Portal:Mathematics
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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.
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Johannes Kepler Image credit: User:ArtMechanic 
Johannes Kepler (1571 – 1630) was an Austrian Lutheran mathematician, astronomer and a key figure in the 17th century astronomical revolution. He is best known for his laws of planetary motion, based on his works Astronomia nova and Harmonice Mundi; Kepler's laws provided one of the foundations of Isaac Newton's theory of universal gravitation. Before Kepler, planets' paths were computed by combinations of the circular motions of the celestial orbs; after Kepler astronomers shifted their attention from orbs to orbits—paths that could be represented mathematically as an ellipse.
During his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and adviser to General Wallenstein. He also did fundamental work in the field of optics and helped to legitimize the telescopic discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, while there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of the more prestigious discipline of philosophy).
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Quicksort (also known as the partitionexchange sort) is an efficient sorting algorithm that works for items of any type for which a total order (i.e., "≤") relation is defined. This animation shows how the algorithm partitions the input array (here a random permutation of the numbers 1 through 33) into two smaller arrays based on a selected pivot element (bar marked in red, here always chosen to be the last element in the array under consideration), by swapping elements between the two subarrays so that those in the first (on the left) end up all smaller than the pivot element's value (horizontal blue line) and those in the second (on the right) all larger. The pivot element is then moved to a position between the two subarrays; at this point, the pivot element is in its final position and will never be moved again. The algorithm then proceeds to recursively apply the same procedure to each of the smaller arrays, partitioning and rearranging the elements until there are no subarrays longer than one element left to process. (As can be seen in the animation, the algorithm actually sorts all lefthand subarrays first, and then starts to process the righthand subarrays.) First developed by Tony Hoare in 1959, quicksort is still a commonly used algorithm for sorting in computer applications. On the average, it requires O(n log n) comparisons to sort n items, which compares favorably to other popular sorting methods, including merge sort and heapsort. Unfortunately, on rare occasions (including cases where the input is already sorted or contains items that are all equal) quicksort requires a worstcase O(n^{2}) comparisons, while the other two methods remain O(n log n) in their worst cases. Still, when implemented well, quicksort can be about two or three times faster than its main competitors. Unlike merge sort, the standard implementation of quicksort does not preserve the order of equal input items (it is not stable), although stable versions of the algorithm do exist at the expense of requiring O(n) additional storage space. Other variations are based on different ways of choosing the pivot element (for example, choosing a random element instead of always using the last one), using more than one pivot, switching to an insertion sort when the subarrays have shrunk to a sufficiently small length, and using a threeway partitioning scheme (grouping items into those smaller, larger, and equal to the pivot—a modification that can turn the worstcase scenario of allequal input values into the best case). Because of the algorithm's "divide and conquer" approach, parts of it can be done in parallel (in particular, the processing of the left and right subarrays can be done simultaneously). However, other sorting algorithms (including merge sort) experience much greater speed increases when performed in parallel.
Did you know...
 ... 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 breadthfirst traversal of the Calkin–Wilf tree?
 ... that the Hadwiger conjecture implies that the external surface of any threedimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
 ... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
 ... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
 ...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?
 ...that in Floyd's algorithm for cycle detection, the tortoise and hare move at very different speeds, but always finish at the same spot?
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