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#### Definitions of algebra

1. the mathematics of generalized arithmetical operations Scrapingweb Dictionary DB
2. A treatise on this science. Webster Dictionary DB
3. That branch of mathematics which treats of the relations of the sides and angles of triangles, which the methods of deducing from certain given parts other required parts, and also of the general relations which exist between the trigonometrical functions of arcs or angles. Webster Dictionary DB
4. A branch of mathematics using letters and other symbols to represent quantities. The Winston Simplified Dictionary. By William Dodge Lewis, Edgar Arthur Singer. Published 1919.
5. The science of calculating by symbols, thus forming a kind of universal arithmetic. . The american dictionary of the english language. By Daniel Lyons. Published 1899.
6. Algebraic. The Clarendon dictionary. By William Hand Browne, Samuel Stehman Haldeman. Published 1894.
7. The science of quantity in general; calculation by symbols. The Clarendon dictionary. By William Hand Browne, Samuel Stehman Haldeman. Published 1894.
8. Mathematical calculation by letters and symbols; a treatise on this branch. The Concise Standard Dictionary of the English Language. By James Champlin Fernald. Published 1919.
9. Universal arithmetic, in which symbols are employed to denote operation, and letters to represent number and quantity. Nuttall's Standard dictionary of the English language. By Nuttall, P.Austin. Published 1914.
10. 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. Webster Dictionary DB
11. Arithmetic by signs-commonly the letters of the alphabet-the first letters, a, b, c, d, &c., represent known quantities, and the last letters, w, x, y, z, unknown quantities. Etymological and pronouncing dictionary of the English language. By Stormonth, James, Phelp, P. H. Published 1874.
12. 1. A loose term for an algebraicstructure.2. A vector space that is also a ring, where the vectorspace and the ring share the same addition operation and arerelated in certain other ways.An example algebra is the set of 2x2 matrices with realnumbers as entries, with the usual operations of addition andmatrix multiplication, and the usual scalar multiplication.Another example is the set of all polynomials with realcoefficients, 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 ascalar and a member of the underlying set and whose output isa 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 whoseoutput is one such member, just as in a vector space or aring, (5) an operation of multiplication of members of theunderlying set, whose input is an ordered pair of such membersand 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 theunderlying set we have r(AB) = (rA)B = A(rB). In other wordsit doesn't matter whether you multiply members of the algebrafirst and then multiply by the scalar, or multiply one of themby the scalar first and then multiply the two members of thealgebra. Note that the A comes before the B because themultiplication is in some cases not commutative, e.g. thematrix example.Another example (an example of a Banach algebra) is the setof all bounded linear operators on a Hilbert space, withthe usual norm. The multiplication is the operation ofcomposition of operators, and the addition and scalarmultiplication are just what you would expect.Two other examples are tensor algebras and Cliffordalgebras.[I. N. Herstein, "Topics in Algebra"]. foldoc_fs
13. al'je-bra, n. a method of calculating by symbols--by means of letters employed to represent the numbers, and signs to represent their relations, thus forming a kind of universal arithmetic.--adjs. ALGEBR[=A]'IC, -AL, pertaining to algebra.--n. ALGEBR[=A]'IST, one skilled in algebra. [It. and Sp., from Ar. al-jebr, the resetting of anything broken, hence combination; jabara, to reunite.] gutenberg.org/ebooks/37683
14. Investigation of the properties of numbers by means of general symbols; quadruple a., quaternions. Henee algebraic (AL) aa., algebraically adv., algebraist, algebrist, nn. Concise Oxford Dictionary
15. n. [Arabic] The method of investigating the relations and properties of numbers and quantities by means of letters and other symbols. Cabinet Dictionary
16. A peculiar kind of arithmetick . Complete Dictionary

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