Portal:Mathematics
The Mathematics Portal
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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Nicomachus's theorem states that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers. This result is generalized by Faulhaber's formula, which gives the sum of pth powers of the first n natural numbers. The special case of Nicomachus's theorem can be proved by mathematical induction, but a more direct proof can be given which is illustrated by a proof without words, pictured here.
Did you know…
- ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
- ...that the orthocenter, circumcenter, centroid and the centre of the nine-point circle all lie on one line, the Euler line?
- ...that an arbitrary quadrilateral will tessellate?
- ...that it has not been proven whether or not every even integer greater than two can be expressed as the sum of two primes?
- ...that the sum of the first n odd numbers divided by the sum of the next n odd numbers is always equal to one third?
- ...that i to the power of i, where i is the square root of -1, is a real number?
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