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| π | |
|---|---|
| Name | Pi |
| Value | 3.141592653589793... |
| First recorded | Rhind Mathematical Papyrus (approx. 1650 BCE) |
| Type | Mathematical constant |
| Properties | Irrational; transcendental number |
π
π is the mathematical constant representing the ratio of a circle's circumference to its diameter, approximately 3.14159. It appears across Euclidean geometry, trigonometry, complex analysis, number theory and mathematical physics, linking classical results such as the Pythagorean theorem and modern developments like the Riemann zeta function and Quantum electrodynamics.
The constant is defined in Euclidean terms as the ratio of circumference to diameter in a circle, a concept used by Archimedes, Euclid, Apollonius of Perga, Eratosthenes and later by Ludolph van Ceulen. The Greek letter π was popularized by William Jones and adopted by Leonhard Euler in works that connected π to the Euler identity, Euler's formula, infinite series and infinite products. Notation appears in texts by Isaac Newton, Gottfried Wilhelm Leibniz, Joseph Fourier and modern treatises such as those by Carl Friedrich Gauss.
Early approximations appear in the Rhind Mathematical Papyrus, Babylonian mathematics, Indian mathematics (e.g., Aryabhata), Chinese mathematics (e.g., Zu Chongzhi), and Archimedes's polygonal bounds. In the Renaissance, figures like François Viète, John Wallis, James Gregory, Isaac Newton, and Srinivasa Ramanujan developed series and products to compute π. The 20th and 21st centuries saw algorithms by John von Neumann, D. F. Ferguson, Yasumasa Kanada, and teams at Google and Fabrice Bellard using fast Fourier transform implementations on supercomputers and distributed computing projects connected to institutions like University of Tokyo and IBM.
π appears in infinite series such as the Leibniz formula for π, Gregory series, Machin-like formulas, and Ramanujan-like series used by Srinivasa Ramanujan and Harvey Friedlander. It is central to identities like Euler's identity, relations in the Gamma function, the Wallis product, the Basel problem solved by Leonhard Euler, and connections to the Riemann zeta function and Dirichlet eta function. π satisfies transcendental properties proved by Ferdinand von Lindemann building on results of Carl Louis Ferdinand von Lindemann's predecessors and uses techniques from Galois theory and complex analysis developed by Augustin-Louis Cauchy, Bernhard Riemann, and Évariste Galois.
In geometry π underpins formulas for area and circumference of circles used by Euclid and Archimedes; it appears in volume and surface-area formulae for spheres and cylinders in works by Archimedes and Johannes Kepler. In physics π appears in wave mechanics of Max Planck and Erwin Schrödinger, in electromagnetic theory by James Clerk Maxwell, in statistical mechanics and Ludwig Boltzmann's constants, in General relativity formulations by Albert Einstein, and in quantum field theory developed by Paul Dirac and Richard Feynman. Engineering applications involve signal processing innovations by Claude Shannon and instrumentation by Nikola Tesla.
Historic approximations include those of Zu Chongzhi, Archimedes, Aryabhata, Liu Hui, and Madhava of Sangamagrama. Series and algorithms include the arctangent identities used by John Machin, the iterative methods by Srinivasa Ramanujan, the fast convergent algorithms by Richard Brent and Emerson Le Veque adapted from Gauss–Legendre algorithm, and BBP-type formulas discovered by Simon Plouffe enabling digit extraction. High-precision computations have been executed by teams at University of Tokyo, Google, Fujitsu, IBM, and projects involving Yasumasa Kanada, Peter Trueb, and Alexander J. Yee using binary splitting, fast Fourier transform multiplication, and arbitrary-precision arithmetic libraries.
Irrationality of π was proved by Johann Heinrich Lambert in the 18th century; transcendence was established by Ferdinand von Lindemann, settling the ancient problem of squaring the circle addressed by Ancient Greek mathematics and later commentators like Omar Khayyam and Thomas Harriot. Proofs draw on results from transcendental number theory developed by Charles Hermite, Carl Ludwig Siegel, Alexander Ostrowski, and later expositions by Alan Baker and Benedict Roth.
π features in literature by Lewis Carroll and Edgar Allan Poe, in recreational mathematics communities exemplified by Martin Gardner, and in outreach events such as Pi Day popularized by educators and institutions including San Francisco Exploratorium. It appears in artwork commissioned by museums like the Museum of Modern Art, in films like works of Stanley Kubrick and Christopher Nolan when mathematical motifs arise, and in competitions sponsored by organizations such as the American Mathematical Society, Mathematical Association of America, and International Mathematical Olympiad delegations.
Category:Mathematical constants