Portal:Mathematics
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Mathematics is the study of numbers, quantity, space, structure, and change. 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.
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There are approximately 31,444 mathematics articles in Wikipedia.
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Flowcharts are often used to represent algorithms Image credit: User:Booyabazooka 
An algorithm is a procedure (a finite set of welldefined instructions) for accomplishing some task which, given an initial state, will terminate in a defined endstate. The computational complexity and efficient implementation of the algorithm are important in computing, and this depends on suitable data structures.
Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). Algorithms can be composed to create more complex algorithms.
The concept of an algorithm originated as a means of recording procedures for solving mathematical problems such as finding the common divisor of two numbers or multiplying two numbers. The concept was formalized in 1936 through Alan Turing's Turing machines and Alonzo Church's lambda calculus, which in turn formed the foundation of computer science.
Most algorithms can be directly implemented by computer programs; any other algorithms can at least in theory be simulated by computer programs. In many programming languages, algorithms are implemented as functions or procedures.
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A line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). A detailed explanation of the animation is available. The key insight is that line integrals may be reduced to simpler definite integrals. (See also a similar animation illustrating a line integral of a vector field.) Many formulas in elementary physics (for example, W = F · s to find the work done by a constant force F in moving an object through a displacement s) have line integral versions that work for nonconstant quantities (for example, W = ∫_{C} F · ds to find the work done in moving an object along a curve C within a continuously varying gravitational or electric field F). A higherdimensional analog of a line integral is a surface integral, where the (double) integral is taken over a twodimensional surface instead of along a onedimensional curve. Surface integrals can also be thought of as generalizations of multiple integrals. All of these can be seen as special cases of integrating a differential form, a viewpoint which allows multivariable calculus to be done independently of the choice of coordinate system. While the elementary notions upon which integration is based date back centuries before Newton and Leibniz independently invented calculus, line and surface integrals were formalized in the 18th and 19th centuries as the subject was placed on a rigorous mathematical foundation. The modern notion of differential forms, used extensively in differential geometry and quantum physics, was pioneered by Élie Cartan in the late 19th century.
Did you know...
 ...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?
 ...that the six permutations of the vector (1,2,3) form a hexagon in 3D space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
 ...that the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particleantiparticle annihilation reactions?
 ...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
 ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
 ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
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