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Classification is made more difficult by some subjects, often the most active, which straddle the boundary between different areas.

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A traditional division of mathematics is into pure mathematics , mathematics studied for its intrinsic interest, and applied mathematics , mathematics which can be directly applied to real world problems. Broad divisions, such as discrete mathematics and computational mathematics , have emerged more recently. An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline.

For example, the Langlands program has found unexpected connections between areas previously thought unconnected, at least Galois groups , Riemann surfaces and number theory.


Number theory is the study of numbers and the properties of operations between them. Number theory is traditionally concerned with the properties of integers, but more recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. The study of structure begins with numbers , first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of these numbers are studied in number theory.

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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 everyday numbers. Long standing questions about compass and straightedge construction were finally settled by Galois theory. The physically important concept of vectors , generalized to vector spaces , is studied in linear algebra.

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Themes common to all kinds of algebraic structures are studied in universal algebra. Combinatorics is the study of finite or discrete collections of objects that satisfy specified criteria.

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In particular, it is concerned with "counting" the objects in those collections enumerative combinatorics and with deciding whether certain "optimal" objects exist extremal combinatorics. It includes graph theory , used to describe inter-connected objects a graph in this sense is a network, or collection of connected points. See also the list of combinatorics topics , list of graph theory topics and glossary of graph theory. A combinatorial flavour is present in many parts of problem-solving.

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Geometry deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics. Within the world of mathematics, analysis is the branch that focuses on change: rates of change , accumulated change , and multiple things changing relative to or independently of one another.

Modern analysis is a vast and rapidly expanding branch of mathematics that touches almost every other subdivision of the discipline, finding direct and indirect applications in topics as diverse as number theory , cryptography , and abstract algebra. It is also the language of science itself and is used across chemistry , biology , and physics , from astrophysics to X-ray crystallography.

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This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece , and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.

A separate article, South Asian mathematics , focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.

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  5. The substantive branches of mathematics are treated in several articles. See algebra ; analysis ; arithmetic ; combinatorics ; game theory ; geometry ; number theory ; numerical analysis ; optimization ; probability theory ; set theory ; statistics ; trigonometry. You are using an outdated browser.

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