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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. Used for calculation, it is considered the most important subject. 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. (Full article...)

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Mathematics department in Göttingen where Hilbert worked from 1895 until his retirement in 1930
Image credit: Daniel Schwen

David Hilbert (January 23, 1862, Wehlau, Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation as a great mathematician and scientist by inventing or developing a broad range of ideas, such as invariant theory, the axiomization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students supplied significant portions of the mathematic infrastructure required for quantum mechanics and general relativity. He is one of the founders of proof theory, mathematical logic, and the distinction between mathematics and metamathematics, and warmly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century. (Full article...)

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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 zn+1 = zn2 + c is repeatedly applied starting with z0 = 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.
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