<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} "real number", where a real number 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, areal numberb and creal numberd implies a+creal numberb+d; likewise a.creal numberb.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)

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials.

In mathematics, the set of positive real numbers, \R_{>0} = \left\{ x \in \R \mid x > 0 \right\}, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_{\geq 0} = \left\{ x \in \R \mid x \geq 0 \right\}, also include zero.

In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers. Equivalent conditions are that F is generated over Q by one root of an integer polynomial P, all of the roots of P being real; or that the tensor product algebra of F with the real field, over Q, is isomorphic to a tensor power of R.

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language.

The most negative value, not necessarily or even usually the simple negation of plus infinity. In N bit twos-complement arithmetic, infinity is 2^(N-1) - 1 but minus infinity is -(2^(N-1)), not -(2^(N-1) - 1).

In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals.

In mathematics, a negative number represents an opposite."Integers are the set of whole numbers and their opposites.

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value.

In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W.