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
Strictly, real numbers are the equivalence classes
of the Cauchy sequences
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.