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

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)