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Arabic mathematicians merged the concepts of " number " and " magnitude " into a more general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Joseph Liouville showed that neither e nor e 2 can be a root of an integer quadratic equation , and then established the existence of transcendental numbers; Georg Cantor extended and greatly simplified this proof.
Lindemann's proof was much simplified by Weierstrass , still further by David Hilbert , and has finally been made elementary by Adolf Hurwitz  and Paul Gordan.
The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
The first rigorous definition was published by Georg Cantor in In , he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite.
Contrary to widely held beliefs, his first method was not his famous diagonal argument , which he published in See Cantor's first uncountability proof.
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.
All these constructions of the real numbers have been shown to be equivalent, that is the resulting number systems are isomorphic.
Let R denote the set of all real numbers. The last property is what differentiates the reals from the rationals and from other, more exotic ordered fields.
For example, the set of rationals with square less than 2 has a rational upper bound e. These properties imply Archimedean property which is not implied by other definitions of completeness.
That is, the set of integers is not upper-bounded in the reals. The real numbers are uniquely specified by the above properties.
More precisely, given any two Dedekind-complete ordered fields R 1 and R 2 , there exists a unique field isomorphism from R 1 to R 2 , allowing us to think of them as essentially the same mathematical object.
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.
For details and other constructions of real numbers, see construction of the real numbers. More formally, the real numbers have the two basic properties of being an ordered field , and having the least upper bound property.
The first says that real numbers comprise a field , with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.
The second says that, if a non-empty set of real numbers has an upper bound , then it has a real least upper bound.
The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound e.
A main reason for using real numbers is that the reals contain all limits. More precisely, a sequence of real numbers has a limit, which is a real number, if and only if its elements eventually come and remain arbitrarily close to each other.
This is formally defined in the following, and means that the reals are complete in the sense of metric spaces or uniform spaces , which is a different sense than the Dedekind completeness of the order in the previous section.
This definition, originally provided by Cauchy , formalizes the fact that the x n eventually come and remain arbitrarily close to each other.
Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
The set of rational numbers is not complete. For example, the sequence 1; 1. The completeness property of the reals is the basis on which calculus , and, more generally mathematical analysis are built.
In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
For example, the standard series of the exponential function. The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
First, an order can be lattice-complete. Additionally, an order can be Dedekind-complete , as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant.
This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field the rationals and then forms the Dedekind-completion of it in a standard way.
These two notions of completeness ignore the field structure. 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: there are strictly more real numbers than natural numbers , even though both sets are infinite. 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.
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: these two properties make 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 [0;1] has measure 1.
There exist sets of real numbers that are not Lebesgue measurable, e. 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. 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: 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 with the property that every non-empty subset of R has a least element in this ordering.
Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. In mathematical expressions, unknown or unspecified real numbers are usually represented by lowercase italic letters u through z.
The set R gives rise to other sets such as the set of imaginary numbers and the set of complex numbers. The idea of a real number and what makes it "real" is primarily of interest to theoreticians.
Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.
The set of reals is "dense" in the same sense as the set of irrationals. Both sets are nondenumerable. There are more real numbers than is possible to list, even by implication.
The set R is sometimes called the continuum because it is intuitive to think of the elements of R as corresponding one-to-one with the points on a geometric line.
This notion, first proposed by Georg Cantor who also noted the difference between the cardinalities sizes of the sets of rational and irrational numbers, is called the Continuum Hypothesis.
This hypothesis can be either affirmed or denied without causing contradictions in theoretical mathematics.
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