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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This is a graph of a portion of the complex-valued Riemann zeta function along the critical line (the set of complex numbers having real part equal to 1/2). More specifically, it is a graph of Im ζ(1/2 + it) versus Re ζ(1/2 + it) (the imaginary part vs. the real part) for values of the real variable t running from 0 to 34 (the curve starts at its leftmost point, with real part approximately −1.46 and imaginary part 0). The first five zeros along the critical line are visible in this graph as the five times the curve passes through the origin (which occur at t ≈ 14.13, 21.02, 25.01, 30.42, and 32.93 — for a different perspective, see a graph of the real and imaginary parts of this function plotted separately over a wider range of values). In 1914, G. H. Hardy proved that ζ(1/2 + it) has infinitely many zeros. According to the Riemann hypothesis, zeros of this form constitute the only non-trivial zeros of the full zeta function, ζ(s), where s varies over all complex numbers. Riemann's zeta function grew out of Leonhard Euler's study of real-valued infinite series in the early 18th century. In a famous 1859 paper called "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann extended Euler's results to the complex plane and established a relation between the zeros of his zeta function and the distribution of prime numbers. The paper also contained the previously mentioned Riemann hypothesis, which is considered by many mathematicians to be the most important unsolved problem in pure mathematics. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Did you know…
- ...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.
- ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
- ...that every natural number can be written as the sum of four squares?
- ...that the largest known prime number is nearly 25 million digits long?
- ...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
- ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
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