Pi

The number π ( ; spelled out as pi) is a mathematical constant that is the ratio of a circle's circumference to its diameter. It is approximately equal to: 3.141592653589793238462643383279... It appears in many formulae across mathematics and physics. Some of these formulae are used as definitions of π to avoid relying on the definition of the length of a curve. π is an irrational number, meaning that it cannot be expressed exactly as a ratio of integers (although fractions such as ⁠ 22 7 {\displaystyle {\tfrac {22}{7}}} ⁠ are commonly used to approximate it). Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is also a transcendental number, meaning it is not a root of any non-zero polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. Mathematicians have long 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 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 extended the decimal representation of π to hundreds of trillions of digits.