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However, an ordered group in this case, the additive group of the field defines a uniform structure, and uniform structures have a notion of completeness topology ; the description in the previous section Completeness is a special case.
We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces , since the definition of metric space relies on already having a characterization of the real numbers.
It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field , and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field".
Every uniformly complete Archimedean field must also be Dedekind-complete and vice versa , justifying using "the" in the phrase "the complete Archimedean field".
This sense of completeness is most closely related to the construction of the reals from Cauchy sequences the construction carried out in full in this article , since it starts with an Archimedean field the rationals and forms the uniform completion of it in a standard way.
But the original use of the phrase "complete Archimedean field" was by David Hilbert , who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R.
Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field.
This sense of completeness is most closely related to the construction of the reals from surreal numbers , since that construction starts with a proper class that contains every ordered field the surreals and then selects from it the largest Archimedean subfield.
The reals are uncountable ; that is: In fact, the cardinality of the reals equals that of the set of subsets i. Since the set of algebraic numbers is countable, almost all real numbers are transcendental.
The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. 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: 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 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 , although no negative number does. This shows that the order on R is determined by its algebraic structure.
Also, every polynomial of odd degree admits at least one real root: 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 [0;1] has measure 1.
There exist sets of real numbers that are not Lebesgue measurable, e. 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 set of hyperreal numbers satisfies the same first order sentences as R.
Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of 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 , we know that the same statement must also be true of R.
The field R of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen as a vector space over Q.
Zermelo—Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: 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: Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described.
A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not. The real numbers are most often formalized using the Zermelo—Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics.
In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics. The hyperreal numbers as developed by Edwin Hewitt , Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz , Euler , Cauchy and others.
Paul Cohen proved in that it is an axiom independent of the other axioms of set theory; that is: In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers.
In fact, the fundamental physical theories such as classical mechanics , electromagnetism , quantum mechanics , general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces , that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.
Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.
With some exceptions , most calculators do not operate on real numbers. Instead, they work with finite-precision approximations called floating-point numbers.
In fact, most scientific computation uses floating-point arithmetic. Real numbers satisfy the usual rules of arithmetic , but floating-point numbers do not.
Computers cannot directly store arbitrary real numbers with infinitely many digits. The achievable precision is limited by the number of bits allocated to store a number, whether as floating-point numbers or arbitrary-precision numbers.
A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms,  but an uncountable number of reals, almost all real numbers fail to be computable.
Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable.
The set of definable numbers is broader, but still only countable. In set theory , specifically descriptive set theory , the Baire space is used as a surrogate for the real numbers since the latter have some topological properties connectedness that are a technical inconvenience.
Elements of Baire space are referred to as "reals". As this set is naturally endowed with the structure of a field , the expression field of real numbers is frequently used when its algebraic properties are under consideration.
The notation R n refers to the cartesian product of n copies of R , which is an n - dimensional vector space over the field of the real numbers; this vector space may be identified to the n - dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter.
In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers or the real field.
For example, real matrix , real polynomial and real Lie algebra. The word is also used as a noun , meaning a real number as in "the set of all reals".
From Wikipedia, the free encyclopedia. For the real numbers used in descriptive set theory, see Baire space set theory. For the computing datatype, see Floating-point number.
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