1 [[de:Mathematik]][[eo:Matematiko]][[fr:Mathématiques]][[nl:Wiskunde]][[no:Matematikk]][[pl:Matematyka]][[pt:Matemática]][[sh:Matematika]][[sl:Matematika]]
2 '''Mathematics''' ([[Greek language|Greek]] ''mathema'': science, learning; ''mathematikos'': fond of learning) is the study of patterns of quantity, structure, change and space. In the modern view, it is the investigation of [[axiom]]atically defined abstract structures using formal [[symbolic logic|logic]] as the common framework. The specific structures investigated often have their origin in the natural sciences, most commonly in [[physics]], but mathematicians also define and investigate structures for reasons purely internal to mathematics, for instance because they realize that the structure provides a unifying generalization for several subfields or a helpful tool in common calculations.
4 Historically, mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.
6 The study of structure starts with [[number]]s, initially the familiar [[natural number]]s and [[integer]]s. The rules governing arithmetical operations are recorded in [[elementary algebra]], and the deeper properties of whole numbers are studied in [[number theory]]. The investigation of methods to solve equations leads to the field of [[abstract algebra]], which, among other things, studies [[ring (algebra)|rings]] and [[field]]s, structures that generalize the properties possessed by the familiar numbers. The physically important concept of [[vector]], generalized to [[vector space]]s and studied in [[linear algebra]], belongs to the two branches of structure and space.
8 The study of space originates with [[geometry]], first the [[Euclidean geometry]] and [[trigonometry]] of familiar three-dimensional space, but later also generalized to [[Non-euclidean geometry|non-Euclidean geometries]] which play a central role in [[general relativity]]. The modern fields of [[differential geometry]] and [[algebraic geometry]] generalize geometry in different directions: differential geometry emphasizes the concepts of coordinate system, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of equations. [[Mathematical group|Group theory]] investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. [[Topology]] connects the study of space and the study of change by focusing on the concept of [[continuous|continuity]].
10 Understanding and describing the change in measurable quantities is the central topic of the natural sciences, and [[calculus]] was developed as a most useful tool for doing just that. The central concept used to describe a changing variable is that of a [[function]]. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of [[differential equations]]. The numbers used to represent continuous quantities are the [[real numbers]], and the detailed study of their properties and the properties of real-valued functions is known as [[real analysis]]. For several reasons, it is convenient to generalise to the [[complex number]]s which are studied in [[complex analysis]]. [[Functional analysis]] focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for [[quantum mechanics]] among many other things.
12 In order to clarify and investigate the foundations of mathematics, the fields of [[set theory]], [[mathematical logic]] and [[model theory]] were developed.
14 When [[computer]]s were first conceived, several surrounding theoretical questions were tackled by mathematicians, leading to the fields of [[computability theory]], [[computational complexity theory]], [[information theory]] and [[algorithmic information theory]]. Many of these questions are now investigated in theoretical [[computer science]].
16 Computers also aided the new field of [[chaos theory]], which deals with the fact that many [[dynamical systems]] in nature obey laws that cause their behaviour to become unpredictable in practice, though deterministic in theory.
17 Chaos theory is closely related to [[fractal]] geometry.
19 An important field in applied mathematics is [[probability and statistics]], which allows the description, analysis and prediction of random phenomena and is used in all sciences. [[Numerical analysis]] investigates the methods for performing calculations on computers and [[discrete mathematics]] is the common name for those fields of mathematics useful in computer science.
21 This following list of subfields and topics reflects one organizational view of mathematics:
24 :[[Number]]s -- [[Natural number]]s -- [[Integer]]s -- [[Rational number]]s -- [[Real number]]s -- [[Complex number]]s -- [[Quaternion]]s -- [[Octonion]]s -- [[Sedenion]]s -- [[Hyperreal number]]s -- [[Surreal number]]s -- [[Ordinal number]]s -- [[Cardinal number]]s -- [[p-adic number]]s -- [[Integer sequence]]s -- [[Mathematical constant]]s -- [[Number names]] -- [[Infinity]]
27 :[[Calculus]] -- [[Vector Calculus|Vector calculus]] -- [[Mathematical analysis|Analysis]] -- [[Differential equation]] -- [[Dynamical systems and chaos theory]] -- [[List of functions]]
30 :[[Abstract algebra]] -- [[Number theory]] -- [[Algebraic geometry]] -- [[Mathematical group|Group theory]] -- [[Monoid]]s -- [[Mathematical analysis|Analysis]] -- [[Topology]] -- [[Linear algebra]] -- [[Graph theory]] -- [[Universal algebra]] -- [[Category theory]]
33 :[[Topology]] -- [[Geometry]] -- [[Trigonometry]] -- [[Algebraic geometry]] -- [[Differential geometry]] -- [[Differential topology]] -- [[Algebraic topology]] -- [[Linear algebra]]
35 :'''[[Discrete mathematics|Finite Mathematics]]'''
36 :[[Combinatorics]] -- [[Basic set theory]] -- [[Probability and statistics]] -- [[Computation|Theory of computation]] -- [[Discrete mathematics]] -- [[Cryptography]] -- [[Graph theory]] -- [[Game theory]]
38 :'''[[Applied mathematics|Applied Mathematics]]'''
39 :[[Mechanics]] -- [[Numerical analysis]] -- [[Optimization (mathematics)|Optimization]] -- [[Probability and statistics]]
41 :'''Famous Theorems and Conjectures'''
42 :[[Fermats Last Theorem|Fermat's last theorem]] -- [[Riemann hypothesis]] -- [[Continuum hypothesis]] -- [[Complexity classes P and NP|P=NP]] -- [[Goldbachs conjecture|Goldbach's conjecture]] -- [[Twin Prime Conjecture]] -- [[Goedel's incompleteness theorem|Gödel's incompleteness theorems]] -- [[Poincare conjecture|Poincaré conjecture]] -- [[Cantor's diagonal argument]] -- [[Pythagorean Theorem|Pythagorean theorem]] -- [[Central limit theorem]] -- [[Fundamental Theorem of Calculus|Fundamental theorem of calculus]] -- [[Fundamental Theorem of Algebra|Fundamental theorem of algebra]] -- [[Four color theorem]] -- [[Zorns lemma|Zorn's lemma]] -- [[The most remarkable formula in the world|"The most remarkable formula in the world"]]
44 :'''Foundations and Methods'''
45 :[[Philosophy of mathematics]] -- [[Mathematical intuitionism]] -- [[Mathematical constructivism]] -- [[Foundations of mathematics]] -- [[Set theory]] -- [[Symbolic logic]] -- [[Model theory]] -- [[Category theory]] -- [[Theorem-proving]]
47 :'''History and the World of Mathematicians'''
49 :[[History of mathematics]] -- [[Timeline of mathematics]] -- [[Mathematician|Mathematicians]] -- [[Fields Medal|Fields medal]] -- [[Millennium Prize Problems|Millennium Prize Problems (Clay Math Prize)]] -- [[International Mathematical Union]] -- [[Mathematics Competitions|Mathematics competitions]]
52 '''Further Reading:'''
53 * Davis, Philip J.; Hersh, Reuben: ''The Mathematical Experience''. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
54 * Rusin, Dave: ''The Mathematical Atlas'', http://www.math-atlas.org. A tour through the various branches of modern mathematics.
55 * Weisstein, Eric: ''World of Mathematics'', http://www.mathworld.com. An online encyclopedia of mathematics.
56 * ''Planet Math'', http://planetmath.org. An online math encyclopedia under construction. Uses the [[GFDL]] license, allowing article exchange with Wikpedia. Uses [[TeX]] markup.
57 * Mathematical Society of Japan: ''Encyclopedic Dictionary of Mathematics, 2nd ed.''. MIT Press, Cambridge, Mass., 1993. Definitions, theorems and references.
58 * Michiel Hazewinkel (ed.): ''Encyclopaedia of Mathematics''. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authorative work available. Also in paperback and on CD-ROM.
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