Algebra and Calculus

Algebra

Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.

algebra n : the mathematics of generalized arithmetical operations

Algebra \Al"ge*bra\, n. [LL. algebra, fr. Ar. al-jebr reduction of parts to a whole, or fractions to whole numbers, fr. jabara to bind together, consolidate; al-jebr w'almuq[=a]balah reduction and comparison (by equations): cf. F. alg[`e]bre, It. & Sp. algebra.] 1. (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude. [1913 Webster] 2. A treatise on this science. [1913 Webster]

Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic}, and {-ics}.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations. [1913 Webster] Note: Mathematics embraces three departments, namely: 1. {Arithmetic}. 2. {Geometry}, including {Trigonometry} and {Conic Sections}. 3. {Analysis}, in which letters are used, including {Algebra}, {Analytical Geometry}, and {Calculus}. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations. [1913 Webster]

algebra 1. A loose term for an {algebraic structure}. 2. A {vector space} that is also a {ring}, where the vector space and the ring share the same addition operation and are related in certain other ways. An example algebra is the set of 2x2 {matrices} with {real numbers} as entries, with the usual operations of addition and matrix multiplication, and the usual {scalar} multiplication. Another example is the set of all {polynomials} with real coefficients, with the usual operations. In more detail, we have: (1) an underlying {set}, (2) a {field} of {scalars}, (3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a {vector space}, (4) an operation of addition of members of the underlying set, whose input is an {ordered pair} of such members and whose output is one such member, just as in a vector space or a ring, (5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring. This whole thing constitutes an `algebra' iff: (1) it is a vector space if you discard item (5) and (2) it is a ring if you discard (2) and (3) and (3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example. Another example (an example of a {Banach algebra}) is the set of all {bounded} {linear operators} on a {Hilbert space}, with the usual {norm}. The multiplication is the operation of {composition} of operators, and the addition and scalar multiplication are just what you would expect. Two other examples are {tensor algebras} and {Clifford algebras}. [I. N. Herstein, "Topics_in_Algebra"]. (1999-07-14)

Calculus

Calculus (Latin, , a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

calculus n 1: a hard lump produced by the concretion of mineral salts; found in hollow organs or ducts of the body; "renal calculi can be very painful" [syn: {concretion}] 2: an incrustation that forms on the teeth and gums [syn: {tartar}, {tophus}] 3: the branch of mathematics that is concerned with limits and with the differentiation and integration of functions [syn: {the calculus}, {infinitesimal calculus}] [also: {calculi} (pl)]

Calculus \Cal"cu*lus\, n.; pl. {Calculi}. [L, calculus. See {Calculate}, and {Calcule}.] 1. (Med.) Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc. [1913 Webster] 2. (Math.) A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation. [1913 Webster] {Barycentric calculus}, a method of treating geometry by defining a point as the center of gravity of certain other points to which co["e]fficients or weights are ascribed. {Calculus of functions}, that branch of mathematics which treats of the forms of functions that shall satisfy given conditions. {Calculus of operations}, that branch of mathematical logic that treats of all operations that satisfy given conditions. {Calculus of probabilities}, the science that treats of the computation of the probabilities of events, or the application of numbers to chance. {Calculus of variations}, a branch of mathematics in which the laws of dependence which bind the variable quantities together are themselves subject to change. {Differential calculus}, a method of investigating mathematical questions by using the ratio of certain indefinitely small quantities called differentials. The problems are primarily of this form: to find how the change in some variable quantity alters at each instant the value of a quantity dependent upon it. {Exponential calculus}, that part of algebra which treats of exponents. {Imaginary calculus}, a method of investigating the relations of real or imaginary quantities by the use of the imaginary symbols and quantities of algebra. {Integral calculus}, a method which in the reverse of the differential, the primary object of which is to learn from the known ratio of the indefinitely small changes of two or more magnitudes, the relation of the magnitudes themselves, or, in other words, from having the differential of an algebraic expression to find the expression itself. [1913 Webster]

Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic}, and {-ics}.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations. [1913 Webster] Note: Mathematics embraces three departments, namely: 1. {Arithmetic}. 2. {Geometry}, including {Trigonometry} and {Conic Sections}. 3. {Analysis}, in which letters are used, including {Algebra}, {Analytical Geometry}, and {Calculus}. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations. [1913 Webster]

Data Sources:

  • algebra: WordNet (r) 2.0
  • algebra: The Collaborative International Dictionary of English v.0.44
  • algebra: The Collaborative International Dictionary of English v.0.44
  • algebra: The Free On-line Dictionary of Computing (27 SEP 03)
  • calculus: WordNet (r) 2.0
  • calculus: The Collaborative International Dictionary of English v.0.44
  • calculus: The Collaborative International Dictionary of English v.0.44

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