Portal:Mathematics

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Pythagoras

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. Used for calculation, it is considered the most important subject. 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...)

Selected article –

The graph of a real-valued quadratic function of a real variable x, is a parabola. Image credit: Enoch Lau

A quadratic equation is a polynomial equation of degree two. The general form is

$ax^{2}+bx+c=0,\,\!$

where a ≠ 0 (if a = 0, then the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x², the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term.

Quadratic equations are known by that name because quadratus is Latin for "square"; in the leading term the variable is squared.

A quadratic equation has two (not necessarily distinct) solutions, which may be real or complex, given by the quadratic formula:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$

If the discriminant $b^{2}-4ac>0$, then the quadratic equation has two distinct real solutions; if $b^{2}-4ac=0$, the equation has two real solutions which are equal; if $b^{2}-4ac<0$, the equation has two complex solutions.

These solutions are roots of the corresponding quadratic function

$f(x)=ax^{2}+bx+c.\,$ (Full article...)

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Mandelbrot set

Credit: Elphaba

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_n² + c is repeatedly applied starting with z₀ = 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 –

• … that the best known lower bound for the length of the smallest superpermutation was first posted anonymously to the internet imageboard 4chan?
• ...that the mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
• ...that a regular heptagon is the regular polygon with the fewest sides which is not constructible with a compass and straightedge?
• ...that the regular trigonometric functions and the hyperbolic trigonometric functions can be related without using complex numbers through the Gudermannian function?
• ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
• ...that a ball can be cut up and reassembled into two balls, each the same size as the original (Banach-Tarski paradox)?
• ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?

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