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al⋅ge⋅bra
[al-juh-bruh] 
| 1. | the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. |
| 2. | any of several algebraic systems, esp. a ring in which elements can be multiplied by real or complex numbers (linear algebra) as well as by other elements of the ring. |
| 3. | any special system of notation adapted to the study of a special system of relationship: algebra of classes. |
1535–45; < ML < Ar al-jabr lit., restoration
Based on the Random House Dictionary, © Random House, Inc. 2009.
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| al·ge·bra (āl'jə-brə) n.
[Middle English, bone-setting, and Italian, algebra, both from Medieval Latin, from Arabic al-jabr (wa-l-muqābala), the restoration (and the compensation), addition (and subtraction) : al-, the + jabr, bone-setting, restoration (from jabara, to set (bones), force, restore; see gpr in Semitic roots).] al'ge·bra'ist (-brā'ĭst) n. |
Copyright © 2009 by Houghton Mifflin Company.
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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. 2. A treatise on this science.Cite This Source
algebra
A branch of mathematics marked chiefly by the use of symbols to represent numbers, as in the use of a2 + b2 = c2 to express the Pythagorean theorem.
Copyright © 2005 by Houghton Mifflin Company.
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algebra
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| algebra (āl'jə-brə) Pronunciation Key
A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or quantities and express general relationships that hold for all members of a specified set. |
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algebra mathematics, logic
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)
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