Portal:Mathematics

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The Mathematics Portal

Carl Friedrich Gauss

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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  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 also proved Noether's first and second theorems, which are fundamental in mathematical physics. Noether 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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  General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever is
  present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.
  
  Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data. (Full article...)
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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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  Portrait by Jakob Emanuel Handmann, 1753

  
  Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] ^ⓘ, Swiss Standard German: [ˈleːɔnhard ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
  
  Euler is credited for popularizing the Greek letter $\pi$ (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation $f(x)$ for the value of a function, the letter $i$ to express the imaginary unit ${\sqrt {-1}}$, the Greek letter $\Sigma$ (capital sigma) to express summations, the Greek letter $\Delta$ (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant $e$, the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the design of microscopes and telescopes, and he studied the bending of beams and the critical load of columns. (Full article...)
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  Elementary algebra studies which values solve equations formed using arithmetical operations.

  
  Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
  
  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 investigates linear equations and combinations of them called systems 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...)
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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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  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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  Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.
  
  Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and is widely used. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
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  Title page of the first edition of Wright's Certaine Errors in Navigation (1599)

  
  Edward Wright (baptised 8 October 1561; died November 1615) was an English mathematician and cartographer noted for his book Certaine Errors in Navigation (1599; 2nd ed., 1610), which for the first time explained the mathematical basis of the Mercator projection by building on the works of Pedro Nunes, and set out a reference table giving the linear scale multiplication factor as a function of latitude, calculated for each minute of arc up to a latitude of 75°. This was in fact a table of values of the integral of the secant function, and was the essential step needed to make practical both the making and the navigational use of Mercator charts.
  
  Wright was born at Garveston in Norfolk and educated at Gonville and Caius College, Cambridge, where he became a fellow from 1587 to 1596. In 1589 the college granted him leave after Elizabeth I requested that he carry out navigational studies with a raiding expedition organised by the Earl of Cumberland to the Azores to capture Spanish galleons. The expedition's route was the subject of the first map to be prepared according to Wright's projection, which was published in Certaine Errors in 1599. The same year, Wright created and published the first world map produced in England and the first to use the Mercator projection since Gerardus Mercator's original 1569 map. (Full article...)
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  Archimedes Thoughtful by Fetti (1620)

  
  Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kim-EE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is considered one of the leading scientists in classical antiquity. Regarded as the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove many geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
  
  Archimedes' other mathematical achievements include deriving an approximation of pi (π), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (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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  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 opposing all forms of technology, rejecting leftism and fascism, and advocating a nature-centered form of anarchism. (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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  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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  The manipulations of the Rubik's Cube form the Rubik's Cube group.

  
  In mathematics, a group is a set with a binary operation that satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
  
  Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group, which is generated by a single element called ⁠$1$⁠ (these properties characterize the integers in a unique way). (Full article...)

Selected image –

animation of patterns of black pixels moving on a white background

Conway's Game of Life

Credit: User:Protious (animation) & Hyperdeath (original still image)

Conway's Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is an example of a zero-player game, meaning that its evolution is completely determined by its initial state, requiring no further input as the game progresses. After an initial pattern of filled-in squares ("live cells") is set up in a two-dimensional grid, the fate of each cell (including empty, or "dead", ones) is determined at each step of the game by considering its interaction with its eight nearest neighbors (the cells that are horizontally, vertically, or diagonally adjacent to it) according to the following rules: (1) any live cell with fewer than two live neighbors dies, as if caused by under-population; (2) any live cell with two or three live neighbors lives on to the next generation; (3) any live cell with more than three live neighbors dies, as if by overcrowding; (4) any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction. By repeatedly applying these simple rules, extremely complex patterns can emerge. In this animation, a breeder (in this instance called a puffer train, colored red in the final frame of the animation) leaves guns (green) in its wake, which in turn "fire out" gliders (blue). Many more complex patterns are possible. Conway developed his rules as a simplified model of a hypothetical machine that could build copies of itself, a more complicated version of which was discovered by John von Neumann in the 1940s. Variations on the Game of Life use different rules for cell birth and death, use more than two states (resulting in evolving multicolored patterns), or are played on a different type of grid (e.g., a hexagonal grid or a three-dimensional one). After making its first public appearance in the October 1970 issue of Scientific American, the Game of Life popularized a whole new field of mathematical research called cellular automata, which has been applied to problems in cryptography and error-correction coding, and has even been suggested as the basis for new discrete models of the universe.

Good articles –

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

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  Fredrik Carl Mülertz Størmer (Norwegian pronunciation: [fʁɛdʁɪk kaːl ˈmʏlɐːt͡s ̍ˈʃtøːmɐː]) (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of π and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, From the Depths of Space to the Heart of the Atom. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the Moon is named after him. (Full article...)
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  Rosa Margaret Morris (16 July 1914 – 15 October 2011) was a Welsh applied mathematician, working in potential theory and aerodynamics. When she was 23, her research and examination results made national and international news. She received several fellowships and awards before graduating with a PhD in mathematics from the University of Cambridge in 1940. In her later career, she taught at the University College of South Wales and Monmouthshire (now Cardiff University), where she co-authored a successful textbook on Mathematical Methods of Physics with Roy Chisholm and became one of the first female Heads of School of Mathematics in the United Kingdom. (Full article...)
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  Berlin, 1959

  
  Andrew Mattei Gleason (1921–2008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in math­e­mat­ics teaching at all levels. Gleason's theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him.
  
  As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career at Harvard University, from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and the Harvard Society of Fellows, and presidency of the American Mathematical Society. He continued to advise the United States government on cryptographic security, and the Commonwealth of Massachusetts on math­e­mat­ics education for children, almost until the end of his life. (Full article...)
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  A polyhedron and its midsphere. The red circles are the boundaries of spherical caps within which the surface of the sphere can be seen from each vertex.

  
  In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals (Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.
  
  When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing. (Full article...)
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  An optimal drawing of K_4,7, with 18 crossings (red dots)

  
  In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.
  
  A drawing method found by Kazimierz Zarankiewicz has been conjectured to give the correct answer for every complete bipartite graph, and the statement that this is true has come to be known as the Zarankiewicz crossing number conjecture. The conjecture remains open, with only some special cases solved. (Full article...)
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  The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

  
  In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
  
  There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances. (Full article...)
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  The Theil–Sen estimator of a set of sample points with outliers (black line) compared to the non-robust ordinary least squares line for the same set (blue). The dashed green line represents the ground truth from which the samples were generated.

  
  In non-parametric statistics, the Theil–Sen estimator is a method for robustly fitting a line to sample points in the plane (simple linear regression) by choosing the median of the slopes of all lines through pairs of points. It has also been called Sen's slope estimator, slope selection, the single median method, the Kendall robust line-fit method, and the Kendall–Theil robust line. It is named after Henri Theil and Pranab K. Sen, who published papers on this method in 1950 and 1968 respectively, and after Maurice Kendall because of its relation to the Kendall tau rank correlation coefficient.
  
  Theil–Sen regression has several advantages over Ordinary least squares regression. It is insensitive to outliers. It can be used for significance tests even when residuals are not normally distributed. It can be significantly more accurate than non-robust simple linear regression (least squares) for skewed and heteroskedastic data, and competes well against least squares even for normally distributed data in terms of statistical power. It has been called "the most popular nonparametric technique for estimating a linear trend". There are fast algorithms for efficiently computing the parameters. (Full article...)
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  The Math Myth: And Other STEM Delusions is a 2016 nonfiction book by Queens College political scientist Andrew Hacker analyzing and critiquing the United States educational system's teaching of mathematics as a linear progression towards more advanced fields. Based on a 2012 New York Times op-ed by Hacker titled "Is Algebra Necessary", Hacker argues that the teaching of advanced algebra, trigonometry, and calculus is not useful to the majority of students. He further claims that the requirement of advanced mathematics courses in secondary education contributes to dropout rates and impedes socioeconomically disadvantaged students from pursuing further education. Hacker critiques the Common Core system and American focus on STEM education in lieu of social sciences, arguing that the educational system should prioritize numeracy over pure mathematics education.
  
  The Math Myth received broadly critical coverage from critics and mathematicians, some citing Hacker's arguments as "disingenuous" and contributing to an elitist attitude towards mathematics, with many citing a lack of exploration on mathematics in early childhood and primary education. Others praised Hacker's work, describing the book as offering a convincing critique of STEM education in the United States and empowering to students struggling in mathematics. (Full article...)
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  Graphical demonstration of the convergence of the sum 1/2 + 1/3 + 1/7 + 1/43 + ... to 1. Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. Squares with side lengths 1/1807 or smaller are too small to see in the figure and are not shown.

  
  In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. Its first few terms are
  :2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
  
  Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms. (Full article...)
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  69 (sixty-nine; LXIX) is the natural number following 68 and preceding 70. An odd number and a composite number, 69 is divisible by 1, 3, 23 and 69.
  
  The number and its pictograph give its name to the sexual position of the same name. The association of the number with this sex position has resulted in it being associated in meme culture with sex. People knowledgeable of the meme may respond "nice" in response to the appearance of the number, whether intentionally an innuendo or not. (Full article...)
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  A set of 20 points in a 10 × 10 grid, with no three points in a line.

  
  The no-three-in-line problem in discrete geometry asks how many points can be placed in the $n\times n$ grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".
  
  At most $2n$ points can be placed, because $2n+1$ points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with $2n$ points for every $n$ up to $46$, it is conjectured that fewer than $2n$ 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 $1.5n$ points, not $2n$. (Full article...)
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  8x8 Rook's graph

  
  In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape. Although rook's graphs have only minor significance in chess lore, they are more important in the abstract mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete bipartite graphs. The square rook's graphs constitute the two-dimensional Hamming graphs.
  
  Rook's graphs are highly symmetric, having symmetries taking every vertex to every other vertex. In rook's graphs defined from square chessboards, more strongly, every two edges are symmetric, and every pair of vertices is symmetric to every other pair at the same distance in moves (making the graph distance-transitive). For rectangular chessboards whose width and height are relatively prime, the rook's graphs are circulant graphs. With one exception, the rook's graphs can be distinguished from all other graphs using only two properties: the numbers of triangles each edge belongs to, and the existence of a unique 4-cycle connecting each nonadjacent pair of vertices. (Full article...)

Did you know (auto-generated) –

• ... that ten-sided gaming dice have kite-shaped faces?
• ... that more than 60 scientific papers authored by mathematician Paul Erdős were published posthumously?
• ... that multiple mathematics competitions have made use of Sophie Germain's identity?
• ... that Kit Nascimento, a spokesperson for the government of Guyana during the aftermath of Jonestown, disagrees with current proposals to open the former Jonestown site as a tourist attraction?
• ... that Fathimath Dheema Ali is the first Olympic qualifier from the Maldives?
• ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
• ... that Green Day's "Wake Me Up When September Ends" became closely associated with the aftermath of Hurricane Katrina?
• ... that two members of the French parliament were killed when a delayed-action German bomb exploded in the town hall at Bapaume on 25 March 1917?

More did you know –

Did you know...

• ...that people are significantly slower to identify the parity of zero than other whole numbers, regardless of age, language spoken, or whether the symbol or word for zero is used?
• ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
• ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
• ...work in artificial intelligence makes use of swarm intelligence, which has foundations in the behavioral examples found in nature of ants, birds, bees, and fish among others?
• ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
• ...that modular arithmetic has application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to mathematics?
• ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?

Selected article –

An example of a map projection: the area-preserving Mollweide projection of the earth. Image credit: NASA

A map projection is any method used in cartography (mapmaking) to represent the dimensional surface of the earth or other bodies. The term "projection" here refers to any function defined on the earth's surface and with values on the plane, and not necessarily a geometric projection.

Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. One can see this mathematically as a consequence of Gauss's Theorema Egregium. Flat maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the earth's surface at once; and they are cheaper to produce and transport. These useful traits of flat maps motivate the development of map projections. (Full article...)

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