real number

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both  rational numbers, such as 42 and −23/129, and  irrational numbers, such as π and the square root of 2, and can be represented as points along an infinitely long number line.
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<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] |  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)

Noun
1. any rational or irrational number
(synonym) real
(hypernym) complex number, complex quantity, imaginary number
(hyponym) scalar product, inner product, dot product