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

Although mathematics itself is not usually considered a natural science, 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 the structures 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. Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is considered absolute, without any reference.

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.


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

NumberNatural numberPiIntegersRational 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


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 theoryGame theory

Applied mathematics

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

MechanicsNumerical analysisOptimizationProbabilityStatisticsFinancial mathematics

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 architecture – Mathematics and education – Mathematics of musical scales

Mathematical coincidences

List of mathematical coincidences

Mathematical tools



Mathematics is not...

Mathematics is not numerology. Although numerology uses modular arithmetic to boil names and dates down to single digit numbers, numerology arbitrarily assigns emotions or traits to numbers without bothering to prove the assignments in a logical manner. Mathematics is concerned with proving or disproving ideas in a logical manner, but numerology is not. The interactions between the arbitrarily assigned emotions of the numbers are intuitively estimated rather than calculated in a thoroughgoing manner.

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.

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

Mathematics is not a science. Science has often aspired to formulate ideas into an axiomatic logical framework, like that of mathematics. However, philosophers have long ago determined that such an endeavour is not possible, or even a realistic point of view to take.


  • 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

Wikibooks provides a collection of electronic books on Mathematics.
Wikiversity, is developing courses on Mathematics.
Wikiquote has a collection of quotations about Mathematics.</div>

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