Portal:Mathematics

Topics: Culture; Geography; Health; History; Mathematics; Nature; People; Philosophy; Religion; Society; Technology

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.

More about Mathematics...

There are approximately 31,444 mathematics articles in Wikipedia.

View new selections below (purge)

edit

Selected article

The four charts each map part of the circle to an open interval, and together cover the whole circle.
Image credit: User:KSmrq

A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional.

In a one dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.

Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.

View all selected articles

Read More...

edit

Selected picture

Credit: John Reid

Pi, represented by the Greek letter π, is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space (i.e., on a flat plane); it is also the ratio of a circle's area to the square of its radius. (These facts are reflected in the familiar formulas from geometry, $C = π d$ and $A = π r 2$ .) In this animation, the circle has a diameter of 1 unit, giving it a circumference of π. The rolling shows that the distance a point on the circle moves linearly in one complete revolution is equal to π. Pi is an irrational number and so cannot be expressed as the ratio of two integers; as a result, the decimal expansion of π is nonterminating and nonrepeating. To 50 decimal places, π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, a value of sufficient precision to allow the calculation of the volume of a sphere the size of the orbit of Neptune around the Sun (assuming an exact value for this radius) to within 1 cubic angstrom. According to the Lindemann–Weierstrass theorem, first proved in 1882, π is also a transcendental (or non-algebraic) number, meaning it is not the root of any non-zero polynomial with rational coefficients. (This implies that it cannot be expressed using any closed-form algebraic expression—and also that solving the ancient problem of squaring the circle using a compass and straightedge construction is impossible). Perhaps the simplest non-algebraic closed-form expression for π is $4 arctan 1$ , based on the inverse tangent function (a transcendental function). There are also many infinite series and some infinite products that converge to π or to a simple function of it, like 2/π; one of these is the infinite series representation of the inverse-tangent expression just mentioned. Such iterative approaches to approximating π first appeared in 15th-century India and were later rediscovered (perhaps not independently) in 17th- and 18th-century Europe (along with several continued fractions representations). Although these methods often suffer from an impractically slow convergence rate, one modern infinite series that converges to 1/π very quickly is given by the Chudnovsky algorithm, first published in 1989; each term of this series gives an astonishing 14 additional decimal places of accuracy. In addition to geometry and trigonometry, π appears in many other areas of mathematics, including number theory, calculus, and probability.

More selected pictures...

Read more...

edit

Did you know...

...that the six permutations of the vector (1,2,3) form a hexagon in 3d space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
...that the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particle-antiparticle annihilation reactions?
...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
...that the Pythagorean Theorem generalizes to any three similar shapes on the three sides of a right-angled triangle?
...that the orthocenter, circumcenter, centroid and the centre of the nine-point circle all line on one line, the Euler line?