# real number

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## real number:

see number**number,**

entity describing the magnitude or position of a mathematical object or extensions of these concepts.

**The Natural Numbers**

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their

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## Real Number

any positive number, negative number, or zero. Real numbers are divided into rational and irrational numbers. A rational number is representable both in the form of a rational fraction, that is, the fraction *p/q*, where *p* and *q* are integers and *q* ≠ 0, and in the form of a finite or infinite repeating decimal. An irrational number is representable only in the form of an infinite nonrepeating decimal.

A rigorous theory of real numbers, which makes it possible to define irrational numbers by proceeding from rational numbers, was developed only in the second half of the 19th century by K. Weierstrass, R. Dedekind, and G. Cantor. The set of all real numbers is called the number line and is designated by *R*. This set is ordered linearly and forms a field with respect to the fundamental arithmetic operations (addition and multiplication). The set of rational numbers is everywhere dense in *R*, and *R* is its completion. The number line *R* is similar to a geometric line, in the sense that it is possible a one-to-one order preserving correspondence between the numbers of *R* and the points on a line. The most important feature of the number line is its continuity. The principle of the continuity of the number line has several different formulations owing to Weierstrass, Dedekind, and Cantor. The Weierstrass principle: any nonempty set of numbers bounded from above has a (unique) least upper bound. The Dedekind principle: any cut in the domain of real numbers has a boundary. The Cantor principle (the principle of nested sequences): the intersection of a nested sequence of intervals {[*a _{n}*,

*b*]} of the number whose lengths tend to zero consists of a single real number.

_{n}The theory of real numbers is one of the most important issues of mathematics. The properties of the number line are the foundation of the theory of limits and thus of the entire structure of modern mathematical analysis.

S. B. STECHKIN

## real number

[′rēl ′nəm·bər]## real number

## real number

(mathematics)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] | < 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.