# Portal:Mathematics

## The Mathematics Portal

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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An animation showing how an obliquely cut torus reveals a pair of intersecting circles known as Villarceau circles, named after the French astronomer and mathematician Yvon Villarceau. These are two of the four circles that can be drawn through any given point on the torus. (The other two are oriented horizontally and vertically, and are the analogs of lines of latitude and longitude drawn through the given point.) The circles have no known practical application and seem to be merely a curious characteristic of the torus. However, Villarceau circles appear as the fibers in the Hopf fibration of the 3-sphere over the ordinary 2-sphere, and the Hopf fibration itself has interesting connections to fluid dynamics, particle physics, and quantum theory.

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 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

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

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:

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

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

These solutions are roots of the corresponding quadratic function

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

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