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)
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