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| {{About||the real numbers used in descriptive set theory|Baire space (set theory)|the computing datatype|Floating-point number}}
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| [[File:Latex real numbers.svg|right|thumb|120px|A symbol of the set of '''real numbers''' (ℝ)]]
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| [[File:Number-line.svg|right|thumb|350px|Real numbers can be thought of as points on an infinitely long [[number line]].]]
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| In [[mathematics]], a '''real number''' is a value that represents a quantity along a continuous line. The real numbers include all the [[rational number]]s, such as the [[integer]] −5 and the [[fraction (mathematics)|fraction]] 4/3, and all the [[irrational number]]s such as √{{overline|2}} (1.41421356… the [[square root of two]], an irrational [[algebraic number]]) and [[pi|π]] (3.14159265…, a [[transcendental number]]). Real numbers can be thought of as points on an infinitely long [[line (geometry)|line]] called the [[number line]] or [[real line]], where the points corresponding to [[integers]] are equally spaced. Any real number can be determined by a possibly infinite [[decimal representation]] such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The [[real line]] can be thought of as a part of the [[complex plane]], and correspondingly, [[complex number]]s include real numbers as a special case.
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| These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique [[Archimedean property|Archimedean]] [[complete space|complete]] [[total order|totally ordered]] [[field (mathematics)|field]] {{nowrap|('''R''',+,·,<),}} [[up to]] an [[isomorphism]],<ref>More precisely, given two complete totally ordered fields, there is a ''unique'' isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering.</ref> whereas popular constructive definitions of real numbers include declaring them as [[equivalence class]]es of [[Cauchy sequence]]s of rational numbers, [[Dedekind cut]]s, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of [[classical mathematics]].
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| The reals are [[uncountable set|uncountable]]; that is, while both the set of all [[natural number]]s and the set of all real numbers are [[infinite set]]s, there can be no [[one-to-one function]] from the real numbers to the natural numbers: the [[cardinality]] of the set of all real numbers (denoted <math>\mathfrak c</math> and called [[cardinality of the continuum]]) is strictly greater than the cardinality of the set of all natural numbers (denoted [[aleph number#Aleph-naught|<math>\aleph_0</math>]]). The statement that there is no subset of the reals with cardinality strictly greater than <math>\aleph_0</math> and strictly smaller than <math>\mathfrak c</math> is known as the [[continuum hypothesis]]. It is known to be neither provable nor refutable using the axioms of [[Zermelo–Fraenkel set theory]], the standard foundation of modern mathematics, provided ZF set theory is [[consistency|consistent]].
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| ==Basic properties==
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| A real number may be either [[rational number|rational]] or [[irrational number|irrational]]; either [[algebraic number|algebraic]] or [[transcendental number|transcendental]]; and either [[positive number|positive]], [[negative number|negative]], or [[0 (number)|zero]]. Real numbers are used to measure [[Continuous function|continuous]] quantities. They may be expressed by [[decimal representation]]s that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147… The [[Ellipsis#In mathematical notation|ellipsis]] (three dots) indicates that there would still be more digits to come.
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| More formally, real numbers have the two basic properties of being an [[ordered field]], and having the [[least upper bound axiom|least upper bound]] property. The first says that real numbers comprise a [[Field (mathematics)|field]], with addition and multiplication as well as division by nonzero numbers, which can be [[total order|totally ordered]] on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an [[Upper and lower bounds|upper bound]], then it has a real [[supremum|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.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.
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| ==In physics==
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| 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]].
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| In some recent developments of theoretical physics stemming from the [[holographic principle]], the Universe is seen fundamentally as an information store, essentially zeroes and ones, organized in much less geometrical fashion and manifesting itself as space-time and particle fields only on a more superficial level. This approach removes the real number system from its foundational role in physics and even prohibits the existence of infinite precision real numbers in the physical universe by considerations based on the [[Bekenstein bound]].<ref>[[Scott Aaronson]], ''[http://arxiv.org/abs/quant-ph/0502072 NP-complete Problems and Physical Reality]'', ACM [[SIGACT]] News, Vol. 36, No. 1. (March 2005), pp. 30–52.</ref>
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| ==In computation==
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| Computer arithmetic cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them, whether as [[floating-point number]]s or [[arbitrary precision arithmetic|arbitrary precision numbers]]. However, [[computer algebra system]]s can operate on [[irrational number|irrational quantities]] exactly by manipulating formulas for them (such as <math>\textstyle\sqrt 2</math>, <math>\textstyle\arcsin \left({{2}\over{23}}\right)</math>, or<math>\textstyle\int_{0}^{1} {x^{x}}\;dx</math>) rather than their rational or decimal approximation;<ref>{{Citation |publisher=A K Peters, Ltd. |isbn=978-1-56881-158-1 |volume=1 |last=Cohen |first=Joel S. |title=Computer algebra and symbolic computation: elementary algorithms |year=2002 |page=32}}</ref> however, it is not in general possible to determine whether two such expressions are equal (the [[constant problem]]).
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| A real number is called ''[[computable number|computable]]'' if there exists an algorithm that yields its digits. Because there are only [[countably infinite|countably]] many algorithms,<ref>James L. Hein, [http://books.google.com/books?id=vmlcc2IH9dEC ''Discrete Structures, Logic, and Computability''], 3rd edition (Jones and Bartlett Publishers: Sudbury, MA), section 14.1.1 (2010).</ref> 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 [[constructivism (mathematics)|constructivists]] accept the existence of only those reals that are computable. The set of [[definable number]]s is broader, but still only countable.
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| ==Notation==
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| Mathematicians use the symbol '''R''' (or alternatively, <math>\mathbb{R}</math>, the letter "[[R]]" in [[blackboard bold]], Unicode <big>ℝ</big> – U+211D, named DOUBLE-STRUCK CAPITAL R) to represent the [[Set (mathematics)|set]] of all real numbers (as this set is naturally endowed with the structure of a [[field (mathematics)|field]], the expression ''field of real numbers'' is frequently used when its algebraic properties are under consideration).
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| The set of positive (resp. negative) real numbers is often denoted by '''R'''<sup>+</sup> (resp. '''R'''<sup>-</sup>), or '''R'''<sub>>0</sub> (resp. '''R'''<sub><0</sub>).<ref>{{harvnb|Schumacher|1996|loc=pp. 114-115}}</ref> While the notation '''R'''<sub>≥0</sub> may be used for the set of non-negative real numbers.
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| The notation '''R'''<sup>''n''</sup> refers to the [[Cartesian product]] of ''n'' copies of '''R''', which is an ''n''-[[dimension]]al [[vector space]] over the field of the real numbers; this vector space may be identified to the ''n''-[[dimension]]al space of [[Euclidean geometry]] as soon as a [[coordinate system]] has been chosen in the latter. For example, a value from '''R'''<sup>3</sup> consists of three real numbers and specifies the [[coordinates]] of a [[Point (geometry)|point]] in 3-dimensional space.
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| 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 (mathematics)|matrix]]'', ''real [[polynomial]]'' and ''real [[Lie algebra]]''. As a substantive, the term is used almost strictly in reference to the real numbers themselves (e.g., The "set of all reals").
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| ==History==
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| [[Fraction (mathematics)#Common, vulgar, or simple fractions|Simple fractions]] have been used by the [[History of Egypt|Egyptians]] around 1000 BC; the [[Vedic civilization|Vedic]] "[[Sulba Sutras]]" ("The rules of chords") in, {{nowrap|c. 600 BC}}, include what may be the first "use" of [[irrational number]]s. The concept of irrationality was implicitly accepted by early [[Indian mathematics|Indian mathematicians]] since [[Manava]] {{nowrap|(c. 750–690 BC)}}, who were aware that the [[square root]]s of certain numbers such as 2 and 61 could not be exactly determined.<ref>T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 410–1, in {{citation |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media|Springer]] |isbn=1-4020-0260-2}}</ref> Around 500 BC, the [[Greek mathematics|Greek mathematicians]] led by [[Pythagoras]] realized the need for irrational numbers, in particular the irrationality of the [[square root of 2]].
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| The [[Middle Ages]] brought the acceptance of [[zero]], [[negative number|negative]], [[Integer|integral]], and [[Fraction (mathematics)|fractional]] numbers, first by [[Indian mathematics|Indian]] and [[Chinese mathematics|Chinese mathematicians]], and then by [[Mathematics in medieval Islam|Arabic mathematicians]], who were also the first to treat irrational numbers as algebraic objects,<ref>{{MacTutor |class=HistTopics |id=Arabic_mathematics |title=Arabic mathematics: forgotten brilliance? |year=1999}}</ref> which was made possible by the development of [[algebra]]. Arabic mathematicians merged the concepts of "[[number]]" and "[[Magnitude (mathematics)|magnitude]]" into a more general idea of real numbers.<ref>{{citation |last=Matvievskaya |first=Galina |year=1987 |title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics |journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]] |volume=500 |pages=253–277 [254] |doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> The [[Egypt]]ian mathematician [[Abū Kāmil Shujā ibn Aslam]] {{nowrap|(c. 850–930)}} was the first to accept irrational numbers as solutions to [[quadratic equation]]s or as [[coefficient]]s in an [[equation]], often in the form of square roots, [[cube root]]s and [[Nth root|fourth roots]].<ref>Jacques Sesiano, "Islamic mathematics", p. 148, in {{citation |title=Mathematics Across Cultures: The History of Non-western Mathematics |first1=Helaine |last1=Selin |first2=Ubiratan |last2=D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media|Springer]] |isbn=1-4020-0260-2}}</ref>
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| 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.
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| In the 17th century, [[Descartes]] introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.
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| In the 18th and 19th centuries there was much work on irrational and [[transcendental number]]s. [[Johann Heinrich Lambert]] (1761) gave the first flawed proof that π cannot be rational;{{citation needed|date=January 2012}} [[Adrien-Marie Legendre]] (1794) completed the proof, and showed that π is not the square root of a rational number. [[Paolo Ruffini]] (1799) and [[Niels Henrik Abel]] (1842) both constructed proofs of the [[Abel–Ruffini theorem]]: that the general [[Quintic equation|quintic]] or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
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| [[Évariste Galois]] (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of [[Galois theory]]. [[Joseph Liouville]] (1840) showed that neither ''e'' nor ''e''<sup>2</sup> can be a root of an integer [[quadratic equation]], and then established the existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873). [[Charles Hermite]] (1873) first proved that [[e (mathematical constant)|''e'']] is transcendental, and [[Ferdinand von Lindemann]] (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by [[David Hilbert]] (1893), and has finally been made elementary by [[Adolf Hurwitz]] and [[Paul Gordan]].
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| 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 given by [[Georg Cantor]] in 1871. In 1874 he showed that the set of all real numbers is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]]. Contrary to widely held beliefs, his first method was not his famous [[Cantor's diagonal argument|diagonal argument]], which he published in 1891. See [[Cantor's first uncountability proof]].
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| ==Definition==
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| {{Main|Construction of the real numbers}}
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| The real number system <math>(\mathbb R,+,\cdot,<)</math> can be defined [[Axiomatic system|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 sequence]]s or as [[Dedekind cut]]s, 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 [[Structuralism (philosophy of mathematics)|structuralist]] point of view all these constructions are on equal footing.
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| ===Axiomatic approach===
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| Let ℝ denote the [[Set (mathematics)|set]] of all real numbers. Then:
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| *The set ℝ is a [[field (mathematics)|field]], meaning that [[addition]] and [[multiplication]] are defined and have the usual properties.
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| *The field ℝ is [[ordered field|ordered]], meaning that there is a [[total order]] ≥ such that, for all real numbers ''x'', ''y'' and ''z'':
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| **if ''x'' ≥ ''y'' then ''x'' + ''z'' ≥ ''y'' + ''z'';
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| **if ''x'' ≥ 0 and ''y'' ≥ 0 then ''xy'' ≥ 0.
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| *The order is [[Dedekind completion|Dedekind-complete]]; that is, every [[empty set|non-empty]] subset ''S'' of ℝ with an [[upper bound]] in ℝ has a [[supremum|least upper bound]] (also called supremum) in ℝ.
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| The last property is what differentiates the reals from the [[rational number|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.
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| The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields ℝ<sub>1</sub> and ℝ<sub>2</sub>, there exists a unique field [[isomorphism]] from ℝ<sub>1</sub> to ℝ<sub>2</sub>, allowing us to think of them as essentially the same mathematical object.
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| For another axiomatization of ℝ, see [[Tarski's axiomatization of the reals]].
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| ===Construction from the rational numbers===
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| 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, …) [[Limit of a sequence|converges]] to a unique real number. For details and other constructions of real numbers, see [[construction of the real numbers]].
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| ==Properties==
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| ===Completeness===
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| A main reason for using real numbers is that the reals contain all [[limit (mathematics)|limits]]. More precisely, every sequence of real numbers having the property that consecutive terms of the sequence become arbitrarily close to each other necessarily has the property that after some term in the sequence the remaining terms are arbitrarily close to some specific real number. In mathematical terminology, this means that the reals are [[completeness (topology)|complete]] (in the sense of [[metric space]]s or [[uniform space]]s, which is a different sense than the Dedekind completeness of the order in the previous section). This is formally defined in the following way:
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| A [[sequence]] (''x''<sub>''n''</sub>) of real numbers is called a ''[[Cauchy sequence]]'' if for any {{nowrap|ε > 0}} there exists an integer ''N'' (possibly depending on ε) such that the [[distance]] {{nowrap|{{!}}''x<sub>n</sub>'' − ''x<sub>m</sub>''{{!}}}} is less than ε for all ''n'' and ''m'' that are both greater than ''N''. In other words, a sequence is a [[Augustin Louis Cauchy|Cauchy]] sequence if its elements ''x''<sub>''n''</sub> eventually come and remain arbitrarily close to each other.
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| A sequence (''x''<sub>''n''</sub>) ''converges to the limit'' ''x'' if for any {{nowrap|ε > 0}} there exists an integer ''N'' (possibly depending on ε) such that the distance {{nowrap|{{!}}''x<sub>n</sub>'' − ''x''{{!}}}} is less than ε provided that ''n'' is greater than ''N''. In other words, a sequence has limit ''x'' if its elements eventually come and remain arbitrarily close to ''x''.
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| Notice that every convergent sequence is a Cauchy sequence. The converse is also true:
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| :'''Every Cauchy sequence of real numbers is convergent to a real number.'''
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| That is, the reals are complete.
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| Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, …), where each term adds a digit of the decimal expansion of the positive [[square root]] of 2, is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the positive [[square root]] of 2.)
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| The existence of limits of Cauchy sequences is what makes [[calculus]] work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.
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| For example, the standard series of the [[exponential function]]
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| :<math>\mathrm{e}^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math>
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| converges to a real number because for every ''x'' the sums
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| :<math>\sum_{n=N}^{M} \frac{x^n}{n!}</math>
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| can be made arbitrarily small by choosing ''N'' sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if the limit is not known in advance.
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| ==="The complete ordered field"===
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| The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
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| First, an order can be [[complete lattice|lattice-complete]]. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', {{nowrap|''z'' + 1}} is larger), so this is not the sense that is meant.
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| Additionally, an order can be [[Dedekind completion|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.
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| 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 space|uniform]] structure, and uniform structures have a notion of [[completeness (topology)]]; the description in the section '''Completeness''' above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for [[metric space]]s, 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, of course), 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.
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| 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 number]]s, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
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| ===Advanced properties===
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| {{See also|Real line}}
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| The reals are [[uncountable]]; that is, there are strictly more real numbers than [[natural number]]s, even though both sets are [[Infinite set|infinite]]. In fact, the [[cardinality of the continuum|cardinality of the reals]] equals that of the set of subsets (i.e., the power set) of the natural numbers, and [[Cantor's diagonal argument]] states that the latter set's cardinality is strictly greater than the cardinality of '''N'''. Since only a countable set of real numbers can be [[algebraic number|algebraic]], [[almost all]] real numbers are [[transcendental number|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 [[logical independence|independent]] from the [[axiomatic set theory|axioms of set theory]].
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| As a topological space, the real numbers are [[separable space|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.
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| The real numbers form a [[metric space]]: the distance between ''x'' and ''y'' is defined as the [[absolute value]] {{nowrap|{{!}}''x'' − ''y''{{!}}}}. By virtue of being a [[total order|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 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 are a [[contractible]] (hence [[connected space|connected]] and [[simply connected]]), [[separable space|separable]] and [[complete space|complete]] metric space of [[Hausdorff dimension]] 1. The real numbers are [[local compactness|locally compact]] but not [[compact space|compact]]. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable [[total order|order topologies]] are necessarily [[homeomorphic]] to the reals.
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| 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]].
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| The reals carry a canonical [[Measure (mathematics)|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.g. [[Vitali set]]s.
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| 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 number]]s <!--is equal in cardinality to '''R''' and also-->satisfies the same first order sentences as '''R'''. Ordered fields that satisfy the same first-order sentences as '''R''' are called [[nonstandard model]]s 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'''.
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| The [[field (mathematics)|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 (linear algebra)|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.
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| The [[well-ordering theorem]] implies that the real numbers can be [[well-order]]ed if the axiom of choice is assumed: there exists a [[total order]] on '''R''' with the property that every [[empty set|non-empty]] [[subset]] of '''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 cannot be explicitly described.
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| ==Generalizations and extensions==
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| The real numbers can be generalized and extended in several different directions:
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| *The [[complex number]]s contain solutions to all [[polynomial]] equations and hence are an [[algebraically closed field]] unlike the real numbers. However, the complex numbers are not an [[ordered field]].
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| *The [[affinely extended real number system]] adds two elements +∞ and −∞. It is a [[compact space]]. It is no longer a field, not even an additive group, but it still has a [[total order]]; moreover, it is a [[complete lattice]].
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| *The [[real projective line]] adds only one value ∞. It is also a compact space. Again, it is no longer a field, not even an additive group. However, it allows division of a non-zero element by zero. It is not ordered anymore.
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| *The [[Long line (topology)|long real line]] pastes together ℵ<sub>1</sub>* + ℵ<sub>1</sub> copies of the real line plus a single point (here ℵ<sub>1</sub>* denotes the reversed ordering of ℵ<sub>1</sub>) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ<sub>1</sub> in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
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| *Ordered fields extending the reals are the [[hyperreal number]]s and the [[surreal number]]s; both of them contain [[infinitesimal]] and infinitely large numbers and are therefore [[non-Archimedean ordered field]]s.
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| *[[Self-adjoint operator]]s on a [[Hilbert space]] (for example, self-adjoint square complex [[matrix (math)|matrices]]) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their [[eigenvector|eigenvalues]] are real and they form a real [[associative algebra]]. [[Positive-definite]] operators correspond to the positive reals and [[normal operator]]s correspond to the complex numbers.
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| =="Reals" in set theory==
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| In [[set theory]], specifically [[descriptive set theory]], the [[Baire space (set theory)|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".
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| ==Real numbers and logic==
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| 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 [[Constructivism (mathematics)|constructive mathematics]].<ref>{{Citation |last1=Bishop |first1=Errett |last2=Bridges |first2=Douglas |title=Constructive analysis |publisher=[[Springer-Verlag]] |location=Berlin, New York |series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] |isbn=978-3-540-15066-4 |year=1985 |volume=279}}, chapter 2.</ref>
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| [[Abraham Robinson]]'s theory of [[nonstandard real number|nonstandard]] or [[hyperreal number]]s extends the set of the real numbers by infinitesimal numbers, which allows building [[infinitesimal calculus]] in a way closer to the usual intuition of the notion of [[Limit (mathematics)|limit]]. [[Edward Nelson]]'s [[internal set theory]] is a non-[[Zermelo–Fraenkel]] set theory that considers non-standard real numbers as elements of the set of the reals (and not of an extension of it, as in Robinson's theory).
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| The [[continuum hypothesis]] posits that the cardinality of the set of the real numbers is <math>\aleph_1</math>; i.e. the smallest infinite [[cardinal number]] after <math>\aleph_0</math>, the cardinality of the integers. [[Paul Cohen (mathematician)|Paul Cohen]] proved in 1963 that it is an axiom independent of the other axioms of set theory; that is, one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.
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| ==See also==
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| {{Colbegin|2}}
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| *[[Completeness#Mathematical completeness|Completeness]]
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| *[[Continued fraction]]
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| *[[Limit of a sequence]]
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| *[[Real analysis]]
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| *[[Imaginary number]]
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| *[[Complex number]]
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| {{Colend}}
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *[[Georg Cantor]], 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", ''Journal für die Reine und Angewandte Mathematik'', volume 77, pages 258–262.
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| *[[Solomon Feferman]],1989, ''The Numbers Systems: Foundations of Algebra and Analysis'', AMS Chelsea, ISBN 0-8218-2915-7.
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| *Robert Katz, 1964, ''Axiomatic Analysis'', D. C. Heath and Company.
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| *[[Edmund Landau]], 2001, ISBN 0-8218-2693-X, ''Foundations of Analysis'', [[American Mathematical Society]].
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| *Howie, John M., ''Real Analysis'', Springer, 2005, ISBN 1-85233-314-6
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| * {{citation|first=Carol|last=Schumacher|title=ChapterZero / Fundamental Notions of Abstract Mathematics|year=1996|publisher=Addison-Wesley|isbn=0-201-82653-4}}
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| ==External links==
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| *{{springer|title=Real number|id=p/r080060}}
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| *[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_1.html The real numbers: Pythagoras to Stevin]
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| *[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_2.html The real numbers: Stevin to Hilbert]
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| *[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_3.html The real numbers: Attempts to understand]
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| *[http://www.math.vanderbilt.edu/~schectex/courses/thereals/ What are the "real numbers," really?]
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| {{Number Systems}}
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| [[Category:Real numbers| ]]
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| [[Category:Real algebraic geometry]]
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| [[Category:Elementary mathematics]]
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