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4 definitions found
 for Algebra
From The Collaborative International Dictionary of English v.0.48 :

  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 :

  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) 3.0 (2006) :

  algebra
      n 1: the mathematics of generalized arithmetical operations

From The Free On-line Dictionary of Computing (30 December 2018) :

  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)
  

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