Portal:Mathematics

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.

---

• 
  
  

  

  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...)
• 
  
  

  

  A stamp of Zhang Heng issued by China Post in 1955

  
  Zhang Heng (Chinese: 張衡; AD 78–139), formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer, cartographer, ethnographer, artist, poet, philosopher, politician, and literary scholar.
  
  Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in present-day Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...)
• 

  

  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...)
• 

  

  The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

  
  
  In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
  $$\sum _{n=1}^{m}n(-1)^{n-1}.$$
  
  The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
  $$1-2+3-4+\cdots ={\frac {1}{4}}.$$ (Full article...)
• 

  

  Plots of logarithm functions, with three commonly used bases. The special points log_b b = 1 are indicated by dotted lines, and all curves intersect in log_b 1 = 0.

  
  
  In mathematics, the logarithm to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 10³, the logarithm base $10$ of 1000 is 3, or log₁₀ (1000) = 3. The logarithm of x to base b is denoted as log_b (x), or without parentheses, log_b x. When the base is clear from the context or is irrelevant it is sometimes written log x.
  
  The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science. (Full article...)
• 

  

  The manipulations of the Rubik's Cube form the Rubik's Cube group.

  
  In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and 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...)
• 
  
  

  

  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...)
• 
  The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\tfrac {22}{7}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
  
  For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. (Full article...)
• 
  
  

  

  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...)
• 
  
  

  

  Portrait by August Köhler, c. 1910, after 1627 original

  
  Johannes Kepler (/ˈkɛplər/; German: [joˈhanəs ˈkɛplɐ, -nɛs -] ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae, influencing among others Isaac Newton, providing one of the foundations for his theory of universal gravitation. The variety and impact of his work made Kepler one of the founders and fathers of modern astronomy, the scientific method, natural and modern science. He has been described as the "father of science fiction" for his novel Somnium.
  
  Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein.
  Additionally, he did fundamental work in the field of optics, being named the father of modern optics, in particular for his Astronomiae pars optica. He also invented an improved version of the refracting telescope, the Keplerian telescope, which became the foundation of the modern refracting telescope, while also improving on the telescope design by Galileo Galilei, who mentioned Kepler's discoveries in his work. (Full article...)
• 

  

  
  Josiah Willard Gibbs (/ɡɪbz/; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.
  
  In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical 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 by Albert 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, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics". (Full article...)
• 
  
  

  

  Archimedes Thoughtful
  by Domenico 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 regarded as 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 a range of 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, 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. 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...)
• 
  
  

  

  Portrait by Jakob Emanuel Handmann, 1753

  
  Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhar​d ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
  
  Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. (Full article...)
• 
  
  

  

  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...)
• 

  

  
  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...)

Selected image –

graph showing two sets of 4 points, each set perfectly fit by a trend line with positive slope; the set of points on the left is higher and the set on the right lower, so the entire collection of points is best fit by a trend line with negative slope

Simpson's paradox

Credit: Schutz

Simpson's paradox (also known as the Yule–Simpson effect) states that an observed association between two variables can reverse when considered at separate levels of a third variable (or, conversely, that the association can reverse when separate groups are combined). Shown here is an illustration of the paradox for quantitative data. In the graph the overall association between X and Y is negative (as X increases, Y tends to decrease when all of the data is considered, as indicated by the negative slope of the dashed line); but when the blue and red points are considered separately (two levels of a third variable, color), the association between X and Y appears to be positive in each subgroup (positive slopes on the blue and red lines — note that the effect in real-world data is rarely this extreme). Named after British statistician Edward H. Simpson, who first described the paradox in 1951 (in the context of qualitative data), similar effects had been mentioned by Karl Pearson (and coauthors) in 1899, and by Udny Yule in 1903. One famous real-life instance of Simpson's paradox occurred in the UC Berkeley gender-bias case of the 1970s, in which the university was sued for gender discrimination because it had a higher admission rate for male applicants to its graduate schools than for female applicants (and the effect was statistically significant). The effect was reversed, however, when the data was split by department: most departments showed a small but significant bias in favor of women. The explanation was that women tended to apply to competitive departments with low rates of admission even among qualified applicants, whereas men tended to apply to less-competitive departments with high rates of admission among qualified applicants. (Note that splitting by department was a more appropriate way of looking at the data since it is individual departments, not the university as a whole, that admit graduate students.)

Good articles –

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

---

• 

  

  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...)
• 

  

  Dyadic rationals in the interval from 0 to 1

  
  In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number.
  
  The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted $\mathbb {Z} [{\tfrac {1}{2}}]$. (Full article...)
• 

  

  The Rado graph, as numbered by Ackermann (1937) and Rado (1964).

  
  In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
  
  Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose. (Full article...)
• 
  Basic Math is an educational video game for the Atari Video Computer System (Atari VCS). The game was developed at Atari, Inc. by Gary Palmer. The game involves a series of ten arithmetic problems involving addition, subtraction, multiplication, or division. The player can edit different gameplay modes to alter how the numbers in the problem are chosen, or if their questions are timed. The game was released in 1977 as one of the earliest releases for the console.
  
  The game is the only known game developed for the VCS by Palmer, who initially worked at Atari creating debugging stations for game developers, and later assisted with work on the Atari 400/800 line of computers. It was the first educational video game for the VCS, with others companies also releasing arithmetic-themed cartridges in the same year for the Fairchild Channel F and RCA Studio II. The game was also released under the title Fun with Numbers and Math. Both contemporary and retrospective reviews were generally unenthused by the game with common criticism being that it had poor quality graphics and was not appealing in terms of gameplay or control. (Full article...)
• 

  

  A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points.

  
  In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the curve.
  
  Bounded convex curves have a well-defined length, which can be obtained by approximating them with polygons, or from the average length of their projections onto a line. The maximum number of grid points that can belong to a single curve is controlled by its length. The points at which a convex curve has a unique supporting line are dense within the curve, and the distance of these lines from the origin defines a continuous support function. A smooth simple closed curve is convex if and only if its curvature has a consistent sign, which happens if and only if its total curvature equals its total absolute curvature. (Full article...)
• 

  

  A spiral staircase in the Cathedral of St. John the Divine. Several helical curves in the staircase project to hyperbolic spirals in its photograph.

  
  
  A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. As this curve widens, it approaches an asymptotic line. It can be found in the view up a spiral staircase and the starting arrangement of certain footraces, and is used to model spiral galaxies and architectural volutes.
  
  As a plane curve, a hyperbolic spiral can be described in polar coordinates $(r,\varphi )$ by the equation
  $$r={\frac {a}{\varphi }},$$
  for an arbitrary choice of the scale factor $a.$ (Full article...)
• 
  
  

  

  Francis Amasa Walker

  
  Francis Amasa Walker (July 2, 1840 – January 5, 1897) was an American economist, statistician, journalist, educator, academic administrator, and an officer in the Union Army.
  
  Walker was born into a prominent Boston family, the son of the economist and politician Amasa Walker, and he graduated from Amherst College at the age of 20. He received a commission to join the 15th Massachusetts Infantry and quickly rose through the ranks as an assistant adjutant general. Walker fought in the Peninsula, Bristoe, Overland, and Richmond-Petersburg Campaigns before being captured by Confederate forces and held at the infamous Libby Prison. In July 1866, he was awarded the honorary grade of brevet brigadier general United States Volunteers, to rank from March 13, 1865, when he was 24 years old. (Full article...)
• 
  
  

  

  Equilateral form of the Cairo tiling

  
  In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes. John Horton Conway called it a 4-fold pentille.
  
  Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges. (Full article...)
• 

  

  The logo of the International Mathematical Olympiad

  
  
  The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. It is "the most prestigious" mathematical competition in the world. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries participate. Each country sends a team of up to six students, plus one team leader, one deputy leader, and observers.
  
  The content ranges from extremely difficult algebra and pre-calculus problems to problems in branches of mathematics not conventionally covered in secondary or high school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required. Calculus, though allowed in solutions, is never required, as there is a principle that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge. Supporters of this principle claim that this allows more universality and creates an incentive to find elegant, deceptively simple-looking problems which nevertheless require a certain level of ingenuity, often times a great deal of ingenuity to net all points for a given IMO problem. (Full article...)
• 

  

  A Pythagorean tiling

  
  A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern,
  but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
  
  This tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have also been studied. (Full article...)
• 

  

  Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + ⁠96/2⁠ − 1 = 48

  
  In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. (Full article...)
• 

  

  
  In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid, an example of a Johnson solid.
  
  Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. (Full article...)

Did you know (auto-generated) –

• ... that multiple mathematics competitions have made use of Sophie Germain's identity?
• ... 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 in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
• ... that a folded paper lantern shows that certain mathematical definitions of surface area are incorrect?
• ... that although the problem of squaring the circle with compass and straightedge goes back to Greek mathematics, it was not proven impossible until 1882?
• ... that subgroup distortion theory, introduced by Misha Gromov in 1993, can help encode text?
• ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
• ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?

More did you know –

Did you know...

• ... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
• ... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
• ... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
• ...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?
• ...that in Floyd's algorithm for cycle detection, the tortoise and hare move at very different speeds, but always finish at the same spot?
• ...that in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
• ...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?

Selected article –

In this shear transformation of the Mona Lisa, the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an eigenvector of this particular transformation and the blue vector is not. Image credit: User:Voyajer

In mathematics, an eigenvector of a transformation is a vector, different from the zero vector, which that transformation simply multiplies by a constant factor, called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue, together with the zero vector.

In the specific case of linear algebra, the eigenvalue problem is this: given an n by n matrix A, what nonzero vectors x in $R^{n}$ exist, such that Ax is a scalar multiple of x?

The scalar multiple is denoted by the Greek letter λ and is called an eigenvalue of the matrix A, while x is called the eigenvector of A corresponding to λ. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.

It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency. (Full article...)

View all selected articles

Subcategories

Algebra | Arithmetic | Analysis | Complex analysis | Applied mathematics | Calculus | Category theory | Chaos theory | Combinatorics | Dynamical systems | Fractals | Game theory | Geometry | Algebraic geometry | Graph theory | Group theory | Linear algebra | Mathematical logic | Model theory | Multi-dimensional geometry | Number theory | Numerical analysis | Optimization | Order theory | Probability and statistics | Set theory | Statistics | Topology | Algebraic topology | Trigonometry | Linear programming

---

Mathematics | History of mathematics | Mathematicians | Awards | Education | Literature | Notation | Organizations | Theorems | Proofs | Unsolved problems

Full category tree. Select [►] to view subcategories.

Mathematics

Mathematics by country

Fields of mathematics

Mathematics-related lists

Mathematicians

Works about mathematics

Mathematics and art

Mathematical classification systems

Mathematical concepts

Mathematical constants

Mathematics and culture

Eponyms in mathematics

Mathematical examples

FL-Class Polyhedra articles

History of mathematics

Mathematical notation

Mathematical science occupations

Outlines of mathematics and logic

Philosophy of mathematics

Mathematical projects

Mathematical proofs

Pseudomathematics

Set index articles on mathematics

Mathematical terminology

Mathematical theorems

Mathematical tools

Mathematics stubs

Topics in mathematics

General	Foundations	Number theory	Discrete mathematics
Mathematicians History of mathematics Philosophy of mathematics Mathematical notation Mathematical beauty Mathematics education Glossary of mathematical symbols	Foundations of mathematics Mathematical logic Proof theory Gödel's incompleteness theorems Model theory Recursion theory Set theory Naive set theory Axiomatic set theory Category theory Topos theory	Number theory Algebraic number theory Analytic number theory Arithmetic Fundamental theorem of arithmetic Numbers Natural numbers Prime numbers Rational numbers Algebraic numbers	Discrete mathematics Combinatorics Combinatorial geometry Coding theory Combinatorial design Enumerative combinatorics Combinatorial optimization Graph theory Order theory Lattice theory Digital Signal Processing Theoretical computer science Computability theory Complexity theory
Algebra	Analysis	Geometry and topology	Applied mathematics
Algebra Elementary algebra Abstract algebra Group theory Ring theory Field theory Commutative algebra Geometric algebra Linear algebra Matrix theory Multilinear algebra Universal algebra Fundamental theorem of algebra	Analysis Calculus Fundamental theorem of calculus Vector calculus Geometric calculus Measure theory Real analysis Complex analysis Differential equations Ordinary differential equations Partial differential equations Integral equations Approximation theory Special functions Potential theory Harmonic analysis Fourier analysis Functional analysis Operator theory	Geometry Euclidean geometry Trigonometry Analytic geometry Non-Euclidean geometry Affine geometry Projective geometry Convex geometry Discrete geometry Algebraic geometry Differential geometry K theory Gauge theory Riemannian geometry Lie groups Topology General topology Algebraic topology Geometric topology Differential topology	Applied mathematics Mathematical modeling Mathematical physics Dynamical systems Control theory Calculus of variations Optimization Mathematical economics Game theory Mathematical finance Statistics Probability theory Stochastic processes Numerical analysis Cryptography Information theory

Index of mathematics articles

ARTICLE INDEX:	0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
MATHEMATICIANS:	A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Related portals

WikiProjects

The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.

Project pages

• Participants
• Current activity
• Manual of style
• Conventions

Essays

• Advice on using Wikipedia for mathematics self-study
• Proofs

Subprojects

• PlanetMath Exchange (Talk)
• Graphics (Talk)
• Article assessment (Talk)

Related projects

Computer science | Cryptography | Game theory | Numbers | Physics | Science | Statistics

Things you can do

• Create a new article: List of requested articles
• Add a new image: Image requests
• Expand an article: Mathematics stubs
• Lend your expertise: Expert attention needed

In other Wikimedia projects

The following Wikimedia Foundation sister projects provide more on this subject:

• 

  Commons
  Free media repository

• 

  Wikibooks
  Free textbooks and manuals

• 

  Wikidata
  Free knowledge base

• 

  Wikinews
  Free-content news

• 

  Wikiquote
  Collection of quotations

• 

  Wikisource
  Free-content library

• 

  Wikiversity
  Free learning tools

• 

  Wiktionary
  Dictionary and thesaurus

More portals

Discover Wikipedia using portals

• 

  List of all portals

• 

  The arts portal

• 

  Biography portal

• 

  Current events portal

• 

  Geography portal

• 

  History portal

• 

  Mathematics portal

• 

  Science portal

• 

  Society portal

• 

  Technology portal

• 

  Random portal

• 

  WikiProject Portals