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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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A dodecahedron, one of the five Platonic solids
Image credit: User:DTR

A regular polytope is a geometric figure with a high degree of symmetry. Examples in two dimensions include the square, the regular pentagon and hexagon, and so on. In three dimensions the regular polytopes include the cube, the dodecahedron, and all other Platonic solids. Other Platonic solids include the tetrahedron, the octahedron, the icosahedron. Examples exist in higher dimensions also, such as the 5-dimensional hendecatope. Circles and spheres, although highly symmetric, are not considered polytopes because they do not have flat faces. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.

Many regular polytopes, at least in two and three dimensions, exist in nature and have been known since prehistory. The earliest surviving mathematical treatment of these objects comes to us from ancient Greek mathematicians such as Euclid. Indeed, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids. (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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WikiProjects The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.

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