457 relations: Abraham Sharp, Absolute value, Adriaan van Roomen, Adrien-Marie Legendre, Aerial (album), Akira Haraguchi, Albert Eagle, Algebra, Algebraic equation, Algebraic number, Algorithm, Almagest, American Mathematical Monthly, Ancient Egypt, Ancient Egyptian mathematics, Apache Hadoop, Apollonius of Perga, Approximations of π, Arc length, Archimedes, Area of a circle, Arithmetic–geometric mean, Aryabhata, Aryabhatiya, Automorphic form, Automorphism, Babylon, Babylonian mathematics, Bailey–Borwein–Plouffe formula, Ball (mathematics), Basel problem, BBC, BBC Four, Bellard's formula, Bill Gosper, Binomial distribution, Bit, Bohr model, Boundary value problem, Bronshtein and Semendyayev, Brownian motion, Buckling, Buffon's needle, Bulletin of the American Mathematical Society, C. Stanley Ogilvy, Cadaeic Cadenza, Calculus, Calculus of variations, Cambridge University Press, Cao Wei, ..., Carl Friedrich Gauss, Carl Sagan, Cartesian coordinate system, Cauchy distribution, Cauchy principal value, Cauchy's integral formula, Central limit theorem, Character (mathematics), Characteristic class, Cheering, Chern–Weil homomorphism, Chinese mathematics, Christiaan Huygens, Christoph Grienberger, Chudnovsky algorithm, Chudnovsky brothers, Circle, Circle group, Circumference, Classical antiquity, Classical mechanics, Clay tablet, Closed-form expression, Coefficient, Compact space, Compass-and-straightedge construction, Complex analysis, Complex number, Complex plane, Computer science, Computer scientist, Cone, Constrained writing, Constructible number, Contact (1997 American film), Contact (novel), Continued fraction, Continuous function, Contour integration, Convergent series, Convex body, Convex set, Convolution, Coprime integers, Cosmological constant, Cosmology, Coulomb's law, Cubit, Curvature, Dante Alighieri, David Gregory (mathematician), Decimal representation, Definite quadratic form, Delta (letter), Diameter, Differential calculus, Differential equation, Differential geometry of surfaces, Dirac delta function, Dirichlet eigenvalue, Dirichlet energy, Distributed computing, Divergence theorem, Divisor, Donald Knuth, Drag (physics), E (mathematical constant), Edmund Landau, Ehrhart's volume conjecture, Eigenvalues and eigenvectors, Einstein field equations, Elastic modulus, Electric field, Electromagnetism, Elias M. 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Abraham Sharp (1653 – 18 July 1742) was an English mathematician and astronomer.
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
Adriaan van Roomen (29 September 1561 – 4 May 1615), also known as Adrianus Romanus, was a Flemish mathematician.
Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.
Aerial is the eighth studio album by the English singer-songwriter and musician Kate Bush, released in 2005, twelve years after her 1993 album The Red Shoes.
(born 1946, Miyagi Prefecture), a retired Japanese engineer, is known for memorizing and reciting digits of pi.
Albert Eagle was an English mathematician who wrote several books (some of them privately published) giving his forcefully expressed and somewhat eccentric views on science and mathematics.
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
The Almagest is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy. One of the most influential scientific texts of all time, its geocentric model was accepted for more than 1200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus.
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River - geographically Lower Egypt and Upper Egypt, in the place that is now occupied by the countries of Egypt and Sudan.
Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. 300 BC, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt.
Apache Hadoop is a collection of open-source software utilities that facilitate using a network of many computers to solve problems involving massive amounts of data and computation.
Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes).
Determining the length of an irregular arc segment is also called rectification of a curve.
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
In geometry, the area enclosed by a circle of radius is.
In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers and is defined as follows: Call and and: \end Then define the two interdependent sequences and as \end where the square root takes the principal value.
Aryabhata (IAST) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.
Aryabhatiya (IAST) or Aryabhatiyam, a Sanskrit astronomical treatise, is the magnum opus and only known surviving work of the 5th century Indian mathematician Aryabhata.
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group.
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
Babylon (KA2.DIĜIR.RAKI Bābili(m); Aramaic: בבל, Babel; بَابِل, Bābil; בָּבֶל, Bavel; ܒܒܠ, Bāwēl) was a key kingdom in ancient Mesopotamia from the 18th to 6th centuries BC.
Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC.
The Bailey–Borwein–Plouffe formula (BBP formula) is a spigot algorithm for computing the nth binary digit of the mathematical constant pi using base-16 representation.
In mathematics, a ball is the space bounded by a sphere.
The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''.
The British Broadcasting Corporation (BBC) is a British public service broadcaster.
BBC Four is a British television channel operated by the British Broadcasting Corporation and available to digital television viewers on Freeview, IPTV, satellite, and cable.
Bellard's formula, as used by PiHex, the now-completed distributed computing project, is used to calculate the nth digit of π in base 16.
Ralph William Gosper Jr. (born April 26, 1943), known as Bill Gosper, is an American mathematician and programmer.
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q.
The bit (a portmanteau of binary digit) is a basic unit of information used in computing and digital communications.
In atomic physics, the Rutherford–Bohr model or Bohr model or Bohr diagram, introduced by Niels Bohr and Ernest Rutherford in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar to the structure of the Solar System, but with attraction provided by electrostatic forces rather than gravity.
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.
Bronshtein and Semendyayev (often just Bronshtein or Bronstein) is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Russian mathematician Ilya Nikolaevich Bronshtein and engineer Konstantin Adolfovic Semendyayev.
Brownian motion or pedesis (from πήδησις "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid.
In science, buckling is a mathematical instability that leads to a failure mode.
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry.
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.
Charles Stanley Ogilvy (1913–2000) was an American mathematician, sailor, and author.
"Cadaeic Cadenza" is a 1996 short story by Mike Keith.
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
Wei (220–266), also known as Cao Wei, was one of the three major states that competed for supremacy over China in the Three Kingdoms period (220–280).
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
Carl Edward Sagan (November 9, 1934 – December 20, 1996) was an American astronomer, cosmologist, astrophysicist, astrobiologist, author, science popularizer, and science communicator in astronomy and other natural sciences.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
Cheering involves the uttering or making of sounds and may be used to encourage, excite to action, indicate approval, or welcome.
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry.
Mathematics in China emerged independently by the 11th century BC.
Christiaan Huygens (Hugenius; 14 April 1629 – 8 July 1695) was a Dutch physicist, mathematician, astronomer and inventor, who is widely regarded as one of the greatest scientists of all time and a major figure in the scientific revolution.
Christoph (Christophorus) Grienberger (also variously spelled Gruemberger, Bamberga, Bamberger, Banbergiera, Gamberger, Ghambergier, Granberger, Panberger) (2 July 1561 – 11 March 1636) was an Austrian Jesuit astronomer, after whom the crater Gruemberger on the Moon is named.
The Chudnovsky algorithm is a fast method for calculating the digits of pi.
David Volfovich Chudnovsky (born 1947 in Kiev) and Gregory Volfovich Chudnovsky (born 1952 in Kiev) are American mathematicians and engineers known for their world-record mathematical calculations and developing the Chudnovsky algorithm used to calculate the digits of pi with extreme precision.
A circle is a simple closed shape.
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th or 6th century AD centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world.
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
In the Ancient Near East, clay tablets (Akkadian ṭuppu(m) 𒁾) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age.
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.
A computer scientist is a person who has acquired the knowledge of computer science, the study of the theoretical foundations of information and computation and their application.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
Constrained writing is a literary technique in which the writer is bound by some condition that forbids certain things or imposes a pattern.
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length || can be constructed with compass and straightedge in a finite number of steps.
Contact is a 1997 American science fiction drama film directed by Robert Zemeckis.
Contact is a 1985 hard science fiction novel by American scientist Carl Sagan.
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior.
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space.
Cosmology (from the Greek κόσμος, kosmos "world" and -λογία, -logia "study of") is the study of the origin, evolution, and eventual fate of the universe.
Coulomb's law, or Coulomb's inverse-square law, is a law of physics for quantifying the amount of force with which stationary electrically charged particles repel or attract each other.
The cubit is an ancient unit of length that had several definitions according to each of the various different cultures that used the unit.
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
Durante degli Alighieri, commonly known as Dante Alighieri or simply Dante (c. 1265 – 1321), was a major Italian poet of the Late Middle Ages.
David Gregory (originally spelt Gregorie) FRS (1661 – 10 October 1708) was a Scottish mathematician and astronomer.
A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,...
In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.
Delta (uppercase Δ, lowercase δ or 𝛿; δέλτα délta) is the fourth letter of the Greek alphabet.
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.
A differential equation is a mathematical equation that relates some function with its derivatives.
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape.
In mathematics, the Dirichlet energy is a measure of how variable a function is.
Distributed computing is a field of computer science that studies distributed systems.
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid.
The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
In the geometry of numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it.
An electric field is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them.
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.
Elias Menachem Stein (born January 13, 1931) is a mathematician.
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object.
ENIAC (Electronic Numerical Integrator and Computer) was amongst the earliest electronic general-purpose computers made.
Information entropy is the average rate at which information is produced by a stochastic source of data.
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
Eugene Salamin is a mathematician who discovered (independently with Richard Brent) the Salamin–Brent algorithm, used in high-precision calculation of pi.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
In mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler.
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.
In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects.
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.
Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that pi (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.
Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted (the Greek letter ''alpha''), is a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles.
In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress.
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.
Flux describes the quantity which passes through a surface or substance.
Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.
The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.
In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.
A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.
François Viète (Franciscus Vieta; 1540 – 23 February 1603), Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations.
In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself.
In mathematics, a functional equation is any equation in which the unknown represents a function.
In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions.
In potential theory, a discipline within applied mathematics, the Furstenberg boundary is a notion of boundary associated with a group.
In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field.
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π. However, the drawback is that it is computer memory-intensive and therefore sometimes Machin-like formulas are used instead.
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: for arbitrary real constants, and.
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.
The word geek is a slang term originally used to describe eccentric or non-mainstream people; in current use, the word typically connotes an expert or enthusiast or a person obsessed with a hobby or intellectual pursuit, with a general pejorative meaning of a "peculiar person, especially one who is perceived to be overly intellectual, unfashionable, boring, or socially awkward".
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e, that is, e raised to the power pi.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
Gerald Budge Folland is an American mathematician and a professor of mathematics at the University of Washington.
Google LLC is an American multinational technology company that specializes in Internet-related services and products, which include online advertising technologies, search engine, cloud computing, software, and hardware.
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
In mathematics, the gradient is a multi-variable generalization of the derivative.
The gravitational constant (also known as the "universal gravitational constant", the "Newtonian constant of gravitation", or the "Cavendish gravitational constant"), denoted by the letter, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.
In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.
Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.
The gravity of Earth, which is denoted by, refers to the acceleration that is imparted to objects due to the distribution of mass within Earth.
The Great Pyramid of Giza (also known as the Pyramid of Khufu or the Pyramid of Cheops) is the oldest and largest of the three pyramids in the Giza pyramid complex bordering what is now El Giza, Egypt.
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.
Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
Guinness World Records, known from its inception in 1955 until 2000 as The Guinness Book of Records and in previous United States editions as The Guinness Book of World Records, is a reference book published annually, listing world records both of human achievements and the extremes of the natural world.
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
In mathematics, a function u(x,\,y) defined on some open domain \Omega\subset\R^2 is said to have as a conjugate a function v(x,\,y) if and only if they are respectively real and imaginary parts of a holomorphic function f(z) of the complex variable z.
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: where k is a positive constant.
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.
In mathematics and computing, hexadecimal (also base, or hex) is a positional numeral system with a radix, or base, of 16.
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).
The earliest known written records of the history of China date from as early as 1250 BC,William G. Boltz, Early Chinese Writing, World Archaeology, Vol.
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1.
The imaginary unit or unit imaginary number is a solution to the quadratic equation.
An in-joke, also known as an inside joke or a private joke, is a joke whose humour is understandable only to members of an ingroup, that is, people who are in a particular social group, occupation, or other community of shared interest.
India (IAST), also called the Republic of India (IAST), is a country in South Asia.
Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century.
The Indiana General Assembly is the state legislature, or legislative branch, of the state of Indiana.
A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897 |citation.
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.
In mathematics, for a sequence of complex numbers a1, a2, a3,...
Infographics (a clipped compound of "information" and "graphics") are graphic visual representations of information, data or knowledge intended to present information quickly and clearly.
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t.
An integer relation between a set of real numbers x1, x2,..., xn is a set of integers a1, a2,..., an, not all 0, such that An integer relation algorithm is an algorithm for finding integer relations.
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain.
In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.
Internet culture, or cyberculture, is the culture that has emerged, or is emerging, from the use of computer networks for communication, entertainment, and business.
Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis.
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus.
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.
In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
Ivan Morton Niven (October 25, 1915 – May 9, 1999) was a Canadian-American mathematician, specializing in number theory.
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for defined on the upper half-plane of complex numbers.
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group H^_R.
James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer.
Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician.
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (غیاث الدین جمشید کاشانی Ghiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician.
Jean-Paul Delahaye (born June 29, 1952 in Saint-Mandé Seine) is a French computer scientist and mathematician.
Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.
John Machin (bapt. c. 1686 – June 9, 1751), a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.
John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.
John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
John William Wrench, Jr. (October 13, 1911 – February 27, 2009) was an American mathematician who worked primarily in numerical analysis.
Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia.
In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting continuous loop in the plane.
The Karatsuba algorithm is a fast multiplication algorithm.
Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".
Catherine "Kate" Bush (born 30 July 1958) is an English singer-songwriter, musician, dancer and record producer.
In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.
In mathematical optimization, the method of Lagrange multipliers (named after Joseph-Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables).
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis.
In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.
Lawrence Berkeley National Laboratory (LBNL), commonly referred to as Berkeley Lab, is a United States national laboratory located in the Berkeley Hills near Berkeley, California that conducts scientific research on behalf of the United States Department of Energy (DOE).
In mathematics, the Leibniz formula for pi, named after Gottfried Leibniz, states that It is also called Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.
In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I.
In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers.
The following is a list of significant formulae involving the mathematical constant pi.
Liu Hui was a Chinese mathematician who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China.
Liu Hui's algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the Cao Wei Kingdom.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
Ludolph van Ceulen (28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim.
In mathematics, Machin-like formulae are a popular technique for computing pi to a large number of digits.
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland.
Mādhava of Sangamagrāma, was a mathematician and astronomer from the town of Sangamagrama (believed to be present-day Aloor, Irinjalakuda in Thrissur District), Kerala, India.
In mathematics, a Leibniz or Madhava series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics and later by Gottfried Wilhelm Leibniz, among others.
The Mandelbrot set is the set of complex numbers c for which the function f_c(z).
Marcus Peter Francis du Sautoy (born 26 August 1965) is a British mathematician, author, and populariser of science and mathematics.
The Massachusetts Institute of Technology (MIT) is a private research university located in Cambridge, Massachusetts, United States.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
A mathematical constant is a special number that is "significantly interesting in some way".
As the term is understood by mathematicians, folk mathematics or mathematical folklore is the body of theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word of mouth but have not appeared in print, either in books or in scholarly journals.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
In the classical physics observed in everyday life, matter is any substance that has mass and takes up space by having volume.
In mathematics, mean has several different definitions depending on the context.
A meander is one of a series of regular sinuous curves, bends, loops, turns, or windings in the channel of a river, stream, or other watercourse.
Mechanica (Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler, which describes analytically the mathematics governing movement.
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.
The method of loci (loci being Latin for "places") is a method of memory enhancement which uses visualizations with the use of spatial memory, familiar information about one's environment, to quickly and efficiently recall information.
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.
Michael Brant Shermer (born September 8, 1954) is an American science writer, historian of science, founder of The Skeptics Society, and editor-in-chief of its magazine Skeptic, which is largely devoted to investigating pseudoscientific and supernatural claims.
Michael David Spivak (born May 25, 1940)Biographical sketch in, Vol.
The name Milü ("close ratio"), also known as Zulü (Zu's ratio), is given to an approximation to pi (pi) found by Chinese mathematician and astronomer, Zǔ Chōngzhī (祖沖之).
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces.
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem.
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object.
Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.
The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
Newton's law of universal gravitation states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity.
Kelallur Nilakantha Somayaji (also referred to as Kelallur Comatiri; 14 June 1444 – 1544) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics in India.
In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation.
In mathematics, a normal number is a real number whose infinite sequence of digits in every positive integer base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc.
Nortel Networks Corporation, formerly known as Northern Telecom Limited, Northern Electric and sometimes known simply as Nortel, was a multinational telecommunications and data networking equipment manufacturer headquartered in Mississauga, Ontario, Canada.
In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
The observable universe is a spherical region of the Universe comprising all matter that can be observed from Earth at the present time, because electromagnetic radiation from these objects has had time to reach Earth since the beginning of the cosmological expansion.
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7.
The Old Kingdom, in ancient Egyptian history, is the period in the third millennium (c. 2686–2181 BC) also known as the 'Age of the Pyramids' or 'Age of the Pyramid Builders' as it includes the great 4th Dynasty when King Sneferu perfected the art of pyramid building and the pyramids of Giza were constructed under the kings Khufu, Khafre and Menkaure.
The Open University (OU) is a public distance learning and research university, and one of the biggest universities in the UK for undergraduate education.
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil.
An oxbow lake is a U-shaped lake that forms when a wide meander from the main stem of a river is cut off, creating a free-standing body of water.
The Palais de la Découverte ("Discovery Palace") is a science museum located in the Grand Palais, in the 8th arrondissement on Avenue Franklin D. Roosevelt, Paris, France.
Perfums Givenchy is a French brand of perfumes and cosmetics, known for fragrances L'Interdit, Amarige, Organza, Pi, and Givenchy III.
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.
A pendulum is a weight suspended from a pivot so that it can swing freely.
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x.
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.
Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953) is a Canadian mathematician and a professor at Simon Fraser University.
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time.
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
The number is a mathematical constant.
Pi (stylized as) is a 1998 American surrealist psychological thriller film written and directed by Darren Aronofsky in his directorial debut.
Pi (uppercase Π, lowercase π; πι) is the sixteenth letter of the Greek alphabet, representing the sound.
Pi Day is an annual celebration of the mathematical constant pi (pi).
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of pi.
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant pi.
The Planck constant (denoted, also called Planck's constant) is a physical constant that is the quantum of action, central in quantum mechanics.
In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.
In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.
Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium.
In physics, potential energy is the energy possessed by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
Probability is the measure of the likelihood that an event will occur.
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
In the 18th century, Johann Heinrich Lambert proved that the number pi (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer.
Claudius Ptolemy (Κλαύδιος Πτολεμαῖος, Klaúdios Ptolemaîos; Claudius Ptolemaeus) was a Greco-Roman mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology.
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers.
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.
In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.
In differential geometry, the radius of curvature,, is the reciprocal of the curvature.
In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system.
In mathematics, the Radon–Nikodym theorem is a result in measure theory.
In mathematics, a Ramanujan–Sato seriesHeng Huat Chan, Song Heng Chan, and Zhiguo Liu, "Domb's numbers and Ramanujan–Sato type series for 1/Pi" (2004)Gert Almkvist and Jesus Guillera, Ramanujan–Sato Like Series (2012) generalizes Ramanujan’s pi formulas such as, to the form by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels.
Random number generation is the generation of a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance, usually through a hardware random-number generator (RNG).
The concept of a random sequence is essential in probability theory and statistics.
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.
A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity".
In logic, reductio ad absurdum ("reduction to absurdity"; also argumentum ad absurdum, "argument to absurdity") is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible.
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero.
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well.
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of Egyptian mathematics.
Rho (uppercase Ρ, lowercase ρ or ϱ; ῥῶ) is the 17th letter of the Greek alphabet.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a small wedge of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.
Richard Phillips Feynman (May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model.
Richard Peirce Brent (born 20 April 1946, Melbourne) is an Australian mathematician and computer scientist.
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
A rotation is a circular movement of an object around a center (or point) of rotation.
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value due to rounding.
In typography and lettering, a sans-serif, sans serif, gothic, or simply sans letterform is one that does not have extending features called "serifs" at the end of strokes.
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.
Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.
The 2nd moment of area, also known as moment of inertia of plane area, area moment of inertia, or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
Sexagesimal (base 60) is a numeral system with sixty as its base.
The Shatapatha Brahmana (IAST:, "Brāhmaṇa of one hundred parts") is a prose text describing Vedic rituals, history and mythology associated with the Śukla Yajurveda.
Simon Plouffe (born June 11, 1956, Saint-Jovite, Quebec) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the nth binary digit of π, in 1995.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
In mathematics, the sine is a trigonometric function of an angle.
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations.
In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces X and Y, are the square roots of the eigenvalues of the non-negative self-adjoint operator (where T* denotes the adjoint of T).
Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve.
A sequence of six 9s occurs in the decimal representation of π, starting at the 762nd decimal place.
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces.
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.
The speed of light in vacuum, commonly denoted, is a universal physical constant important in many areas of physics.
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
A spigot algorithm is an algorithm for computing the value of a mathematical constant such as pi or ''e'' which generates output digits in some base (usually 2 or a power of 2) from left to right, with limited intermediate storage.
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
Squaring the circle is a problem proposed by ancient geometers.
Srinivasa Ramanujan (22 December 188726 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable.
Stanley Wagon is a Canadian-American mathematician, a professor of mathematics at Macalester College in Minnesota.
In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables.
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness.
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.
--> In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables.
In 1851, George Gabriel Stokes derived an expression, now known as Stokes's law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid.
In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.
The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics.
A vibration in a string is a wave.
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset.
In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.
A supercomputer is a computer with a high level of performance compared to a general-purpose computer.
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.
Tantrasamgraha, or Tantrasangraha, (literally, A Compilation of the System) is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics.
A tap (also spigot or faucet: see usage variations) is a valve controlling the release of a liquid or gas.
Tau (uppercase Τ, lowercase τ; ταυ) is the 19th letter of the Greek alphabet.
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
TeX (see below), stylized within the system as TeX, is a typesetting system (or "formatting system") designed and mostly written by Donald Knuth and released in 1978.
The Independent is a British online newspaper.
The Japan Times is Japan's largest and oldest English-language daily newspaper.
The Story of Maths is a four-part British television series outlining aspects of the history of mathematics.
In mathematics, theta functions are special functions of several complex variables.
Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.
A turn is a unit of plane angle measurement equal to 2pi radians, 360 degrees or 400 gradians.
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known.
In mathematics, a unit circle is a circle with a radius of one.
The University of Oregon (also referred to as UO, U of O or Oregon) is a public flagship research university in Eugene, Oregon.
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.
Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
The Verlag Harri Deutsch (VHD, HD) with headquarters in Frankfurt am Main, Germany, as well as in Zürich and Thun, Switzerland, was a German publishing house founded in 1961 and closed in 2013.
In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant pi: It is named after François Viète (1540–1603), who published it in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII.
The viscosity of a fluid is the measure of its resistance to gradual deformation by shear stress or tensile stress.
In mathematics, Wallis' product for pi, written down in 1655 by John Wallis, states that \prod_^ \left(\frac \cdot \frac\right).
In the physical sciences, the wavenumber (also wave number or repetency) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance.
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes.
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ(G) of a simply connected simple algebraic group defined over a number field is 1.
The Western world refers to various nations depending on the context, most often including at least part of Europe and the Americas.
Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, known in the English-speaking world as Snell.
William Jones, FRS (1675 – 3 July 1749) was a Welsh mathematician, most noted for his use of the symbol (the Greek letter pi) to represent the ratio of the circumference of a circle to its diameter.
William Oughtred (5 March 1574 – 30 June 1660) was an English mathematician and Anglican clergyman.
William Shanks (25 January 1812 – June 1882) was a British amateur mathematician.
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point.
In mathematics, historically Wirtinger's inequality for real functions was an inequality used in Fourier analysis.
Wolfram Alpha (also styled WolframAlpha, and Wolfram|Alpha) is a computational knowledge engine or answer engine developed by Wolfram Alpha LLC, a subsidiary of Wolfram Research.
Yahoo! is a web services provider headquartered in Sunnyvale, California and wholly owned by Verizon Communications through Oath Inc..
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of pi.
Yuktibhāṣā (യുക്തിഭാഷ; "Rationale in the Malayalam/Sanskrit language") also known as Gaṇitanyāyasaṅgraha ("Compendium of astronomical rationale"), is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530.
Johann Martin Zacharias Dase (June 23, 1824, Hamburg – September 11, 1861, Hamburg) was a German mental calculator.
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
In mathematics, a zero of a function is a value such that.
Zu Chongzhi (429–500 AD), courtesy name Wenyuan, was a Chinese mathematician, astronomer, writer and politician during the Liu Song and Southern Qi dynasties.
The 1st millennium BC encompasses the Iron Age and sees the rise of many successive empires, and spanned from 1000 BC to 1 BC.
3 (three) is a number, numeral, and glyph.
2 Pi Day, 2pi, 2π, 3,14, 3,14..., 3.14, 3.14 (number), 3.14..., 3.141, 3.1415, 3.1415..., 3.14159, 3.14159..., 3.141592, 3.1415926, 3.14159265, 3.141592653, 3.1415926535, 3.14159265358, 3.141592653589, 3.1415926535897, 3.14159265358979, 3.141592653589793, 3.1415926535897932, 3.14159265358979323, 3.141592653589793238, 3.1415926535897932384, 3.14159265358979323846, 3.141592653589793238462, 3.1415926535897932384626, 3.14159265358979323846264, 3.141592653589793238462643383279, 3.1415926535897932384626433832795, 3.1415926535897932385, 3.14159265358979324, 3.1415926535898, 3.14159265359, 3.1415926536, 3.141592654, 3.1415927, 3.141593, 3.1416, 3.142, A000796, A001203, A060294, A063673, A063674, Archimedean constant, Archimedes constant, Archimedes' Constant, Archimedes' constant, Archimedes's constant, Circumference-to-diameter ratio, History of Pi, History of human knowledge about pi, History of pi, History of π, Irrationality of π, List of digits in pi, Ludolph constant, Ludolph transcendental number, Ludolph's constant, Ludolph's number, Ludolphian, Ludolphian number, Ludolphine, Ludolphine number, Number of pi, Numeric pi, Perimeter/diameter, Pi (definition), Pi (mathematical constant), Pi (mathematics), Pi (number), Pi Continued Fraction, Pi Digits, Pi Number, Pi Value, Pi continued fraction, Pi digits, Pi number, Tau against pi debate, Tau versus pi debate, The value of pi, Twice pi, Π (number), ℼ.