real number definition
<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)
Nearby terms:
real hack « reality check « real mode « real number » real operating system » Real Programmer » Real Programmers Don't Use Pascal
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