Portal:Mathematics

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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). 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. (Full article...)

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animation showing a right triangle being duplicated and rearranged in a way that illustrates the Pythagorean theorem
animation showing a right triangle being duplicated and rearranged in a way that illustrates the Pythagorean theorem
An animated geometric proof of the Pythagorean theorem, which states that among the three sides of a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, written as a2 + b2 = c2. A large square is formed with area c2, from four identical right triangles with sides a, b and c, fitted around a small central square (of side length ba). Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2 and b2, which together must have the same area as the initial large square. This is a somewhat subtle example of a proof without words.

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The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages
Image credit: Matt Crypto

Cryptography (or cryptology) is derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write". In modern times, it has become a branch of information theory, as the mathematical study of information and especially its transmission from place to place. The noted cryptographer Ron Rivest has observed that "cryptography is about communication in the presence of adversaries." It is a central contributor to several fields: information security and related issues, particularly, authentication, and access control. One of cryptography's primary purposes is hiding the meaning of messages, not usually the existence of such messages. In modern times, cryptography also contributes to computer science. Cryptography is central to the techniques used in computer and network security for such things as access control and information confidentiality. Cryptography is also used in many applications encountered in everyday life; the security of ATM cards, computer passwords, and electronic commerce all depend on cryptography. (Full article...)

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Topics in mathematics

General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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