witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), ringsring,
in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law; associative law; distributive law).
..... Click the link for more information. (of which the integers, or whole numbers, constitute a basic example), and fieldsfield,
in algebra, set of elements (usually numbers) that may be combined under the operations of addition and multiplication so that it constitutes an additive group, the nonzero elements form a multiplicative group, and multiplication distributes over addition.
..... Click the link for more information. (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.
Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmeticarithmetic,
branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with numbers for practical purposes, e.g.
..... Click the link for more information. and number theorynumber theory,
branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers.
..... Click the link for more information. , which are concerned with special properties of the integers—e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences—are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems.
The essential ingredient of analysisanalysis,
branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral
..... Click the link for more information. is the use of infinite processes, involving passage to a limitlimit,
in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1-2, 1-4, 1-8, 1-16, … are obviously getting smaller and smaller; since, if enough
..... Click the link for more information. . For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. . The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vectorvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
..... Click the link for more information. analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and i