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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Alan Turing memorial statue in Sackville Park Image credit: User:Lmno |
Alan Mathison Turing, OBE (June 23, 1912 – June 7, 1954), was an English mathematician, logician, and cryptographer.
Turing is often considered to be the father of modern computer science. Turing provided an influential formalisation of the concept of the algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine. With the Turing test, he made a significant and characteristically provocative contribution to the debate regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He later worked at the National Physical Laboratory, creating one of the first designs for a stored-program computer, although it was never actually built. In 1947 he moved to the University of Manchester to work, largely on software, on the Manchester Mark I then emerging as one of the world's earliest true computers.
During World War II, Turing worked at Bletchley Park, Britain's codebreaking centre, and was for a time head of Hut 8, the section responsible for German Naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine which could find settings for the Enigma machine.
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This is a modern reproduction of the first published image of the Mandelbrot set, which appeared in 1978 in a technical paper on Kleinian groups by Robert W. Brooks and Peter Matelski. The Mandelbrot set consists of the points c in the complex plane that generate a bounded sequence of values when the recursive relation z_{n+1} = z_{n}^{2} + c is repeatedly applied starting with z_{0} = 0. The boundary of the set is a highly complicated fractal, revealing ever finer detail at increasing magnifications. The boundary also incorporates smaller near-copies of the overall shape, a phenomenon known as quasi-self-similarity. The ASCII-art depiction seen in this image only hints at the complexity of the boundary of the set. Advances in computing power and computer graphics in the 1980s resulted in the publication of high-resolution color images of the set (in which the colors of points outside the set reflect how quickly the corresponding sequences of complex numbers diverge), and made the Mandelbrot set widely known by the general public. Named by mathematicians Adrien Douady and John H. Hubbard in honor of Benoit Mandelbrot, one of the first mathematicians to study the set in detail, the Mandelbrot set is closely related to the Julia set, which was studied by Gaston Julia beginning in the 1910s.
Did you know…
- ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
- ... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?
- ... that the clique problem of programming a computer to find complete subgraphs in an undirected graph was first studied as a way to find groups of people who all know each other in social networks?
- ... that the Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle?
- ... that the Life without Death cellular automaton, a mathematical model of pattern formation, is a variant of Conway's Game of Life in which cells, once brought to life, never die?
- ... that one can list every positive rational number without repetition by breadth-first traversal of the Calkin–Wilf tree?
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