mathematics(redirected from Mathmetics)
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mathematics,deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science.
Branches of 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 rests (see logiclogic,
the systematic study of valid inference. A distinction is drawn between logical validity and truth. Validity merely refers to formal properties of the process of inference.
..... Click the link for more information. ; symbolic logicsymbolic logic
or mathematical logic,
formalized system of deductive logic, employing abstract symbols for the various aspects of natural language. Symbolic logic draws on the concepts and techniques of mathematics, notably set theory, and in turn has contributed to
..... Click the link for more information. ). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axiomsaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
..... Click the link for more information. effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of setsset,
in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
..... Click the link for more information. , originated by Georg Cantor, which now constitutes a universal mathematical language.
branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and
..... Click the link for more information. is the study of solutions of one or several algebraic equations, involving the polynomialpolynomial,
mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a1x
..... Click the link for more information. functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.
Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groupsgroup,
in mathematics, system consisting of a set of elements and a binary operation a+b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e.
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