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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.

The Mathematics Portal

Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of 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.

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A Hilbert space is a real or complex vector space with a positive-definite Hermitian form, that is complete under its norm. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the limits exist when expected, which facilitates various definitions from calculus. A typical example of a Hilbert space is the space of square summable sequences.

Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.

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The Lorenz attractor, named for Edward N. Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.

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Did you know...

...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?
...that the primality of a number can be determined using only a single division using Wilson's Theorem?
...that the line separating the numerator and denominator of a fraction is called a solidus if written as a diagonal line or a vinculum if written as a horizontal line.
...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare.

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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