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Mathematics is commonly defined as the study of patterns ofstructure, change , and space ; more informally,one might say it is the study of "figures and numbers". In the formalist view, itis the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation ; other views are described in Philosophy of mathematics . Mathematics might beseen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for thepurpose of describing and exploring physical and conceptual relationships.

Although mathematics itself is not usually considered a naturalscience , the specific structures that are investigated by mathematicians often have their origin in the natural sciences,most commonly in physics . However, mathematicians also define and investigate structures for reasons purely internal to mathematics, because thestructures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations.Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science . Einsteinreffered to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is consideredabsolute, without any reference.

Mathematics is often abbreviated as math ( AmericanEnglish ) or maths ( British English ).


Overview and history of mathematics

See the article on the history of mathematics for details.

The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning";μαθηματικός (mathematikós) means "fond of learning".

The major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predictastronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study ofstructure, space and change.

The study of structure starts with numbers , first the familiar natural numbers and integers andtheir arithmetical operations, which are recorded in elementary algebra . The deeper properties of whole numbers are studiedin number theory . The investigation of methods to solve equations leadsto the field of abstract algebra , which, among other things,studies rings and fields , structures that generalize the properties possessed by the familiar numbers. Thephysically important concept of vectors , generalized to vector spaces and studied in linear algebra , belongs to the two branches of structure and space.

The study of space originates with geometry , first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and less dimensions), later alsogeneralized to non-Euclidean geometries which play acentral role in general relativity . Several long standingquestions about ruler and compassconstructions were finally settled by Galois theory . The modernfields of differential geometry and algebraic geometry generalize geometry in different directions:differential geometry emphasizes the concepts of functions, fiber bundles , derivatives , smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies ofspace and structure. Topology connects the study of space and the study of changeby focusing on the concept of continuity .

Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that. The central concept usedto 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 ofreal-valued functions is known as real analysis . For several reasons, itis convenient to generalise to the complex numbers which are studied in complex analysis . Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, layingthe groundwork for quantum mechanics among many other things.Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibitunpredictable yet deterministic behavior.

In order to clarify and investigate the foundations of mathematics, the fields of set theory , mathematical logic and model theory were developed.

When computers were first conceived, several essential theoretical conceptswere shaped 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 . Discrete mathematics is the common name for those fields of mathematics useful in computerscience.

An important field in applied mathematics is statistics , which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and isused in all sciences. Numerical analysis investigates themethods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.

Topics in mathematics

An alphabetical and subclassified list ofmathematical topics is available. The following list of subfields and topics reflects one organizational view ofmathematics.


In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find suchmeasurements.

Number - Naturalnumber - Pi - Integers - Rational numbers - Real numbers - Complex numbers - Hypercomplex numbers - Quaternions - Octonions - Sedenions - Hyperreal numbers - Surreal numbers - Ordinal numbers - Cardinal numbers - p-adic numbers - Integer sequences - Mathematicalconstants - Number names - Infinity - Base


These topics give ways to measure change in mathematical functions, and changes between numbers.

Arithmetic - Calculus - Vector calculus - Analysis - Differential equations - Dynamical systems and chaos theory - List of functions


These branches of mathematics measure size and symmetry of numbers, and various constructs.

Abstract algebra - Number theory - Algebraic geometry - Group theory - Monoids - Analysis - Topology - Linearalgebra - Graph theory - Universal algebra - Categorytheory - Order theory


These topics tend to quantify a more visual approach to mathematics than others.

Topology - Geometry - Trigonometry - Algebraic geometry - Differential geometry - Differential topology - Algebraictopology - Linear algebra - Fractal geometry

Discrete mathematics

Topics in discrete mathematics deal with branches ofmathematics with objects that can only take on specific, separated values.

Combinatorics - Naive set theory - Probability - Theory of computation - Finite mathematics - Cryptography - Graph theory - Gametheory

Applied mathematics

Fields in applied mathematics use knowledge of mathematicsto real world problems.

Mechanics - Numerical analysis - Optimization - Probability - Statistics - Financial mathematics

Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.

Fermat's last theorem - Goldbach's conjecture - Twin Prime Conjecture - Gödel's incompleteness theorems - Poincaré conjecture - Cantor's diagonal argument - Four color theorem - Zorn's lemma - Euler's identity - Scholz Conjecture - Church-Turing thesis

Important theorems

These are theorems that have changed the face of mathematics throughout history.

Riemann hypothesis - Continuum hypothesis - P=NP - Pythagorean theorem - Central limit theorem - Fundamental theorem of calculus - Fundamental theorem of algebra - Fundamental theorem of arithmetic - Fundamental theoremof projective geometry - classification theorems of surfaces - Gauss-Bonnet theorem

Foundations and methods

Such topics are approaches to mathematics, and influence the way mathematicians study their subject.

Philosophy of mathematics - Mathematical intuitionism - Mathematical constructivism - Foundations of mathematics - Set theory - Symboliclogic - Model theory - Category theory - Theorem-proving - Logic - Reverse Mathematics - Table of mathematical symbols

History and the world of mathematicians

History of mathematics - Timeline of mathematics - Mathematicians - Fields medal - Abel Prize - Millennium Prize Problems (Clay Math Prize) - International Mathematical Union - Mathematics competitions - Lateral thinking - Mathematical abilities andgender issues

Mathematics and other fields

Mathematics and architecture - Mathematicsand education - Mathematics ofmusical scales

Mathematical coincidences

List of mathematicalcoincidences

Mathematical tools




Referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequencesthereof are then logically derived, Bertrand Russell said:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we aresaying is true.

This may explain why John Von Neumann once said:

In mathematics you don't understand things. You just get used to them.

About the beauty of Mathematics, Bertrand Russell said inStudy of Mathematics:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that ofsculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimelypure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, thesense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely aspoetry.

Elucidating the symmetry between the creative and logical aspects of mathematics, W.S. Anglin observed, in Mathematics andHistory:

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where theexplorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers havegone elsewhere.

Mathematics is not...

Mathematics is not numerology . Although numerology uses modular arithmeticto boil names and dates down to single digit numbers, numerology arbitrarily assigns emotions or traits to numbers withoutbothering to prove the assignments in a logical manner. Mathematics is concerned with proving or disproving ideas in a logicalmanner, but numerology is not. The interactions between the arbitrarily assigned emotions of the numbers are intuitivelyestimated rather than calculated in a thoroughgoing manner.

Mathematics is not accountancy . Although arithmetic computation is crucialto the work of accountants, they are mainly concerned with proving that the computations are true and correct through a system ofdoublechecks. The proving or disproving of hypotheses is very important to mathematicians, but not so much to accountants.Advances in abstract mathematics are irrelevant to accountancy if the discoveries can't be applied to improving the efficiency ofconcrete bookkeeping.

Mathematics is not physics , despite the number of historical and philosophicalrelations between the two.


  • Courant, R. and H. Robbins, What Is Mathematics? ( 1941 );
  • Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980 . A gentle introduction to the world of mathematics.
  • Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996 . An encyclopedic overview of mathematics presented in clear, simple language.
  • Hazewinkel, Michiel (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 authoritative work available. Also in paperback and on CD-ROM.
  • Kline, M., Mathematical Thought from Ancient to Modern Times ( 1973 );

External links

Our sister project, Wikibooks , provides a collection of electronic books on Mathematics .

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