Real number
Other properties

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

The real numbers form a metric space: the distance between x and y is defined as the absolute value | x − y |. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in R {\displaystyle \mathbb {R} }, although no negative number does. This shows that the order on R {\displaystyle \mathbb {R} } is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R {\displaystyle \mathbb {R} } the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R {\displaystyle \mathbb {R} }. Ordered fields that satisfy the same first-order sentences as R {\displaystyle \mathbb {R} } are called nonstandard models of R {\displaystyle \mathbb {R} }. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R {\displaystyle \mathbb {R} }), we know that the same statement must also be true of R {\displaystyle \mathbb {R} }.

The field R {\displaystyle \mathbb {R} } of real numbers is an extension field of the field Q {\displaystyle \mathbb {Q} } of rational numbers, and R {\displaystyle \mathbb {R} } can therefore be seen as a vector space over Q {\displaystyle \mathbb {Q} }. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R {\displaystyle \mathbb {R} } with the property that every nonempty subset of R {\displaystyle \mathbb {R} } has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.

A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.
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