Index using other classification systems
Here are the subdivisions of mathematics classified according to some other systems found. These are comparatively small systems (few divisions) with limited or no known application to sorting data. You can go directly back to the index arranged according to the AMS/Zbl Mathematics Subject Classification system used to organize materials at this site.
Thes are the divisions of the branches of mathematics used by the American Mathematical Society to track the employment of new Ph.D.'s. The "Field of Thesis" record uses the following divisions in 1998 (shown here with number of new Ph.D.'s in the US in each field); here the fields are sorted to mimic the divisions used in our tour.
To get another perspective, consider the division into branches of mathematics taken from the entry, "Mathematics (Branches)" of The Concise Columbia Electronic Encyclopedia, Third Edition:
There is a longer article with the Columbia copyright at InformationPlease.com
Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics.
The term foundations is used to refer to the formulation and analysis of the language, AXIOMS, and logical methods on which all of mathematics rest (see LOGIC); SET theory, originated by Georg CANTOR, now constitutes a universal mathematical language.
ALGEBRA, historically, is the study of solutions of one or several algebraic equations, involving POLYNOMIAL functions of one or several variables; ARITHMETIC and NUMBER THEORY are areas of algebra concerned with special properties of the integers.
ANALYSIS applies the concepts and methods of the CALCULUS to various mathematical entities.
GEOMETRY is concerned with the spatial side of mathematics, i.e., the properties of and relationships between points, lines, planes, figures, solids, and surfaces. TOPOLOGY studies the structures of geometric objects in a very general way.
The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes COMPUTER science, mathematical physics, PROBABILITY theory, and mathematical STATISTICS.
Mathematics is often defined as the study of quantity, magnitude, and relations of numbers or symbols. It embraces the subjects of arithmetic, geometry, algebra, calculus, probability, statistics, and many other special areas of research.
There are two major divisions of mathematics: pure and applied. Pure mathematics investigates the subject solely for its theoretical interest. Applied mathematics develops tools and techniques for solving specific problems of business and engineering or for highly theoretical applications in the sciences...
Among the major subdivisions of modern mathematics are the following:
Arithmetic ... is the study of the nature and properties of numbers. It includes study of the algorithms of calculation with numbers, namely the basic operations of addition, subtraction, multiplication, and division, as well as the taking of powers and roots.
...Modern algebra, or abstract algebra, is a more general branch of mathematics that analyzes algebraic axioms and operations with arbitrary sets of symbols. Special areas of abstract algebra include the study of groups, rings, fields, the algebra of matrices, and a large variety of nonassociative and noncommutative algebras. Special algebras of sets and vectors and Boolean algebras arise in the study of logic (see Boole).
...In the 19th century, Euclidean geometry's status as the primary geometry was challenged by the discovery of non-Euclidean geometries. These inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines. This led to many new geometries, including elliptical, hyperbolic, and parabolic geometries. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures. Projective geometry, for example, is an abstract geometry concerned with the geometric properties that remain invariant under the projection of figures onto other figures, as in the case of mathematical perspective.
A very useful approach to geometry is found in topology, the study of the properties of a geometric figure that remain the same when a figure is subjected to continuous transformation without loss of identity of any of its parts. Differential geometry is the study of geometry in terms of infinitesimals.
Analytic Geometry and Trigonometry
Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Analytic geometry addresses geometric problems from an algebraic point of view by associating any curve with variables by means of a coordinate system...
Trigonometry is the study of triangles, angles, and their relations. It also involves the study of trigonometric functions...
...In the 19th century, in response to questions about its rigorous foundations, the calculus was developed in terms of a theory of limits. Analysis--differential and integral calculus--was subsequently approached even more rigorously by those who sought to establish its results by strictly arithmetic means. This required an exact definition of the continuity of the real numbers. Others extended the power of analysis with very general theories of measure.
...Analysis gives primary emphasis to functions, convergence of sequences, series, continuity, differentiability, and questions about the completeness of the real numbers.
Complex analysis extends the methods of analysis from real to complex variables... [T]hey are especially useful in applications whenever two variables must be treated simultaneously. For example, complex analysis has proven particularly valuable in applications to fluid dynamics, where both pressure and velocity vary from point to point. Complex numbers were made more acceptable to many in the 19th century when they were given a geometric interpretation.
...This branch of mathematics involves the study of the properties of numbers and the structure of different number systems. It is concerned with integers, or whole numbers. Many problems in number theory deal with prime numbers...
Questions about highest common factors, least common multiples, decompositions into primes, and the representation of natural numbers in certain forms as well as their divisibility are all the province of number theory. Computers have recently been applied to the solution of certain number-theory problems.
Probability Theory and Statistics
The branch of mathematics concerned with the analysis of random phenomena is called probability theory. The entire set of possible outcomes of a random event is called the sample space. Each outcome in this space is assigned a probability, a number...
Statistics applies probability theory to real cases and involves the analysis of empirical data... general techniques for analyzing data and computing various values, drawing correlations, using methods of sampling, counting, estimating, and ranking data according to certain criteria.
Created in the 19th century by the German mathematician Georg Cantor, set theory was originally meant to provide techniques for the mathematical analysis of the infinite. Set theory deals with the properties of well-defined collections of objects...
Early in the 20th century certain contradictions of set theory concerning infinite sets, transfinite numbers, and purely logical paradoxes brought about attempts to axiomatize set theory in hopes of eliminating such difficulties...
Logic is the study of the way in which valid conclusions may be drawn from given premises... Modern logicians use algebraic and formal methods to study the relations between logical propositions. This has led to model theory and model logic.
Compton's Encyclopedia Online v3.0 © 1998 The Learning Company, Inc.