Mathematics (from Greek Î¼Î¬Î¸Î·Î¼Î± mÃ¡thÄ“ma, â€œknowledge, study, learningâ€) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858â€“1932), David Hilbert (1862â€“1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Galileo Galilei (1564â€“1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss (1777â€“1855) referred to mathematics as "the Queen of the Sciences". Benjamin Peirce (1809â€“1880) called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein (1879â€“1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." French mathematician Claire Voisin states "There is creative drive in mathematics, it's all about movement trying to express itself." Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance 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, which has led 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. (en)

Mathematics

"Maths" and "Math" redirect here. For other uses see Mathematics (disambiguation) and Math (disambiguation). Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

MathematicsFormal sciencesMathematicsGreek loanwords

Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write f(–3) = 9.

Function (mathematics)Elementary mathematicsFunctions and mappingsBasic concepts in set theory

Matrix (mathematics)

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.

Matrix (mathematics)Matrices

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy four conditions called the group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group.

Group (mathematics)Group theorySymmetryAlgebraic structures

Vector space

A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

Vector spaceLinear algebraGroup theoryMathematical structuresVectorsAbstract algebraFundamental physics conceptsVector spaces

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context is a collection of "vertices" or "nodes" and a collection of edges that connect pairs of vertices.

Graph theoryGraph theory

Set (mathematics)

A set is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.

Set (mathematics)Set theoryConcepts in logicMathematical concepts

Field (mathematics)

In abstract algebra, a field is a ring whose nonzero elements form a commutative group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

Field (mathematics)Field theoryAlgebraic structures

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below the x axis subtract from the total.

IntegralLinear operators in calculusFunctions and mappingsIntegrals

Mathematical logic

Mathematical logic (also known as symbolic logic) is a subfield of mathematics with close connections to the foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logicMathematical logic

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". Continuity of functions is one of the core concepts of topology, which is treated in full generality below.

Continuous functionContinuous mappingsCalculusTypes of functions

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.

VolumeVolumeFundamental physics concepts

University of Belgrade

The University of Belgrade is the oldest and largest university of Serbia. Founded in 1808 as the Belgrade Higher School in revolutionary Serbia, by 1838 it merged with the Kragujevac-based departments into a single university. The University has nearly 90,000 students (including around 1,700 postgraduates) and over 4,200 members of teaching staff.

University of BelgradeUniversity of BelgradeUniversities in BelgradeUniversities in SerbiaEducational institutions established in 1808Education in Belgrade

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria, finding "largest", "smallest", or "optimal" objects, and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems.

CombinatoricsCombinatorics

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or – as here – simply a vector) is a geometric object that has a magnitude and direction and can be added according to the parallelogram law of addition.

Euclidean vectorVector calculusLinear algebraVectorsAbstract algebraFundamental physics conceptsIntroductory physics

Reason

Reason is a term that refers to the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions, and beliefs. It is closely associated with such characteristically human activities as philosophy, science, language, mathematics, and art, and is normally considered to be a definitive characteristic of human nature.

ReasonConcepts in logicBeliefThoughtEpistemologyReasoning

Manifold

In mathematics, a manifold is a mathematical object that on a small enough scale resembles Euclidean space. For example, seen from far away, the surface of the planet Earth is not flat and Euclidean, but on a smaller scale, one may describe each region via a geographic map, a projection of the surface onto the Euclidean plane. A precise mathematical definition of a manifold is given below. Lines and circles (but not figure eights) are one-dimensional manifolds (1-manifolds).

ManifoldGeometric topologyDifferential geometryManifoldsDifferential topologyTopology

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are equidistant from a given point, the centre. The distance between any of the points and the centre is called the radius. Circles are simple closed curves which divide the plane into two regions: an interior and an exterior.

CircleElementary shapesCurvesConic sectionsCirclesPi

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s.

Set theorySet theoryFormal methodsMathematical logic

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces.

Compact spaceTopologyProperties of topological spacesCompactness (mathematics)General topology

Mathematical optimization

In mathematics, computer science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.

Mathematical optimizationOperations researchMathematical optimization

Expected value

In probability theory, the expected value (or expectation, or mathematical expectation, or mean, or the first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable.

Expected valueGambling terminologyTheory of probability distributions

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy some basic conditions.

Category theoryHigher category theoryCategory theory

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. Analysis may be conventionally distinguished from geometry.

Mathematical analysisMathematical analysis

Ring (mathematics)

In mathematics, a ring is an algebraic structure which generalizes the main properties of the addition and the multiplication of integers, real and complex numbers, as well as that of square matrices.

Ring (mathematics)Ring theoryMathematical structuresAlgebraic structures

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect.

Chaos theoryChaos theory

Fractal

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far" Fractals may be exactly the same at every scale, or as illustrated in Figure 1, they may be nearly the same at different scales.

FractalTopologyDigital artMathematical structuresFractals

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection(YBC 7289), which gives a sexagesimal numerical approximation of, the length of the diagonal in a unit square.

Numerical analysisComputational scienceNumerical analysisMathematical physics

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra.

Group theoryGroup theory

Nonlinear system

This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation). 50x40px This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations.

Nonlinear systemDynamical systemsFundamental physics conceptsNonlinear systems

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

Graph (mathematics)Graph theory objectsGraph theory

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R.

Measure (mathematics)Measure theoryMathematical analysisMeasures (measure theory)

Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 = 10 × 10 × 10. More generally, if x = b, then y is the logarithm of x to base b, and is written logb(x), so log10(1000) = 3. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations.

LogarithmLogarithmsElementary special functionsScottish inventions

Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point (zero-dimensions), a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

Plane (geometry)SurfacesEuclidean geometryMathematical concepts

Torus

In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit. When the axis is tangent to the circle, the resulting surface is called a horn torus; when the axis is a chord of the circle, it is called a spindle torus.

TorusSurfaces

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.".

Mathematical physicsPhysicsMathematical physics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge.

Applied mathematicsApplied mathematics

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well.

DeterminantLinear algebraHomogeneous polynomialsMatrix theoryDeterminantsAlgebra

Stochastic process

In probability theory, a stochastic process, or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process.

Stochastic processTelecommunication theoryStatistical data typesStatistical modelsStochastic processes

Specialist school

The specialist schools programme was a UK government initiative which encouraged secondary schools in England to specialise in certain areas of the curriculum to boost achievement. The Specialist Schools and Academies Trust was responsible for the delivery of the programme. Currently there are nearly 3,000 specialist schools, or 88% of the state-funded secondary schools in England.

Specialist schoolSchool types

Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

Commutative propertyFunctional analysisRules of inferenceAbstract algebraSymmetryElementary algebraFundamental physics conceptsMathematical relationsBinary operations

Open set

In topology, a set is called an open set if it is a neighborhood of every point . When dealing with metric spaces, this can be intuitively interpreted as saying that every can be "moved" some non-zero distance, in any direction, and it will still lie within . The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

Open setGeneral topology

Group action

In algebra and geometry, a group action is a description of symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set.

Group actionGroup actionsGroup theorySymmetryRepresentation theory of groups

Charles Sanders Peirce

Charles Sanders Peirce (like "purse"; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist, born at 3 Phillips Place in Cambridge, Massachusetts. Peirce was educated as a chemist and employed as a scientist for 30 years. Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism.

Charles Sanders PeircePhilosophers of educationCharles Sanders PeirceFellows of the American Academy of Arts and SciencesHarvard University alumni1839 birthsChristian philosophers20th-century mathematiciansAmerican logicians1914 deathsGeodesistsAmerican statisticiansPragmatistsLattice theorists19th-century mathematiciansModal logiciansMembers of the United States National Academy of SciencesAmerican mathematicians20th-century philosophersAmerican EpiscopaliansAmerican philosophersPhilosophers of languageLogiciansJohns Hopkins University facultySemioticians19th-century philosophersPeople from Cambridge, Massachusetts

Formal language

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols. The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed; frequently it is required to be finite. The strings formed from this alphabet are called words, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas.

Formal languageTheoretical computer scienceCombinatorics on wordsFormal languages

Tensor

whose columns are the forces acting on the,, and faces of the cube. ]] Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values.

TensorTensorsFundamental physics concepts

University of Ljubljana

The University of Ljubljana is the oldest and largest university in Slovenia. With over 63,000 enrolled undergraduate and graduate students, it is among the largest universities in Europe.

University of LjubljanaEducational institutions established in 1919Educational institutions in Ljubljana1919 establishments in SloveniaUniversities in SloveniaCenter DistrictUniversity of Ljubljana

Isomorphism

In abstract algebra, an isomorphism is a bijective homomorphism. Two mathematical structures are said to be isomorphic if there is an isomorphism between them. In category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f: Y → X, with the property that both ff = idX and f f = idY.

IsomorphismMorphisms

Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.

Binary relationMathematical relations

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring. Modules also generalize the notion of abelian groups, which are modules over the ring of integers.

Module (mathematics)Module theoryAlgebraic structures

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