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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The continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:
 There is no set whose size is strictly between that of the integers and that of the real numbers.
Or mathematically speaking, noting that the cardinality for the integers is ("alephnull") and the cardinality of the real numbers is , the continuum hypothesis says
This is equivalent to:
The real numbers have also been called the continuum, hence the name.
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This is a graph of a portion of the complexvalued Riemann zeta function along the critical line (the set of complex numbers having real part equal to 1/2). More specifically, it is a graph of Im ζ(1/2 + it) versus Re ζ(1/2 + it) (the imaginary part vs. the real part) for values of the real variable t running from 0 to 34 (the curve starts at its leftmost point, with real part approximately −1.46 and imaginary part 0). The first five zeros along the critical line are visible in this graph as the five times the curve passes through the origin (which occur at t ≈ 14.13, 21.02, 25.01, 30.42, and 32.93 — for a different perspective, see a graph of the real and imaginary parts of this function plotted separately over a wider range of values). In 1914, G. H. Hardy proved that ζ(1/2 + it) has infinitely many zeros. According to the Riemann hypothesis, zeros of this form constitute the only nontrivial zeros of the full zeta function, ζ(s), where s varies over all complex numbers. Riemann's zeta function grew out of Leonhard Euler's study of realvalued infinite series in the early 18th century. In a famous 1859 paper called "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann extended Euler's results to the complex plane and established a relation between the zeros of his zeta function and the distribution of prime numbers. The paper also contained the previously mentioned Riemann hypothesis, which is considered by many mathematicians to be the most important unsolved problem in pure mathematics. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Did you know...
 ...that some functions can be written as an infinite sum of trigonometric polynomials and that this sum is called the Fourier series of that function?
 ...that the identity elements for arithmetic operations make use of the only two whole numbers that are neither composites nor prime numbers, 0 and 1?
 ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
 ...the Piphilology record (memorizing digits of Pi) is in excess of 67000 as of Apr 2010?
 ...with a Perrin number denoted P(i), i=1,2,3..., when i is prime then P(i) is composite, being divisible by i?
 ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
 ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
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