from
The Free On-line Dictionary of Computing (8 July 2008)
real number
<mathematics> 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)