real number
     n : any rational or irrational number [syn: 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)