Portal:Mathematics

Portal topics: Activities; Culture; Geography; Health; History; Mathematics; Nature; People; Philosophy; Religion; Society; Technology; Random portal

The Mathematics Portal

Mathematics is the study of numbers, quantity, space, pattern, 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. It is used for calculation and considered as 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.

More about Mathematics...

Refresh with new selections below (purge)

edit

Selected article - show another

Fourteen ways of triangulating a hexagon
Image credit: User:Dmharvey

The Catalan numbers, named for the Belgian mathematician Eugène Charles Catalan, are a sequence of natural numbers that are important in combinatorial mathematics. The sequence begins:

1, 1, 2, 5, 14, 42, 132, ...

The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n^th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves. It is also the number of ways of associating n applications of a binary operator as well as the number of ways that a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines.

View all selected articles

Read More...

edit

Selected image - show another

Credit: Pop-up casket & Fool

This spiral diagram represents all ordinal numbers less than $ω ω$ . The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular counting numbers starting with zero. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach infinity, or more precisely $ω$ , the first transfinite ordinal number (identified with the set of all counting numbers, a "countably infinite" set, the cardinality of which corresponds to the first transfinite cardinal number, called $ℵ 0$ ). The ordinal numbers continue from this point in the second turn of the spiral with $ω + 1$ , $ω + 2$ , and so forth. (A special ordinal arithmetic is defined to give meaning to these expressions, since the $+$ symbol here does not represent the addition of two real numbers.) Halfway through the second turn of the spiral (at the bottom) the numbers approach $ω + ω$ , or $ω \cdot 2$ . The ordinal numbers continue with $ω \cdot 2 + 1$ through $ω \cdot 2 + ω = ω \cdot 3$ (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through $ω \cdot 4$ , and so forth, up to $ω \cdot ω = ω 2$ at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) The ordinals continue in the third turn of the spiral with $ω 2 + 1$ through $ω 2 + ω$ , then through $ω 2 + ω 2 = ω 2 \cdot 2$ , up to $ω 2 \cdot ω = ω 3$ at the top of the third turn. Continuing in this way, the ordinals increase by one power of $ω$ for each turn of the spiral, approaching $ω ω$ in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue (not shown in this diagram) through $\omega ^{\omega ^{\omega }}$ and $\omega ^{\omega ^{\omega ^{\omega }}}$ , and so on, approaching the first epsilon number, $ε 0$ . Each of these ordinals is still countable, and therefore equal in cardinality to $ω$ . After uncountably many of these transfinite ordinals, the first uncountable ordinal is reached, corresponding to only the second infinite cardinal $\aleph _{1}$ . The identification of this larger cardinality with the cardinality of the set of real numbers can neither be proved nor disproved within the standard version of axiomatic set theory called Zermelo–Fraenkel set theory, whether or not one also assumes the axiom of choice.

More selected pictures... Read more...

edit

Did you know - view different entries

... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?
...that in Floyd's algorithm for cycle detection, the tortoise and hare move at very different speeds, but always finish at the same spot?
...that in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?