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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e is the unique number such that the slope of y=e^{x} (blue curve) is exactly 1 when x=0 (illustrated by the red tangent line). For comparison, the curves y=2^{x} (dotted curve) and y=4^{x} (dashed curve) are shown. Image credit: Dick Lyon 
The mathematical constant e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:
 The sum of the infinite series
 where n! is the factorial of n.
 The global maximizer of the function
 The limit:
The number e is also the base of the natural logarithm. Since e is transcendental, and therefore irrational, its value can not be given exactly. The numerical value of e truncated to 20 decimal places is 2.71828 18284 59045 23536.
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This is a chart of all prime knots having seven or fewer crossings (not including mirror images) along with the unknot (or "trivial knot"), a closed loop that is not a prime knot. The knots are labeled with AlexanderBriggs notation. Many of these knots have special names, including the trefoil knot (3_{1}) and figureeight knot (4_{1}). Knot theory is the study of knots viewed as different possible embeddings of a 1sphere (a circle) in threedimensional Euclidean space (R^{3}). These mathematical objects are inspired by realworld knots, such as knotted ropes or shoelaces, but don't have any free ends and so cannot be untied. (Two other closely related mathematical objects are braids, which can have loose ends, and links, in which two or more knots may be intertwined.) One way of distinguishing one knot from another is by the number of times its twodimensional depiction crosses itself, leading to the numbering shown in the diagram above. The prime knots play a roll very similar to prime numbers in number theory; in particular, any given (nontrivial) knot can be uniquely expressed as a "sum" of prime knots (a series of prime knots spliced together) or is itself prime. Early knot theory enjoyed a brief period of popularity among physicists in the late 19th century after William Thomson suggested that atoms are knots in the luminiferous aether. This led to the first serious attempts to catalog all possible knots (which, along with links, now number in the billions). In the early 20th century, knot theory was recognized as a subdiscipline within geometric topology. Scientific interest was resurrected in the latter half of the 20th century by the need to understand knotting problems in organic chemistry, including the behavior of DNA, and the recognition of connections between knot theory and quantum field theory.
Did you know...
 ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
 ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
 ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
 ... that economists blame market failures on nonconvexity?
 ... that, according to the pizza theorem, a circular pizza that is sliced offcenter into eight equalangled 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?
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