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2 definitions found
 for real number
From WordNet (r) 3.0 (2006) :

  real number
      n 1: any rational or irrational number [syn: real number,
           real]

From The Free On-line Dictionary of Computing (18 March 2015) :

  real number
  
      One of the infinitely divisible range of values
     between positive and negative infinity, used to represent
     continuous physical quantities such as distance, time and
     temperature.
  
     Between any two real numbers there are infinitely many more
     real numbers.  The integers ("counting numbers") are real
     numbers with no fractional part and real numbers ("measuring
     numbers") are complex numbers with no imaginary part.  Real
     numbers can be divided into rational numbers and irrational
     numbers.
  
     Real numbers are usually represented (approximately) by
     computers as floating point numbers.
  
     Strictly, real numbers are the equivalence classes of the
     Cauchy sequences of rationals under the equivalence
     relation "~", where a ~ b if and only if a-b is Cauchy with
     limit 0.
  
     The real numbers are the minimal topologically closed
     field containing the rational field.
  
     A sequence, r, of rationals (i.e. a function, r, from the
     natural numbers to the rationals) is said to be Cauchy
     precisely if, for any tolerance delta there is a size, N,
     beyond which: for any n, m exceeding N,
  
      | r[n] - r[m] | < delta
  
     A Cauchy sequence, r, has limit x precisely if, for any
     tolerance delta there is a size, N, beyond which: for any n
     exceeding N,
  
      | r[n] - x | < delta
  
     (i.e. r would remain Cauchy if any of its elements, no matter
     how late, were replaced by x).
  
     It is possible to perform addition on the reals, because the
     equivalence class of a sum of two sequences can be shown to be
     the equivalence class of the sum of any two sequences
     equivalent to the given originals: ie, a~b and c~d implies
     a+c~b+d; likewise a.c~b.d so we can perform multiplication.
     Indeed, there is a natural embedding of the rationals in the
     reals (via, for any rational, the sequence which takes no
     other value than that rational) which suffices, when extended
     via continuity, to import most of the algebraic properties of
     the rationals to the reals.
  
     (1997-03-12)
  

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