# Portal:Mathematics

## The Mathematics Portal

**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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Johannes Kepler Image credit: User:ArtMechanic |

**Johannes Kepler** (1571 – 1630) was an Austrian Lutheran mathematician, astronomer and a key figure in the 17th century astronomical revolution. He is best known for his laws of planetary motion, based on his works *Astronomia nova* and *Harmonice Mundi*; Kepler's laws provided one of the foundations of Isaac Newton's theory of universal gravitation. Before Kepler, planets' paths were computed by combinations of the circular motions of the celestial orbs; after Kepler astronomers shifted their attention from *orbs* to *orbits*—paths that could be represented mathematically as an ellipse.

During his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and adviser to General Wallenstein. He also did fundamental work in the field of optics and helped to legitimize the telescopic discoveries of his contemporary Galileo Galilei.

Kepler lived in an era when there was no clear distinction between astronomy and astrology, while there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of the more prestigious discipline of philosophy).

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This is a modern reproduction of the first published image of the **Mandelbrot set**, which appeared in 1978 in a technical paper on Kleinian groups by Robert W. Brooks and Peter Matelski. The Mandelbrot set consists of the points c in the complex plane that generate a bounded sequence of values when the recursive relation *z*_{n+1}=*z*_{n}^{2} + *c* is repeatedly applied starting with *z*_{0}=0. The boundary of the set is a highly complicated fractal, revealing ever finer detail at increasing magnifications. The boundary also incorporates smaller near-copies of the overall shape, a phenomenon known as quasi-self-similarity. The ASCII-art depiction seen in this image only hints at the complexity of the boundary of the set. Advances in computing power and computer graphics in the 1980s resulted in the publication of high-resolution color images of the set (in which the colors of points outside the set reflect how quickly the corresponding sequences of complex numbers diverge), and made the Mandelbrot set widely known by the general public. Named by mathematicians Adrien Douady and John H. Hubbard in honor of Benoit Mandelbrot, one of the first mathematicians to study the set in detail, the Mandelbrot set is closely related to the Julia set, which was studied by Gaston Julia beginning in the 1910s.

## Did you know…

- ...the Piphilology record (memorizing digits of Pi) is 70000 as of Mar 2015?
- ...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?
- ...work in artificial intelligence makes use of Swarm intelligence, which has foundations in the behavorial examples found in nature of ants, birds, bees, and fish among others?
- ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
- ...that Modular arithmetic has application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to 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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