Pi

The number π ( ; spelled out as pi) is a mathematical constant, approximately equal to 3 , that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve. The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ⁠22/7⁠ are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of π appear to be evenly distributed, but no proof of this conjecture has been found. Mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to hundreds of trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors. Because it relates to a circle, π is found in formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is found as well in formulae from cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas such as number theory, statistics, and in modern mathematical analysis: π is ubiquitous.