pi

π, a letter of the Greek alphabet used in mathematics to denote a particular irrational number—the ratio of the circumference to the diameter of a circle. The symbol was probably adopted from the Greek word for “circumference,” or “periphery.” Although it came into general use after a paper by L. Euler in 1736, it was first used by the British mathematician W. Jones in 1706. Like all irrational numbers, π is an infinite nonrepeating decimal fraction:

π = 3.141592653589793238462643…

The requirements of practical calculations involving circles and circular solids long ago made it necessary to find approximations of π by rational numbers. In the second millennium B.C., ancient Egyptian computations of the area of a circle made use of the approximation π ≈ 3, or, more precisely, π ≈ (16/9)²= 3.16049.… In the third century B.C., Archimedes found, by comparing the circumference of a circle to regular inscribed and circumscribed polygons, that π is between the values

The second value is still used in calculations that do not require great accuracy. In the second half of the fifth century, the Chinese mathematician Tsu Ch’ung-chih obtained the approximation 3.1415927, which much later (16th century) was also found in Europe. This approximation is exact for the first six decimal places.

The search for a more exact approximation of π continued in later periods. For example, in the first half of the 15th century, al-Kashi calculated π to 17 places. In the early 17th century, the Dutch mathematician Ludolph van Ceulen obtained 32 places. For practical needs, however, it is sufficient to have values for π and the simplest expressions in which π occurs to only a few decimal places; reference works usually give four- to seven-place approximations for π, 1/π, π², and log π.

The number π appears not only in the solution of geometric problems. Since the time of F. Vieta (16th century), the limits of certain arithmetic sequences generated by simple rules have been known to involve π. An example is Leibniz’ series (1673-74)

This series converges extremely slowly. There exist series for calculating π that converge much more rapidly. An example is the formula

where the values of the arc tangents are calculated by means of the series

The formula was used in 1962 for a computer calculation of π to 100,000 places. This type of calculation is of interest in connection with the concept of random and pseudorandom numbers. Statistical processing has shown that this set of 100,000 digits exhibits many features of a random sequence.

The possibility of a purely analytic definition of π is of fundamental importance for geometry. Thus, in non-Euclidean geometry π also occurs in some formulas but is no longer the ratio of the circumference to the diameter of a circle, for the ratio is not a constant in non-Euclidean geometry. The arithmetic nature of π was finally clarified by analytic means, among which a crucial role was played by the remarkable Euler formula e^2πi = 1, where e is the base of the natural system of logarithms and .

At the end of the 18th century, J. H. Lambert and A. M. Legendre proved that π is irrational. In 1882 the German mathematician F. Lindemann showed it to be transcendental—that is, it cannot satisfy any algebraic equation with integral coefficients. The Lindemann theorem conclusively established that the problem of squaring the circle cannot be solved by means of a compass and straightedge.