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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A number is an abstract entity that represents a count or measurement. A symbol for a number is called a numeral. The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.
Numbers can be classified into sets called number systems. The most familiar numbers are the natural numbers, which to some mean the nonnegative integers and to others mean the positive integers. In everyday parlance the nonnegative integers are commonly referred to as whole numbers, the positive integers as counting numbers, symbolised by . Mathematics is used in many classes throughout the course of one's education.
The integers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by (from the German Zahl, meaning "number").
A rational number is a number that can be expressed as a fraction with an integer numerator and a nonzero natural number denominator. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face (for quotient).
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This image illustrates a failed attempt to comb the "hair" on a ball flat, leaving a tuft sticking out at each pole. The hairy ball theorem of algebraic topology states that whenever one attempts to comb a hairy ball, there will always be at least one point on the ball at which a tuft of hair sticks out. More precisely, it states that there is no nonvanishing continuous tangentvector field on an evendimensional n‑sphere (an ordinary sphere in threedimensional space is known as a "2sphere"). This is not true of certain other threedimensional shapes, such as a torus (doughnut shape) which can be combed flat. The theorem was first stated by Henri Poincaré in the late 19th century and proved in 1912 by L. E. J. Brouwer. If one idealizes the wind in the Earth's atmosphere as a tangentvector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What is true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a nonzero 3D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.
Did you know...
 ... that the Hadwiger conjecture implies that the external surface of any threedimensional 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?
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