Portal:Mathematics
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.
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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This logic diagram of a full adder shows how logic gates can be used in a digital circuit to add two binary inputs (i.e., two input bits), along with a carryinput bit (typically the result of a previous addition), resulting in a final "sum" bit and a carryoutput bit. This particular circuit is implemented with two XOR gates, two AND gates and one OR gate, although equivalent circuits may be composed of only NAND gates or certain combinations of other gates. To illustrate its operation, consider the inputs A = 1 and B = 1 with C_{in} = 0; this means we are adding 1 and 1, and so should get the number 2. The output of the first XOR gate (upperleft) is 0, since the two inputs do not differ (1 XOR 1 = 0). The second XOR gate acts on this result and the carryinput bit, 0, resulting in S = 0 (0 XOR 0 = 0). Meanwhile, the first AND gate (in the middle) acts on the output of the first gate, 0, and the carryinput bit, 0, resulting in 0 (0 AND 0 = 0); and the second AND gate (immediately below the other one) acts on the two original input bits, 1 and 1, resulting in 1 (1 AND 1 = 1). Finally, the OR gate at the lowerright corner acts on the outputs of the two AND gates and results in the carryoutput bit C_{out} = 1 (0 OR 1 = 1). This means the final answer is "0carry1", or "10", which is the binary representation of the number 2. Multiplebit adders (i.e., circuits that can add inputs of 4bit length, 8bit length, or any other desired length) can be implemented by chaining together simpler 1bit adders such as this one. Adders are examples of the kinds of simple digital circuits that are combined in sophisticated ways inside computer CPUs to perform all of the functions necessary to operate a digital computer. The fact that simple electronic switches could implement logical operations—and thus simple arithmetic, as shown here—was realized by Charles Sanders Peirce in 1886, building on the mathematical work of Gottfried Wilhelm Leibniz and George Boole, after whom Boolean algebra was named. The first modern electronic logic gates were produced in the 1920s, leading ultimately to the first digital, generalpurpose (i.e., programmable) computers in the 1940s.
 ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
 ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
 ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
 ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
 ...that a ball can be cut up and reassembled into two balls the same size as the original (BanachTarski paradox)?
 ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
 ...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
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