# real number

## real number

[ree-uhl, reel]
noun Mathematics.
a rational number or the limit of a sequence of rational numbers, as opposed to a complex number.
Also called real.

Origin:
1905–10

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Based on the Random House Dictionary, © Random House, Inc. 2014.
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Collins
World English Dictionary
 real number —n See number a number expressible as a limit of rational numbers

Collins English Dictionary - Complete & Unabridged 10th Edition
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American Heritage
Science Dictionary
 real number   (rē'əl)  Pronunciation Key  A number that can be written as a terminating or nonterminating decimal; a rational or irrational number. The numbers 2, -12.5, 3/7 , and pi (π) are all real numbers.
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FOLDOC
Computing Dictionary

### real number definition

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)
The Free On-line Dictionary of Computing, © Denis Howe 2010 http://foldoc.org
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Example sentences
So, you see, the real number continuum is full of holes.
These winter sports require great balance, and they can do a real number on