Mathematics is the study ofrepresenting andreasoning about abstractobjects (such asnumbers,points,spaces,sets,structures, andgames). Mathematics is used throughout the world as an essential tool in many fields, includingnatural science,engineering,medicine, and thesocial sciences.Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such asstatistics andgame theory. Mathematicians also engage inpure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.(Full article...)
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric groupS̃3 Theaffine symmetric groups are a family ofmathematical structures that describe the symmetries of thenumber line and the regulartriangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections ofpermutations (rearrangements) of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as agroup with certaingenerators and relations. They are studied incombinatorics andrepresentation theory.
A finitesymmetric group consists of all permutations of afinite set. Each affine symmetric group is an infiniteextension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups.Permutation statistics such asdescents andinversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)
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Title page of the first edition of Wright'sCertaine Errors in Navigation (1599)
Edward Wright (baptised 8 October 1561; died November 1615) was an English mathematician andcartographer noted for his bookCertaine Errors in Navigation (1599; 2nd ed., 1610), which for the first time explained the mathematical basis of theMercator projection by building on the works ofPedro Nunes, and set out a reference table giving the linear scale multiplication factor as a function oflatitude, calculated for eachminute of arc up to a latitude of 75°. This was in fact a table of values of theintegral of the secant function, and was the essential step needed to make practical both the making and the navigational use of Mercator charts.
Although elements of the indispensability argument may have originated with thinkers such asGottlob Frege andKurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such asnaturalism,confirmational holism, and the criterion ofontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 bookPhilosophy of Logic. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on theno miracles argument in thephilosophy of science. A standard form of the argument in contemporary philosophy is credited toMark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in theStanford Encyclopedia of Philosophy: (Full article...)
Lemoine is best known for his proof of the existence of theLemoine point (or the symmedian point) of atriangle. Other mathematical work includes a system he calledGéométrographie and a method which relatedalgebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work. (Full article...)
In the 16th century,Adriaan van Roomen solved the problem using intersectinghyperbolas, but this solution uses methods not limited tostraightedge and compass constructions.François Viète found a straightedge and compass solution by exploitinglimiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified byIsaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such asLORAN. (Full article...)
The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,geometric shapes andpolynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. (Full article...)
Noether was born to aJewish family in theFranconian town ofErlangen; her father was the mathematicianMax Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at theUniversity of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision ofPaul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited byDavid Hilbert andFelix Klein to join the mathematics department at theUniversity of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, and she spent four years lecturing under Hilbert's name. Herhabilitation was approved in 1919, allowing her to obtain the rank ofPrivatdozent. (Full article...)
In 1863,Yale University awarded Gibbs the first Americandoctorate inengineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor ofmathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised byAlbert Einstein as "the greatest mind in American history". In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, theCopley Medal of theRoyal Society of London, "for his contributions to mathematical physics". (Full article...)
Robert Hues (1553 – 24 May 1632) was an Englishmathematician andgeographer. He attendedSt. Mary Hall atOxford, and graduated in 1578. Hues became interested ingeography andmathematics, and studiednavigation at a school set up byWalter Raleigh. During a trip toNewfoundland, he made observations which caused him to doubt the accepted published values forvariations of the compass. Between 1586 and 1588, Hues travelled withThomas Cavendish on acircumnavigation of the globe, performingastronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on anothercircumnavigation of the globe. During the voyage, Hues made astronomical observations in the South Atlantic, and continued his observations of the variation of the compass at variouslatitudes and at theEquator. Cavendish died on the journey in 1592, and Hues returned to England the following year.
In 1594, Hues published his discoveries in theLatin workTractatus de globis et eorum usu (Treatise on Globes and Their Use) which was written to explain the use of the terrestrial and celestial globes that had been made and published byEmery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues's work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...)
Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 whenPhilip Candelas,Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool inenumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to countrational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since beenproven rigorously. (Full article...)
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Elementary algebra studies which values solve equations formed using arithmetical operations.
Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables.Linear algebra is a closely related field that investigateslinear equations and combinations of them calledsystems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. (Full article...)
animation of a grid of boxes numbered 2 through 120, where the prime numbers are progressively circled and listed to the side while the composite numbers are struck out
Thesieve of Eratosthenes is a simplealgorithm for finding allprime numbers up to a specified maximum value. It works by identifying the prime numbers in increasing order while removing from considerationcomposite numbers that are multiples of each prime. This animation shows the process of finding all primes no greater than 120. The algorithm begins by identifying 2 asthe first prime number and then crossing out every multiple of 2 up to 120. The next available number, 3, is the next prime number, so then every multiple of 3 is crossed out. (In this version of the algorithm, 6 is not crossed out again since it was just identified as a multiple of 2. The same optimization is used for all subsequent steps of the process: given a primep, only multiples no less thanp2 are considered for crossing out, since any lower multiples must already have been identified as multiples of smaller primes. Larger multiples that just happen to already be crossed out—like 12 when considering multiples of 3—are crossed out again, because checking for such duplicates would impose an unnecessary speed penalty on any real-world implementation of the algorithm.) The next remaining number, 5, is the next prime, so its multiples get crossed out (starting with 25); and so on. The process continues until no more composite numbers could possibly be left in the list (i.e., when the square of the next prime exceeds the specified maximum). The remaining numbers (here starting with 11) are all prime. Note that this procedure is easily extended to find primes in any givenarithmetic progression. One of severalprime number sieves, this ancient algorithm was attributed to theGreek mathematicianEratosthenes (d. c. 194 BCE) byNicomachus in his first-century (CE) workIntroduction to Arithmetic. Other more modern sieves include thesieve of Sundaram (1934) and thesieve of Atkin (2003). The main benefit of sieve methods is the avoidance of costlyprimality tests (or, conversely,divisibility tests). Their main drawback is their restriction to specific ranges of numbers, which makes this type of method inappropriate for applications requiring very large prime numbers, such aspublic-key cryptography.
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Modus ponens is one of the main rules of inference.
Rules of inference are ways of deriving conclusions frompremises. They are integral parts offormal logic, serving as thelogical structure ofvalid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false.Modus ponens, an influential rule of inference, connects two premises of the form "if then" and "" to the conclusion "", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such asmodus tollens,disjunctive syllogism,constructive dilemma, andexistential generalization.
Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, andrules of replacement, which state that two expressions are equivalent and can be freely swapped. They contrast withformal fallacies—invalid argument forms involving logical errors. (Full article...)
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Pell's equation forn = 2 and six of its integer solutions Pell's equation, also called thePell–Fermat equation, is anyDiophantine equation of the form wheren is a given positivenonsquareinteger, and integer solutions are sought forx andy. InCartesian coordinates, the equation is represented by ahyperbola; solutions occur wherever the curve passes through a point whosex andy coordinates are both integers, such as thetrivial solution withx = 1 andy = 0.Joseph Louis Lagrange proved that, as long asn is not aperfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accuratelyapproximate thesquare root of n byrational numbers of the form x/y.
This kind of equation was first studied extensivelyin India starting withBrahmagupta, who found an integer solution to in hisBrāhmasphuṭasiddhānta circa 628.Bhaskara II in the 12th century andNarayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Bhaskara II is generally credited with developing thechakravala method, building on the work ofJayadeva and Brahmagupta. Solutions to specific examples of Pell's equation, such as thePell numbers arising from the equation withn = 2, had been known for much longer, since the time ofPythagoras inGreece and a similar date in India.William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose fromLeonhard Euler mistakenly attributing Brouncker's solution of the equation toJohn Pell. (Full article...)
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A Pythagorean tiling
APythagorean tiling ortwo squares tessellation is atiling of aEuclidean plane bysquares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of thePythagorean theorem are based on it, explaining its name. It is commonly used as a pattern forfloor tiles. When used for this, it is also known as ahopscotch pattern orpinwheel pattern, but it should not be confused with the mathematicalpinwheel tiling, an unrelated pattern.
Suppose a standard 8×8chessboard (orcheckerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31dominoes of size 2×1 so as to cover all of these squares?
Inmathematics, theBorromean rings are threesimple closed curves in three-dimensional space that aretopologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of aVenn diagram,alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.
The Borromean rings are named after the ItalianHouse of Borromeo, who used the circular form of these rings as an element of theircoat of arms, but designs based on the Borromean rings have been used in many cultures, including by theNorsemen and in Japan. They have been used in Christian symbolism as a sign of theTrinity, and in modern commerce as the logo ofBallantine beer, giving them the alternative nameBallantine rings. Physical instances of the Borromean rings have been made from linkedDNA or other molecules, and they have analogues in theEfimov state andBorromean nuclei, both of which have three components bound to each other although no two of them are bound. (Full article...)
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A set of 20 points in a 10 × 10 grid, with no three points in a line. Theno-three-in-line problem indiscrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of allslopes, not only those aligned with the grid. It was introduced byHenry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".
At most points can be placed, because points in a grid would include a row of three or more points, by thepigeonhole principle. Although the problem can be solved with points for every up to and for, it is conjectured that fewer than points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than points,not. (Full article...)
The algorithm accepts a 32-bit floating-point number as the input and stores a halved value for later use. Then, treating the bits representing the floating-point number as a 32-bit integer, alogical shift right by one bit is performed and the result subtracted from the number0x5F3759DF, which is a floating-point representation of an approximation of. This results in the first approximation of the inverse square root of the input. Treating the bits again as a floating-point number, it runs one iteration ofNewton's method, yielding a more precise approximation. (Full article...)
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Title page ofDe quinque corporibus regularibus De quinque corporibus regularibus (sometimes calledLibellus de quinque corporibus regularibus) is a book on thegeometry ofpolyhedra written in the 1480s or early 1490s by Italian painter and mathematicianPiero della Francesca. It is amanuscript, in the Latin language; its title means[the little book] on the five regular solids. It is one of three books known to have been written by della Francesca.
Along with the Platonic solids,De quinque corporibus regularibus includes descriptions of five of the thirteenArchimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, includingLuca Pacioli's 1509Divina proportione (which incorporated without credit an Italian translation of della Francesca's work). (Full article...)
Inmathematics,equality is a relationship between twoquantities orexpressions, stating that they have the same value, or represent the samemathematical object. Equality betweenA andB is denoted with anequals sign asA = B, and read "A equalsB". A written expression of equality is called anequation oridentity depending on the context. Two objects that arenot equal are said to bedistinct.
Equality is often considered aprimitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothingelse"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality likereflexivity,symmetry, andtransitivity have been understood intuitively since at least theancient Greeks, but were not symbolically stated as general properties of relations until the late 19th century byGiuseppe Peano. Other properties likesubstitution andfunction application weren't formally stated until the development ofsymbolic logic. (Full article...)
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Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have. Ingeometry,Descartes' theorem states that for every four kissing, or mutuallytangentcircles, the radii of the circles satisfy a certainquadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named afterRené Descartes, who stated it in 1643.
Frederick Soddy's 1936 poemThe Kiss Precise summarizes the theorem in terms of thebends (signed inverse radii) of the four circles: (Full article...)
A square. The blue ∟ annotations mark equal right angles at its vertices and the red // annotations mark equal side lengths.
Ingeometry, asquare is aregularquadrilateral. It has four straight sides of equal length and four equalangles. Squares are special cases ofrectangles, which have four equal angles, and ofrhombuses, which have four equal sides. As with all rectangles, a square's angles areright angles (90degrees, orπ/2radians), making adjacent sidesperpendicular. Thearea of a square is the side length multiplied by itself, and so inalgebra, multiplying a number by itself is calledsquaring.
... thatKit Nascimento, a spokesperson for thegovernment of Guyana during the aftermath ofJonestown, disagrees with current proposals to open the former Jonestown site as a tourist attraction?
... thatHong Wang's latest paper claims to have resolved theKakeya conjecture, described as "one of the most sought-after open problems in geometric measure theory", in three dimensions?
... that the identity ofCleo, who provided online answers to complex mathematics problems without showing any work, was revealed over a decade later in 2025?
... that Leonardo da Vinci invented a device to solveAlhazen's problem, instead of finding a mathematical solution?
... thatEugene Parker described the mathematics behind his theory of solar wind as just "four lines of algebra"?
... that anequitable coloring of agraph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than agraph coloring without this constraint?
Turing is often considered to be the father of moderncomputer science. Turing provided an influential formalisation of the concept of thealgorithm and computation with theTuring machine, formulating the now widely accepted "Turing" version of theChurch–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine. With theTuring test, he made a significant and characteristically provocative contribution to the debate regardingartificial intelligence: whether it will ever be possible to say that a machine isconscious and canthink. He later worked at theNational Physical Laboratory, creating one of the first designs for a stored-program computer, although it was never actually built. In 1947 he moved to theUniversity of Manchester to work, largely on software, on theManchester Mark I then emerging as one of the world's earliest true computers.
DuringWorld War II, Turing worked atBletchley Park, Britain'scodebreaking centre, and was for a time head ofHut 8, the section responsible for German Naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of thebombe, an electromechanical machine which could find settings for theEnigma machine.(Full article...)
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