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Encyclopedia > Mathematics


Mathematics is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation. It is commonly defined as the study of patterns of structure, change, and space; even more informally, one might say it is the study of "figures and numbers". Because it is not empirical, it is not a science.

Mathematical knowledge is constantly growing, through research and application. Mathematics is usually regarded as a tool for science, even though the development of mathematics is not necessarily done with science in mind.

The specific structures that are investigated by mathematicians sometimes do have their origin in natural and social sciences, including physics and economics. Some contemporary mathematics also has its origins in computer science and communication theory.

In the formalist view, widely accepted as a description by professionals in the field, the definition used is the one given at the beginning of this article. Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. There are other views, and some are described in the article on the philosophy of mathematics.

Mathematics itself is usually considered absolute, without any reference. Mathematicians define and investigate some structures for reasons purely internal to mathematics; they may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians work for purely aesthetic reasons, viewing mathematics more as an art ('pure mathematics') rather than for its practical application ('applied mathematics'); this is the same kind of motivation as poets and philosophers may experience, and no more explicable. Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions.

Mathematics is often abbreviated as math (American English) 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 predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.

The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. 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 rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically 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 also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The modern fields 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 of space and structure. Topology connects the study of space and the study of change by 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 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 numbers 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. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable 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 concepts were 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 computer science.

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 is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.

Topics in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of subfields and topics reflects one organizational view of mathematics. For a fuller treatment, see Areas of mathematics


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

NumberNatural numberIntegersRational numbersReal numbersComplex numbersHypercomplex numbersQuaternionsOctonionsSedenionsHyperreal numbersSurreal numbersOrdinal numbersCardinal numbersp-adic numbersInteger sequencesMathematical constantsNumber namesInfinityBase


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

ArithmeticCalculusVector calculusAnalysisDifferential equationsDynamical systems and chaos theoryList of functions


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

Abstract algebraNumber theoryAlgebraic geometryGroup theoryMonoidsAnalysisTopologyLinear algebraGraph theoryUniversal algebraCategory theoryOrder theory

Spatial relations

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

TopologyGeometryTrigonometryAlgebraic geometryDifferential geometryDifferential topologyAlgebraic topologyLinear algebraFractal geometry

Discrete mathematics

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

CombinatoricsNaive set theoryProbabilityTheory of computationFinite mathematicsCryptographyGraph theory

Applied mathematics

Fields in applied mathematics use knowledge of mathematics to solve real world problems.

MechanicsNumerical analysisOptimizationProbabilityStatisticsFinancial mathematicsGame theoryMathematical biologyCryptographyInformation theoryFluid dynamics

Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.

Pythagorean theoremFermat's last theoremGoldbach's conjectureTwin Prime ConjectureGödel's incompleteness theoremsPoincaré conjectureCantor's diagonal argumentFour color theoremZorn's lemmaEuler's identityScholz ConjectureChurch-Turing thesis

Important theorems and conjectures

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

Riemann hypothesisContinuum hypothesisP=NPPythagorean theoremCentral limit theoremFundamental theorem of calculusFundamental theorem of algebraFundamental theorem of arithmeticFundamental theorem of projective geometryclassification theorems of surfacesGauss-Bonnet theorem

Foundations and methods

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

Philosophy of mathematicsMathematical intuitionismMathematical constructivismFoundations of mathematicsSet theorySymbolic logicModel theoryCategory theoryLogicReverse MathematicsTable of mathematical symbols

History and the world of mathematicians

History of mathematicsTimeline of mathematicsMathematiciansFields medalAbel PrizeMillennium Prize Problems (Clay Math Prize)International Mathematical UnionMathematics competitionsLateral thinkingMathematical abilities and gender issues

Mathematics and other fields

Mathematics and architectureMathematics and educationMathematics of musical scales

Mathematical coincidences

List of mathematical coincidences

Mathematical tools



Mathematics is not...

Although Einstein called it "the Queen of the Sciences", mathematics itself is not a science, because it is not empirical.

Mathematics is not physics, as physics is a science and mathematics is not.

Mathematics is not numerology. Although numerology uses modular arithmetic to reduce names and dates down to numbers, numerology assigns emotions or traits to these numbers without proving the assignments in a logical manner, or providing exact definitions for the emotions or traits. The interactions between the assigned emotions of the numbers are established by intuitive estimation rather than rigorous calculation.

Mathematics is not accountancy. Although arithmetic computation is crucial to the work of accountants, they are mainly concerned with proving that the computations are true and correct through a system of doublechecks. 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 of concrete bookkeeping.


  • 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.
  • Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary 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

Wikibooks has more about this subject:
Wikibooks Wikiversity has more about this subject:
Wikiquote has a collection of quotations related to:
  • Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles (http://www.cut-the-knot.org/). A collection of articles on various math topics, with interactive Java illustrations
  • Rusin, Dave: The Mathematical Atlas (http://www.math-atlas.org/). A guided tour through the various branches of modern mathematics.
  • Stefanov, Alexandre: Textbooks in Mathematics (http://us.geocities.com/alex_stef/mylist.html). A list of free online textbooks and lecture notes in mathematics.
  • Weisstein, Eric et al.: MathWorld: World of Mathematics (http://www.mathworld.com/). An online encyclopedia of mathematics, focusing on classical mathematics.
  • A mathematical thesaurus maintained by the NRICH (http://nrich.maths.org/) project at the University of Cambridge (UK), Connecting Mathematics (http://thesaurus.maths.org/)
  • Planet Math (http://planetmath.org/). An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
  • Mathforge (http://www.mathforge.net/). A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
  • Young Mathematicians Network (YMN) (http://www.youngmath.net/concerns). A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
  • Metamath (http://metamath.org/). A site and a language, that formalize math from its foundations.
  • Math in the Movies (http://world.std.com/~reinhold/dir/mathmovies.html). A guide to major motion pictures with scenes of real mathematics
  • Mathematics in fiction (http://math.cofc.edu/faculty/kasman/MATHFICT/default.html). Links to works of fiction that refer to mathematics or mathematicians.



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