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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Fourteen ways of triangulating a hexagon Image credit: User:Dmharvey 
The Catalan numbers, named for the Belgian mathematician Eugène Charles Catalan, are a sequence of natural numbers that are important in combinatorial mathematics. The sequence begins:
The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n^{th} Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves. It is also the number of ways of associating n applications of a binary operator as well as the number of ways that a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines.
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This is a handdrawn graph of the absolute value (or modulus) of the gamma function on the complex plane, as published in the 1909 book Tables of Higher Functions, by Eugene Jahnke and Fritz Emde. Such threedimensional graphs of complicated functions were rare before the advent of highresolution computer graphics (even today, tables of values are used in many contexts to look up function values instead of consulting graphs directly). Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist Michael Berry. See a similar computergenerated image for comparison. When restricted to positive integers, the gamma function generates the factorials through the relation Γ(n) = (n − 1)!, which is the product of all positive integers from n − 1 down to 1 (0! is defined to be equal to 1). For real and complex numbers, the function is defined by the improper integral . This integral diverges when t is a negative integer, which is causing the spikes in the left half of the graph (these are simple poles of the function, where its values increase to infinity, analogous to asymptotes in twodimensional graphs). The gamma function has applications in quantum physics, astrophysics, and fluid dynamics, as well as in number theory and probability.
Did you know 
 ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?
 ... that, while the crisscross algorithm visits all eight corners of the Klee–Minty cube when started at a worst corner, it visits only three more corners on average when started at a random corner?
 ...that in senary, all prime numbers other than 2 and 3 end in 1 or a 5?
 ... if the integer n is prime, then the nth Perrin number is divisible by n?
 ...that it is impossible to trisect a general angle using only a ruler and a compass?
 ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
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