Portal:Mathematics

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Mathematics is the study of numbers, quantity, space, pattern, 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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Game theory is a branch of mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modelling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimisation approach developed in neoclassical economics.

The field of game theory came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies.

Game theory has played, and continues to play a large role in the social sciences, and is now also used in many diverse academic fields. Beginning in the 1970s, game theory has been applied to animal behaviour, including evolutionary theory. Many games, especially the prisoner's dilemma, are used to illustrate ideas in political science and ethics. Game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.

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animation showing a roughly star-shaped graph being traced out as a smaller circle rolls around inside of a larger circle

A hypotrochoid is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. A special case is a hypotrochoid with the inner circle exactly one-half the radius of the outer circle, resulting in an ellipse (see an animation showing this). Mathematical analysis of closely-related curves called hypocycloids lead to special Lie groups. Both hypotrochoids and epitrochoids (where the moving circle rolls around on the outside of the fixed circle) can be created using the Spirograph drawing toy. These curves have applications in the "real world" in epicyclic and hypocycloidal gearing, which were used in World War II in the construction of portable radar gear and may be used today in 3D printing.

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General Foundations Number theory Discrete mathematics
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Algebra Analysis Geometry and topology Applied mathematics
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