Algebra

4 definitions found

From The Collaborative International Dictionary of English v.0.48 [gcide]:

  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]

From The Collaborative International Dictionary of English v.0.48 [gcide]:

  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] Algebraic

From WordNet (r) 2.0 [wn]:

  algebra
       n : the mathematics of generalized arithmetical operations

From The Free On-line Dictionary of Computing (27 SEP 03) [foldoc]:

  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)