Portal:Mathematics

The Mathematics Portal

Euler's Identity

Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social 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 as statistics and game theory. Mathematicians also engage in pure 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...)

Featured articles –

  Featured articles are displayed here, which represent some of the best content on English Wikipedia.

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  High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of spacetime (blue lines) due to the Sun's mass.

  
  
  General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.
  
  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...)
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  Noether c. 1900–1910

  
  Amalie Emmy Noether (US: /ˈnʌtər/, UK: /ˈnɜːtə/; German: [ˈnøːtɐ]; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
  
  Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul 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 by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent. (Full article...)
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  Bust of Shen at the Beijing Ancient Observatory

  
  Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Song dynasty (960–1279). Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).
  
  In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole 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 Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination). (Full article...)
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  Rejewski, c. 1932

  
  Marian Adam Rejewski (Polish: [ˈmarjan rɛˈjɛfskʲi] ^ⓘ; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine, aided by limited documents obtained by French military intelligence.
  
  Over the next nearly seven years, Rejewski and fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski, working at the Polish General Staff's Cipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included the cryptologic card catalog and the cryptologic bomb. (Full article...)
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  Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).

  
  
  In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
  It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,
  and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
  
  The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)
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  Logic studies valid forms of inference like modus ponens.

  
  
  Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
  
  Logic studies arguments, which consist of a set of premises that leads to a conclusion. 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 express propositions 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 by logical vocabulary like $\land$ (and) or $\to$ (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...)
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  One of Molyneux's celestial globes, which is displayed in Middle Temple Library – from the frontispiece of the Hakluyt Society's 1889 reprint of A Learned Treatise of Globes, both Cœlestiall and Terrestriall, one of the English editions of Robert Hues' Latin work Tractatus de Globis (1594)

  
  Emery Molyneux (/ˈɛməri ˈmɒlɪnoʊ/ EM-ər-ee MOL-in-oh; died June 1598) was an English Elizabethan maker of globes, mathematical instruments and ordnance. His terrestrial and celestial globes, first published in 1592, were the first to be made in England and the first to be made by an Englishman.
  
  Molyneux was known as a mathematician and maker of mathematical instruments such as compasses and hourglasses. He became acquainted with many prominent men of the day, including the writer Richard Hakluyt and the mathematicians Robert Hues and Edward Wright. He also knew the explorers Thomas Cavendish, Francis Drake, Walter Raleigh and John Davis. Davis probably introduced Molyneux to his own patron, the London merchant William Sanderson, who largely financed the construction of the globes. When completed, the globes were presented to Elizabeth I. Larger globes were acquired by royalty, noblemen and academic institutions, while smaller ones were purchased as practical navigation aids for sailors and students. The globes were the first to be made in such a way that they were unaffected by the humidity at sea, and they came into general use on ships. (Full article...)
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  Cantor, c. 1910

  
  Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
  
  Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). (Full article...)
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  In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
  
  Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative 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 count rational 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 been proven rigorously. (Full article...)
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  Hilary Putnam

  
  The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Van Orman Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.
  
  Although elements of the indispensability argument may have originated with thinkers such as Gottlob Frege and Kurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as naturalism, confirmational holism, and the criterion of ontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 book Philosophy of Logic. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on the no miracles argument in the philosophy of science. A standard form of the argument in contemporary philosophy is credited to Mark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in the Stanford Encyclopedia of Philosophy: (Full article...)
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  Kaczynski after his arrest in 1996

  
  Theodore John Kaczynski (/kəˈzɪnski/ ^ⓘ kə-ZIN-skee; May 22, 1942 – June 10, 2023), also known as the Unabomber (/ˈjuːnəbɒmər/ ^ⓘ YOO-nə-bom-ər), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a reclusive primitive lifestyle.
  
  Kaczynski murdered three people and injured 23 others between 1978 and 1995 in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored Industrial Society and Its Future, a 35,000-word manifesto and social critique which opposes all forms of technology, rejecting leftism, and advocating a nature-centered form of anarchism. (Full article...)
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  Damage from Hurricane Katrina in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriate reserves, and design appropriate reinsurance and capital management strategies.

  
  An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline is actuarial science.
  
  While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in the 17th century studies of probability and annuities. Actuaries of the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks, does not exceed the expected cost of those risks actualized. The steps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (Full article...)
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  Émile Michel Hyacinthe Lemoine (French: [emil ləmwan]; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school.
  
  Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called Géométrographie and a method which related algebraic 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...)
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  Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.

  
  
  In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).
  
  In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting 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 by Isaac 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 as LORAN. (Full article...)
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  The title page of a 1634 version of Hues' Tractatus de globis in the collection of the Biblioteca Nacional de Portugal

  
  Robert Hues (1553 – 24 May 1632) was an English mathematician and geographer. He attended St. Mary Hall at Oxford, and graduated in 1578. Hues became interested in geography and mathematics, and studied navigation at a school set up by Walter Raleigh. During a trip to Newfoundland, he made observations which caused him to doubt the accepted published values for variations of the compass. Between 1586 and 1588, Hues travelled with Thomas Cavendish on a circumnavigation of the globe, performing astronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on another circumnavigation 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 various latitudes and at the Equator. Cavendish died on the journey in 1592, and Hues returned to England the following year.
  
  In 1594, Hues published his discoveries in the Latin work Tractatus 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 by Emery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues' work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...)

Selected image –

truncated icosahedron with black pentagonal faces and white hexagonal faces, beside a similar-looking 1970s soccer ball

Truncated icosahedron

Credit: Dyfsunctional

Here a polyhedron called a truncated icosahedron (left) is compared to the classic Adidas Telstar–style football (or soccer ball). The familiar 32-panel ball design, consisting of 12 black pentagonal and 20 white hexagonal panels, was first introduced by the Danish manufacturer Select Sport, based loosely on the geodesic dome designs of Buckminster Fuller; it was popularized by the selection of the Adidas Telstar as the official match ball of the 1970 FIFA World Cup. The polyhedron is also the shape of the Buckminsterfullerene (or "Buckyball") carbon molecule initially predicted theoretically in the late 1960s and first generated in the laboratory in 1985. Like all polyhedra, the vertices (corner points), edges (lines between these points), and faces (flat surfaces bounded by the lines) of this solid obey the Euler characteristic, V − E + F = 2 (here, 60 − 90 + 32 = 2). The icosahedron from which this solid is obtained by truncating (or "cutting off") each vertex (replacing each by a pentagonal face), has 12 vertices, 30 edges, and 20 faces; it is one of the five regular solids, or Platonic solids—named after Plato, whose school of philosophy in ancient Greece held that the classical elements (earth, water, air, fire, and a fifth element called aether) were associated with these regular solids. The fifth element was known in Latin as the "quintessence", a hypothesized uncorruptible material (in contrast to the other four terrestrial elements) filling the heavens and responsible for celestial phenomena. That such idealized mathematical shapes as polyhedra actually occur in nature (e.g., in crystals and other molecular structures) was discovered by naturalists and physicists in the 19th and 20th centuries, largely independently of the ancient philosophies.

Good articles –

  These are Good articles, which meet a core set of high editorial standards.

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  As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.

  
  In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
  
  The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk). (Full article...)
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  In this graph, the widest path from Maldon to Feering has bandwidth 29, and passes through Clacton, Tiptree, Harwich, and Blaxhall.

  
  In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum capacity path problem. It is possible to adapt most shortest path algorithms to compute widest paths, by modifying them to use the bottleneck distance instead of path length. However, in many cases even faster algorithms are possible.
  
  For instance, in a graph that represents connections between routers in the Internet, where the weight of an edge represents the bandwidth of a connection between two routers, the widest path problem is the problem of finding an end-to-end path between two Internet nodes that has the maximum possible bandwidth. The smallest edge weight on this path is known as the capacity or bandwidth of the path. As well as its applications in network routing, the widest path problem is also an important component of the Schulze method for deciding the winner of a multiway election, and has been applied to digital compositing, metabolic pathway analysis, and the computation of maximum flows. (Full article...)
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  In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
  $$\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .$$
  
  The first $n$ terms of the series sum to approximately $\ln n+\gamma$, where $\ln$ is the natural logarithm and $\gamma \approx 0.577$ is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
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  Srinivasa Ramanujan Aiyangar
  (22 December 1887 – 26 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, 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 then considered unsolvable.
  
  Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results. (Full article...)
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  In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free $\mathbb {Z}$-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.
  
  The elements of a free abelian group with basis $B$ may be described in several equivalent ways. These include formal sums over $B$, which are expressions of the form ${\textstyle \sum a_{i}b_{i}}$ where each $a_{i}$ is a nonzero integer, each $b_{i}$ is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of $B$, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum.
  Another way to represent an element of a free abelian group is as a function from $B$ to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. (Full article...)
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  16 polygonalizations of a set of six points

  
  In computational geometry, a polygonalization of a finite set of points in the Euclidean plane is a simple polygon with the given points as its vertices. A polygonalization may also be called a polygonization, simple polygonalization, Hamiltonian polygon, non-crossing Hamiltonian cycle, or crossing-free straight-edge spanning cycle.
  
  Every point set that does not lie on a single line has at least one polygonalization, which can be found in polynomial time. For points in convex position, there is only one, but for some other point sets there can be exponentially many. Finding an optimal polygonalization under several natural optimization criteria is a hard problem, including as a special case the travelling salesman problem. The complexity of counting all polygonalizations remains unknown. (Full article...)
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  Fleiss' kappa (named after Joseph L. Fleiss) is a statistical measure for assessing the reliability of agreement between a fixed number of raters when assigning categorical ratings to a number of items or classifying items. This contrasts with other kappas such as Cohen's kappa, which only work when assessing the agreement between not more than two raters or the intra-rater reliability (for one appraiser versus themself). The measure calculates the degree of agreement in classification over that which would be expected by chance.
  
  Fleiss' kappa can be used with binary or nominal-scale. It can also be applied to ordinal data (ranked data): the MiniTab online documentation gives an example. However, this document notes: "When you have ordinal ratings, such as defect severity ratings on a scale of 1–5, Kendall's coefficients, which account for ordering, are usually more appropriate statistics to determine association than kappa alone." Keep in mind however, that Kendall rank coefficients are only appropriate for rank data. (Full article...)
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  Graph of log₂ x as a function of a positive real number x

  
  
  In mathematics, the binary logarithm (log₂ n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x,
  :$x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.$
  For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5.
  
  The binary logarithm is the logarithm to the base 2 and is the inverse function of the power of two function. As well as log₂, an alternative notation for the binary logarithm is lb (the notation preferred by ISO 80000-2). (Full article...)
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  Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root.

  
  
  In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is linear with respect to the height of the tree.
  
  Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. (Full article...)
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  Numbers written with Cistercian numerals. From left to right: 1 in units place, 2 in tens place (20), 3 in hundreds place (300), 4 in thousands place (4,000), then compound numbers 5,555, 6,789, 9,394.

  
  The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999.
  
  Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century. (Full article...)
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  Title page of De quinque corporibus regularibus

  
  De quinque corporibus regularibus (sometimes called Libellus de quinque corporibus regularibus) is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, 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 thirteen Archimedean 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, including Luca Pacioli's 1509 Divina proportione (which incorporated without credit an Italian translation of della Francesca's work). (Full article...)
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  The main arithmetic operations are addition, subtraction, multiplication, and division.

  
  
  Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
  
  Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. (Full article...)

Did you know (auto-generated) –

• ... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?
• ... that ten-sided gaming dice have kite-shaped faces?
• ... that Ukrainian baritone Danylo Matviienko, who holds a master's degree in mathematics, appeared as Demetrius in Britten's opera A Midsummer Night's Dream at the Oper Frankfurt?
• ... that the word algebra is derived from an Arabic term for the surgical treatment of bonesetting?
• ... that the British National Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
• ... that more than 60 scientific papers authored by mathematician Paul Erdős were published posthumously?
• ... that Fairleigh Dickinson's upset victory over Purdue was the biggest upset in terms of point spread in NCAA tournament history, with Purdue being a 23+1⁄2-point favorite?
• ... that Latvian-Soviet artist Karlis Johansons exhibited a skeletal tensegrity form of the Schönhardt polyhedron seven years before Erich Schönhardt's 1928 paper on its mathematics?

More did you know –

Did you know...

• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area, such as Gabriel's Horn?
• ... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
• ...that the primality of a number can be determined using only a single division using Wilson's Theorem?
• ...that the line separating the numerator and denominator of a fraction is called a solidus if written as a diagonal line or a vinculum if written as a horizontal line?
• ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare?
• ... that there are 115,200 solutions to the ménage problem of permuting six female-male couples at a twelve-person table so that men and women alternate and are seated away from their partners?

Selected article –

A number is an abstract object that represents a count or measurement. A symbol for a number is called a numeral. The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

Numbers can be classified into sets called number systems. The most familiar numbers are the natural numbers, which to some mean the non-negative integers and to others mean the positive integers. In everyday parlance the non-negative integers are commonly referred to as whole numbers, the positive integers as counting numbers, symbolised by $\mathbb {N}$. Mathematics is used in many classes throughout the course of one's education.

The integers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by $\mathbb {Z}$ (from the German Zahl, meaning "number").

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. 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 $\mathbb {Q}$ (for quotient). (Full article...)

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