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 irrationalnumbers.

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 equivalencerelation "~", 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)