Pi

Pi, denoted by the Greek letter π, is the mathematical constant defined as the ratio of a circle's circumference to its diameter in Euclidean geometry. Its value is approximately 3.14159, and it has an infinite, non-repeating decimal expansion. The symbol π was introduced by the Welsh mathematician William Jones in 1706 and popularized by Leonhard Euler.

Pi is irrational, meaning it cannot be expressed as a fraction of two integers. It is also transcendental, proven by Ferdinand von Lindemann in 1882, which shows it is not the root of any nonzero polynomial with rational coefficients.

Common formulas involving pi include the area of a circle A = πr^2 and the circumference C = 2πr. More generally, pi appears in formulas for the volumes and surface areas of solid bodies, such as the sphere (V = 4/3 π r^3, A = 4π r^2). It also appears in many areas of mathematics and physics, such as trigonometry, calculus, and quantum mechanics.

Pi has a long history with estimations dating back to ancient civilizations; modern computation has determined trillions of digits. Pi Day is celebrated on March 14. The ubiquity of pi across disciplines reflects its fundamental role in describing circular and periodic phenomena.