Mixed by WIMP, mastered by Maxime Poubanne.
The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes romanized as "pi" (/paɪ/).
Being an irrational number, π cannot be expressed exactly as a ratio of any two integers (fractions such as 22/7 are commonly used to approximate π, but no fraction can be its exact value). Consequently, its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed, although no proof of this has yet been discovered. Also, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge.
For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π. In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that – when combined with increasing computational power – extended the decimal representation of π to, as of late 2011, over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published, the number is celebrated on Pi Day, and record-setting calculations of the digits of π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 67,000 digits.