-Euclidean geometry, but not as the ratio of the length of a circle to its diameter (this ratio is not constant in non-Euclidean geometry). The arithmetic nature of $\pi$ was finally elucidated in analysis, with a decisive part played by Euler's formula:

$$e^{\pi i}=-1.$$

At the end of the 18th century, J. Lambert and A. Legendre established that $\pi$ is an irrational number, while in the 19th century, F. Lindemann showed that $\pi$ is a transcendental number.

Comments

A nice account of Lindemann's proof can be found in [a3], Chapt. 6.

The number of known digits of $\pi$ has increased exponentially in recent times. At the moment (1990), the record seems to be half a billion digits (D.V. Chudnovsky and G.V. Chudnovsky). For an account of such computations see [a1]. Up to the 1960's the standard way to calculate $\pi$ was to use Machin's formula $\pi/4=4\arctan(1/5)-\arctan(1/239)$ and the power series of $\arctan(z)$. Nowadays, some powerful formulas of Ramanujan are used. It is still not known how randomly the digits of $\pi$ are distributed; in particular, whether $\pi$ is a normal number.

References

[a1]	J.M. Borwein, P.B. Borwein, "Pi and the AGM" , Interscience (1987)
[a2]	P. Beckmann, "A history of pi" , The Golem Press , Boulder (Co.) (1971)
[a3]	I. Stewart, "Galois theory" , Chapman & Hall (1979)