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...)
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...)
Image 4
Logic studies valid forms of inference likemodus ponens.
Logic studiesarguments, which consist of a set of premises that leads to aconclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions expresspropositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked bylogical vocabulary like (and) or (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (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...)
High-precision test of general relativity by theCassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) aredelayed by the warping ofspacetime (blue lines) due to theSun's mass.
By the beginning of the 20th century,Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion. (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...)
Image 10
The weighing pans of thisbalance scale contain zero objects, divided into two equal groups. In mathematics,zero is aneven number. In other words, itsparity—the quality of aninteger being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integermultiple of2, specifically0 × 2. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and ify is even theny +x has the same parity asx—indeed,0 +x andx always have the same parity.
Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such aseven −even =even, require 0 to be even. Zero is the additiveidentity element of thegroup of even integers, and it is the starting case from which other evennatural numbers arerecursively defined. Applications of this recursion fromgraph theory tocomputational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by everypower of 2, which is relevant to thebinary numeral system used by computers. In this sense, 0 is the "most even" number of all. (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...)
In hisDream Pool Essays orDream Torrent Essays (夢溪筆談;Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needlecompass, which would be used for navigation (first described in Europe byAlexander Neckam in 1187). Shen discovered the concept oftrue north in terms ofmagnetic declination towards thenorth pole, with experimentation of suspended magnetic needles and "the improvedmeridian determined by Shen's [astronomical] measurement of the distance between thepole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europefor another four hundred years (evidence of German sundials made circa 1450 show markings similar toChinese geomancers' compasses in regard to declination). (Full article...)
These areGood articles, which meet a core set of high editorial standards.
Image 1
In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares. Ingeometry,Keller's conjecture is the conjecture that in anytiling ofn-dimensionalEuclidean space by identicalhypercubes, there are two hypercubes that share an entire(n − 1)-dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration.
This conjecture was introduced byOtt-Heinrich Keller (1930), after whom it is named. A breakthrough byLagarias and Shor (1992) showed that it is false in ten or more dimensions, and after subsequent refinements, it is now known to be true in spaces of dimension at most seven and false in all higher dimensions. The proofs of these results use a reformulation of the problem in terms of theclique number of certain graphs now known asKeller graphs. (Full article...)
In the 19th century, the Belgian physicistJoseph Plateau examinedsoap films, leading him to formulate the concept of aminimal surface. The German biologist and artistErnst Haeckel painted hundreds ofmarine organisms to emphasise theirsymmetry. Scottish biologistD'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematicianAlan Turing predicted mechanisms ofmorphogenesis which give rise topatterns of spots and stripes. The Hungarian biologistAristid Lindenmayer and the French American mathematicianBenoît Mandelbrot showed how the mathematics offractals could create plant growth patterns. (Full article...)
Pythagoras of Samos (Ancient Greek:Πυθαγόρας;c. 570 – c. 495 BC) was an ancientIonianGreek philosopher,polymath, and the eponymous founder ofPythagoreanism. His political and religious teachings were well known inMagna Graecia and influenced the philosophies ofPlato,Aristotle, and, through them,Western philosophy. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled toCroton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal,ascetic lifestyle.
Fibonacci nim is played with a pile of coins. The number of coins in this pile, 21, is a Fibonacci number, so a game starting with this pile and played optimally will be won by the second player. Fibonacci nim is a mathematicalsubtraction game, a variant of the game ofnim. Players alternate removing coins from a pile, on each move taking at most twice as many coins as the previous move, and winning by taking the last coin. TheFibonacci numbers feature heavily in its analysis; in particular, the first player can win if and only if the starting number of coins is not a Fibonacci number. A complete strategy is known for best play in games with a single pile of counters, but not for variants of the game with multiple piles. (Full article...)
An unsolved problem ofPaul Erdős asks how many edges a unit distance graph on vertices can have. The best knownlower bound is slightly above linear in—far from theupper bound, proportional to. The number of colors required tocolor unit distance graphs is also unknown (theHadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For everyalgebraic number α, there is a unit distance graph with two vertices that must be at distance α. According to theBeckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are theisometries. (Full article...)
Image 6
Aderemi Oluyomi KukuNNOMOON (20 March 1941 – 13 February 2022) was a Nigerian mathematician and academic, known for his contributions to the fields ofalgebraic K-theory andnon-commutative geometry. Born inIjebu-Ode, Ogun State, Nigeria, Kuku began his academic journey atMakerere University College and theUniversity of Ibadan, where he earned his B.Sc. in Mathematics, followed by his M.Sc. and Ph.D. under Joshua Leslie andHyman Bass. His doctoral research focused on the Whitehead group ofp-adic integralgroup-rings of finitep-groups. Kuku held positions as a lecturer and professor at various Nigerian universities, including theUniversity of Ife and theUniversity of Ibadan, where he served as Head of the Department of Mathematics and Dean of the Postgraduate School. His research involved developing methods for computing higher K-theory of non-commutative rings and articulating higher algebraic K-theory in the language of Mackey functors. His work on equivariant higher algebraic K-theory and its generalisations impacted the field.
Inmathematics, abinary operation iscommutative if changing the order of theoperands does not change the result. It is a fundamental property of many binary operations, and manymathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g."3 + 4 = 4 + 3" or"2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such asdivision andsubtraction, that do not have it (for example,"3 − 5 ≠ 5 − 3"); such operations arenot commutative, and so are referred to asnoncommutative operations.
The idea that simple operations, such as themultiplication andaddition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when newalgebraic structures started to be studied. (Full article...)
Image 8
AKepler triangle is a right triangle formed by three squares with areas in geometric progression according to thegolden ratio.
AKepler triangle is aspecial right triangle with edge lengths ingeometric progression. The ratio of the progression is where is thegolden ratio, and the progression can be written:, or approximately. Squares on the edges of this triangle have areas in another geometric progression,. Alternative definitions of the same triangle characterize it in terms of the threePythagorean means of two numbers, or via theinradius ofisosceles triangles.
This triangle is named afterJohannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture. (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...)
Image 10
Paterson's worms are a family ofcellular automata devised in 1971 byMike Paterson andJohn Horton Conway to model the behaviour and feeding patterns of certain prehistoric worms. In the model, a worm moves between points on a triangular grid along line segments, representing food. Its turnings are determined by the configuration of eaten and uneaten line segments adjacent to the point at which the worm currently is. Despite being governed by simple rules the behaviour of the worms can be extremely complex, and the ultimate fate of one variant is still unknown.
The worms were studied in the early 1970s by Paterson, Conway and Michael Beeler, described by Beeler in June 1973, and presented in November 1973 inMartin Gardner's "Mathematical Games" column inScientific American. (Full article...)
His first Academy Award was awarded for developing techniques that enabled many technically sophisticated adaptations including the visual effects in 21st century movies in theStar Wars,Harry Potter,Terminator, andPirates of the Caribbeanfranchises. Fedkiw has designed aplatform that has been used to create many of the movie world's most advanced special effects since it was first used on theT-X character inTerminator 3: Rise of the Machines. His second Academy Award was awarded for computer graphics techniques for special effects for large scale destruction. Although he has won an Oscar for his work, he does not design thevisual effects that use his technique. Instead, he has developed a system that other award-winning technicians and engineers have used to create visual effects for some of the world's most expensive and highest-grossing movies. (Full article...)
Image 12
Incomputational complexity theory,Yao's principle (also calledYao's minimax principle orYao's lemma) relates the performance ofrandomized algorithms to deterministic (non-random) algorithms. It states that, for certain classes of algorithms, and certain measures of the performance of the algorithms, the following two quantities are equal:
The optimal performance that can be obtained by a deterministic algorithm on a random input (itsaverage-case complexity), for aprobability distribution on inputs chosen to be as hard as possible and for an algorithm chosen to work as well as possible against that distribution
The optimal performance that can be obtained by a random algorithm on a deterministic input (its expected complexity), for an algorithm chosen to have the best performance on itsworst case inputs, and the worst case input to the algorithm
Yao's principle is often used to prove limitations on the performance of randomized algorithms, by finding a probability distribution on inputs that is difficult for deterministic algorithms, and inferring that randomized algorithms have the same limitation on their worst case performance.
... 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 music ofmath rock bandJyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
... that the wordalgebra is derived from an Arabic term for the surgical treatment ofbonesetting?
... that the identity ofCleo, who provided online answers to complex mathematics problems without showing any work, was revealed over a decade later in 2025?
Numbers can be classified intosets callednumber systems. The most familiar numbers are thenatural numbers, which to some mean thenon-negative integers and to others mean thepositive integers. In everyday parlance the non-negative integers are commonly referred to aswhole numbers, the positive integers ascounting numbers, symbolised by. Mathematics is used in many classes throughout the course of one's education.
Theintegers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by (from the GermanZahl, meaning "number").
Arational number is a number that can be expressed as afraction with an integernumerator and a non-zero natural numberdenominator. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face (forquotient). (Full article...)
.jpg)
.jpg)
.svg)
