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] | 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 | (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)