Portal:Mathematics

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This portal is for the academic discipline of mathematics. For related portals of logic and statistics, please see portals: mathematics, logic, and statistics.

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The Mathematics Portal

Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of quantities (numbers) and their operations, interrelations, combinations, generalizations, and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. It evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

There are approximately 20751 mathematical articles in Wikipedia.

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Flowcharts are often used to represent algorithms

An algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. The computational complexity and efficient implementation of the algorithm are important in computing, and this depends on suitable data structures.

Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). Algorithms can be composed to create more complex algorithms.

The concept of an algorithm originated as a means of recording procedures for solving mathematical problems such as finding the common divisor of two numbers or multiplying two numbers. The concept was formalized in 1936 through Alan Turing's Turing machines and Alonzo Church's lambda calculus, which in turn formed the foundation of computer science.

Most algorithms can be directly implemented by computer programs; any other algorithms can at least in theory be simulated by computer programs. In many programming languages, algorithms are implemented as functions or procedures.

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In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or parallel postulate is equivalent to:

Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line (see 1).

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).

And hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line (see 3).

These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.

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Did you know...

...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
...that every natural number can be written as the sum of four squares?
...that the largest known prime number is over 12 million digits long?
...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
...that it is unknown whether π and e are algebraically independent?
...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?

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Topics in mathematics

General	Foundations	Number theory	Discrete mathematics
Mathematicians History of mathematics Philosophy of mathematics Mathematical notation Mathematical beauty Mathematics education Areas of mathematics List of basic mathematics topics	Foundations of mathematics Mathematical logic Set theory (portal) Naive set theory Axiomatic set theory Proof theory Model theory Recursion theory Category theory (portal) Topos theory Gödel's incompleteness theorems	Number theory (portal) Algebraic number theory Analytic number theory Arithmetic Numbers Natural numbers Prime numbers Rational numbers Algebraic numbers Fundamental theorem of arithmetic	Discrete mathematics (portal) Combinatorics Graph theory Game theory Discrete geometry Theoretical computer science Computability theory Complexity theory Information theory
Analysis	Algebra	Geometry and topology	Applied mathematics
Analysis (portal) Calculus Vector calculus Differential equations Calculus of variations Real analysis Complex analysis Functional analysis Measure theory Harmonic analysis Special functions Fundamental theorem of calculus	Algebra (portal) Elementary algebra Abstract algebra Group theory Ring theory Field theory Commutative algebra Linear algebra Matrix theory Multilinear algebra Universal algebra Order theory Fundamental theorem of algebra	Geometry (portal) Topology (portal) General topology Algebraic topology Geometric topology Euclidean geometry Non-Euclidean geometry Affine geometry Projective geometry Differential geometry Riemannian geometry Lie groups Algebraic geometry Trigonometry	Applied mathematics Numerical analysis Mathematical biology Optimization Dynamical systems Chaos theory Financial mathematics Cryptography (portal) Mathematical physics Classical mechanics Probability Statistics Stochastic processes Operations research