Pi (π)

Pi (symbol π) is a mathematical constant. It equals the ratio of a circle's circumference (the length of the curve that forms its boundary) to its diameter (the straight line passing through the center that connects two points on the circle). This ratio is the same for every circle, regardless of its size.

The value of π begins 3.14159265... and never ends, nor does it repeat in a fixed pattern. For this reason it is called an irrational number—there are no two whole numbers whose ratio can equal π. It is also a transcendental number, meaning it cannot be a solution to any polynomial equation with rational coefficients.

Mathematical applications

π appears in the geometric formulas for circles and spheres: the circumference of a circle is 2πr, the area of a circle is πr², where r is the radius. It also appears in trigonometry, where angles are measured in radians; a complete circle equals 2π radians. Beyond geometry, π emerges in number theory, calculus, and probability theory (for example, in the formula for the normal distribution).

Early history

Ancient mathematicians in Babylon and Egypt used approximations of π to estimate the area of circles. The Greek mathematician Archimedes used a method of inscribing polygons with many sides inside and outside a circle to establish upper and lower bounds for π. Indian and Chinese mathematicians refined the value over centuries. The Greek letter symbol π became standard in the eighteenth century, particularly after its use by the mathematician Leonhard Euler.

Calculation of the value

Today computers calculate π to trillions of decimal places using fast, iterative algorithms. Such calculations are mainly used to test the performance and accuracy of computer systems, since most practical applications require only a few dozen digits.