The real number system can be defined axiomatically up to an isomorphism, which is described below. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing. Axiomatic approach
Let R denote the set of all real numbers. Then:
* The set R is a field, meaning that addition and multiplication are defined and have the usual properties. * The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z: * if x ≥ y then x + z ≥ y + z;
* if x ≥ 0 and y ≥ 0 then xy ≥ 0.
* The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object. For another axiomatization of R, see Tarski's axiomatization of the reals. Construction from the rational numbers
The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like (3, 3.1, 3.14, 3.141, 3.1415,...) converges to a unique real number. For details and other constructions of real numbers, see construction of the real numbers.
A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to...