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 region between two loxodromes on a geometric sphere. Image credit: Karthik Narayanaswami 
The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as
wellbehaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP^{1}.
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically wellbehaved, even near infinity; it is a onedimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.
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This logic diagram of a full adder shows how logic gates can be used in a digital circuit to add two binary inputs (i.e., two input bits), along with a carryinput bit (typically the result of a previous addition), resulting in a final "sum" bit and a carryoutput bit. This particular circuit is implemented with two XOR gates, two AND gates and one OR gate, although equivalent circuits may be composed of only NAND gates or certain combinations of other gates. To illustrate its operation, consider the inputs A = 1 and B = 1 with C_{in} = 0; this means we are adding 1 and 1, and so should get the number 2. The output of the first XOR gate (upperleft) is 0, since the two inputs do not differ (1 XOR 1 = 0). The second XOR gate acts on this result and the carryinput bit, 0, resulting in S = 0 (0 XOR 0 = 0). Meanwhile, the first AND gate (in the middle) acts on the output of the first gate, 0, and the carryinput bit, 0, resulting in 0 (0 AND 0 = 0); and the second AND gate (immediately below the other one) acts on the two original input bits, 1 and 1, resulting in 1 (1 AND 1 = 1). Finally, the OR gate at the lowerright corner acts on the outputs of the two AND gates and results in the carryoutput bit C_{out} = 1 (0 OR 1 = 1). This means the final answer is "0carry1", or "10", which is the binary representation of the number 2. Multiplebit adders (i.e., circuits that can add inputs of 4bit length, 8bit length, or any other desired length) can be implemented by chaining together simpler 1bit adders such as this one. Adders are examples of the kinds of simple digital circuits that are combined in sophisticated ways inside computer CPUs to perform all of the functions necessary to operate a digital computer. The fact that simple electronic switches could implement logical operations—and thus simple arithmetic, as shown here—was realized by Charles Sanders Peirce in 1886, building on the mathematical work of Gottfried Wilhelm Leibniz and George Boole, after whom Boolean algebra was named. The first modern electronic logic gates were produced in the 1920s, leading ultimately to the first digital, generalpurpose (i.e., programmable) computers in the 1940s.
Did you know...
 ...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?
 ...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?
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