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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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proof without words that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers
proof without words that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers
Nicomachus's theorem states that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers. This result is generalized by Faulhaber's formula, which gives the sum of pth powers of the first n natural numbers. The special case of Nicomachus's theorem can be proved by mathematical induction, but a more direct proof can be given which is illustrated by a proof without words, pictured here.

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  • ... that after Archimedes first defined convex curves, mathematicians lost interest in their analysis until the 19th century, more than two millennia later?
  • ... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
  • ... that the music of math rock band Jyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
  • ... that mathematics professor Ari Nagel has fathered more than a hundred children?
  • ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
  • ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
  • ... that subgroup distortion theory, introduced by Misha Gromov in 1993, can help encode text?
  • ... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?

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A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another
Image credit: User:PAR

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, usually denoted by the Greek letter φ (phi).

Expressed algebraically, two quantities a and b (assuming ) are therefore in the golden ratio if

It follows from this property that φ satisfies the quadratic equation φ2 = φ + 1 and is therefore an algebraic irrational number, given by

which is approximately equal to 1.6180339887.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, divine proportion (Italian: proporzionedivina), divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias. (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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