Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Real number

From Wikipedia, the free encyclopedia
Number representing a continuous quantity
For the real numbers used in descriptive set theory, seeBaire space (set theory).
This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(July 2024) (Learn how and when to remove this message)

Inmathematics, areal number is anumber that can be used tomeasure acontinuous one-dimensionalquantity such as adistance,duration ortemperature. Here,continuous means that pairs of values can have arbitrarily small differences.[a] Every real number can be almost uniquely represented by an infinitedecimal expansion.[b][1]

The real numbers are fundamental incalculus (and in many other branches of mathematics), in particular by their role in the classical definitions oflimits,continuity andderivatives.[c]

The set of real numbers, sometimes called "the reals", is traditionallydenoted by a boldR, often usingblackboard bold,⁠R{\displaystyle \mathbb {R} }⁠.[2][3]The adjectivereal, used in the 17th century byRené Descartes, distinguishes real numbers fromimaginary numbers such as thesquare roots of−1.[4]

The real numbers include therational numbers, such as theinteger−5 and thefraction4 / 3. The rest of the real numbers are calledirrational numbers. Some irrational numbers (as well as all the rationals) are theroot of apolynomial with integer coefficients, such as the square root√2 = 1.414...; these are calledalgebraic numbers. There are also real numbers which are not, such asπ = 3.1415...; these are calledtranscendental numbers.[4]

Real numbers can be thought of as all points on a number line

Real numbers can be thought of as all points on aline called thenumber line orreal line, where the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced.

Conversely,analytic geometry is the association of points on lines (especiallyaxis lines) to real numbers such that geometricdisplacements are proportional todifferences between corresponding numbers.

The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs oftheorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of19th-century mathematics and is the foundation ofreal analysis, the study ofreal functions and real-valuedsequences. A currentaxiomatic definition is that real numbers form theunique (up to anisomorphism)Dedekind-completeordered field.[d] Other common definitions of real numbers includeequivalence classes ofCauchy sequences (of rational numbers),Dedekind cuts, and infinitedecimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

Characterizing properties

[edit]

Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form anordered field that isDedekind complete. Here, "completely characterized" means that there is a uniqueisomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. SeeConstruction of the real numbers for details about these formal definitions and the proof of their equivalence.

Arithmetic

[edit]

The real numbers form anordered field. Intuitively, this means that methods and rules ofelementary arithmetic apply to them. More precisely, there are twobinary operations,addition andmultiplication, and atotal order that have the following properties.

Many other properties can be deduced from the above ones. In particular:

Auxiliary operations

[edit]

Several other operations are commonly used, which can be deduced from the above ones.

Auxiliary order relations

[edit]

Thetotal order that is considered above is denoteda<b{\displaystyle a<b} and read as "a isless thanb". Three otherorder relations are also commonly used:

Integers and fractions as real numbers

[edit]

The real numbers0 and1 are commonly identified with thenatural numbers0 and1. This allows identifying any natural numbern with the sum ofn real numbers equal to1.

This identification can be pursued by identifying a negative integer−n{\displaystyle -n} (wheren{\displaystyle n} is a natural number) with the additive inverse−n{\displaystyle -n} of the real number identified withn.{\displaystyle n.} Similarly arational numberp/q{\displaystyle p/q} (wherep andq are integers andq≠0{\displaystyle q\neq 0}) is identified with the division of the real numbers identified withp andq.

These identifications make the setQ{\displaystyle \mathbb {Q} } of the rational numbers an orderedsubfield of the real numbersR.{\displaystyle \mathbb {R} .} TheDedekind completeness described below implies that some real numbers, such as2,{\displaystyle {\sqrt {2}},} are not rational numbers; they are calledirrational numbers.

The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties (axioms). So, the identification of natural numbers with some real numbers is justified by the fact thatPeano axioms are satisfied by these real numbers, with the addition with1 taken as thesuccessor function.

Formally, one has an injectivehomomorphism ofordered monoids from the natural numbersN{\displaystyle \mathbb {N} } to the integersZ,{\displaystyle \mathbb {Z} ,} an injective homomorphism ofordered rings fromZ{\displaystyle \mathbb {Z} } to the rational numbersQ,{\displaystyle \mathbb {Q} ,} and an injective homomorphism ofordered fields fromQ{\displaystyle \mathbb {Q} } to the real numbersR.{\displaystyle \mathbb {R} .} The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write

N⊂Q⊂R.{\displaystyle \mathbb {N} \subset \mathbb {Q} \subset \mathbb {R} .}

These identifications are formallyabuses of notation (since, formally, a rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case inconstructive mathematics andcomputer programming. In the latter case, these homomorphisms are interpreted astype conversions that can often be done automatically by thecompiler.

Dedekind completeness

[edit]

Previous properties do not distinguish real numbers fromrational numbers. This distinction is provided byDedekind completeness, which states that every set of real numbers with anupper bound admits aleast upper bound. This means the following. A set of real numbersS{\displaystyle S} isbounded above if there is a real numberu{\displaystyle u} such thats≤u{\displaystyle s\leq u} for alls∈S{\displaystyle s\in S}; such au{\displaystyle u} is called anupper bound ofS.{\displaystyle S.} So, Dedekind completeness means that, ifS is bounded above, it has an upper bound that is less than any other upper bound.

Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences.

The last two properties are summarized by saying that the real numbers form areal closed field. This implies the real version of thefundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.

Decimal representation

[edit]
Main article:Decimal representation

The most common way of describing a real number is via its decimal representation, a sequence ofdecimal digits each representing the product of an integer between zero and nine times apower of ten, extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a numberx whose decimal representation extendsk places to the left, the standard notation is the juxtaposition of the digitsbkbk−1⋯b0.a1a2⋯,{\displaystyle b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots ,} in descending order by power of ten, with non-negative and negative powers of ten separated by adecimal point, representing theinfinite series

x=bk10k+bk−110k−1+⋯+b0+a110+a2102+⋯.{\displaystyle x=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots .}

For example, for the circle constantπ=3.14159⋯,{\displaystyle \pi =3.14159\cdots ,}k is zero andb0=3,{\displaystyle b_{0}=3,}a1=1,{\displaystyle a_{1}=1,}a2=4,{\displaystyle a_{2}=4,} etc.

More formally, adecimal representation for a nonnegative real numberx consists of a nonnegative integerk and integers between zero and nine in theinfinite sequence

bk,bk−1,…,b0,a1,a2,….{\displaystyle b_{k},b_{k-1},\ldots ,b_{0},a_{1},a_{2},\ldots .}

(Ifk>0,{\displaystyle k>0,} then by conventionbk≠0.{\displaystyle b_{k}\neq 0.})

Such a decimal representation specifies the real number as the least upper bound of thedecimal fractions that are obtained bytruncating the sequence: given a positive integern, the truncation of the sequence at the placen is the finitepartial sum

Dn=bk10k+bk−110k−1+⋯+b0+a110+⋯+an10n=∑i=0kbi10i+∑j=1naj10−j{\displaystyle {\begin{aligned}D_{n}&=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+\cdots +{\frac {a_{n}}{10^{n}}}\\&=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{j=1}^{n}a_{j}10^{-j}\end{aligned}}}

The real numberx defined by the sequence is the least upper bound of theDn,{\displaystyle D_{n},} which exists by Dedekind completeness.

Conversely, given a nonnegative real numberx, one can define a decimal representation ofx byinduction, as follows. Definebk⋯b0{\displaystyle b_{k}\cdots b_{0}} as decimal representation of the largest integerD0{\displaystyle D_{0}} such thatD0≤x{\displaystyle D_{0}\leq x} (this integer exists because of the Archimedean property). Then, supposing byinduction that the decimal fractionDi{\displaystyle D_{i}} has been defined fori<n,{\displaystyle i<n,} one definesan{\displaystyle a_{n}} as the largest digit such thatDn−1+an/10n≤a,{\displaystyle D_{n-1}+a_{n}/10^{n}\leq a,} and one setsDn=Dn−1+an/10n.{\displaystyle D_{n}=D_{n-1}+a_{n}/10^{n}.}

One can use the defining properties of the real numbers to show thatx is the least upper bound of theDn.{\displaystyle D_{n}.} So, the resulting sequence of digits is called adecimal representation ofx.

Another decimal representation can be obtained by replacing≤x{\displaystyle \leq x} with<x{\displaystyle <x} in the preceding construction. These two representations are identical, unlessx is adecimal fraction of the formm10h.{\textstyle {\frac {m}{10^{h}}}.} In this case, in the first decimal representation, allan{\displaystyle a_{n}} are zero forn>h,{\displaystyle n>h,} and, in the second representation, allan{\displaystyle a_{n}} 9. (see0.999... for details).

In summary, there is abijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.

The preceding considerations apply directly for everynumeral baseB≥2,{\displaystyle B\geq 2,} simply by replacing 10 withB{\displaystyle B} and 9 withB−1.{\displaystyle B-1.}

Topological completeness

[edit]
Main article:Completeness of the real numbers

A main reason for using real numbers is so that many sequences havelimits. More formally, the reals arecomplete (in the sense ofmetric spaces oruniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):

Asequence (xn) of real numbers is called aCauchy sequence if for anyε > 0 there exists an integerN (possibly depending on ε) such that thedistance|xn −xm| is less than ε for alln andm that are both greater thanN. This definition, originally provided byCauchy, formalizes the fact that thexn eventually come and remain arbitrarily close to each other.

A sequence (xn)converges to the limitx if its elements eventually come and remain arbitrarily close tox, that is, if for anyε > 0 there exists an integerN (possibly depending on ε) such that the distance|xn −x| is less than ε forn greater thanN.

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that thetopological space of the real numbers is complete.

The set of rational numbers is 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 positivesquare root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positivesquare root of 2).

The completeness property of the reals is the basis on whichcalculus, and more generallymathematical 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 theexponential function

ex=∑n=0∞xnn!{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}

converges to a real number for everyx, because the sums

∑n=NMxnn!{\displaystyle \sum _{n=N}^{M}{\frac {x^{n}}{n!}}}

can be made arbitrarily small (independently ofM) by choosingN sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing thatex{\displaystyle e^{x}} is well defined for everyx.

"The complete ordered field"

[edit]

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can belattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have nolargest element (given any elementz,z + 1 is larger).

Additionally, an order can be Dedekind-complete, see§ Axiomatic approach. 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, anordered group (in this case, the additive group of the field) defines auniform structure, and uniform structures have a notion ofcompleteness; the description in§ Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion formetric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true thatR{\displaystyle \mathbb {R} } is theonly uniformly complete ordered field, but it is the only uniformly completeArchimedean 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 byDavid Hilbert, who meant still something else by it. He meant that the real numbers form thelargest Archimedean field in the sense that every other Archimedean field is a subfield ofR{\displaystyle \mathbb {R} }. ThusR{\displaystyle \mathbb {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 fromsurreal 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.

Cardinality

[edit]

The set of all real numbers isuncountable, in the sense that while both the set of allnatural numbers{1, 2, 3, 4, ...} and the set of all real numbers areinfinite sets, there exists noone-to-one function from the real numbers to the natural numbers. Thecardinality of the set of all real numbers is called thecardinality of the continuum and commonly denoted byc.{\displaystyle {\mathfrak {c}}.} It is strictly greater than the cardinality of the set of all natural numbers, denotedℵ0{\displaystyle \aleph _{0}} and calledAleph-zero oraleph-nought. The cardinality of the continuum equals the cardinality of thepower set of the natural numbers, that is, the set of all subsets of the natural numbers.

The statement that there is no cardinality strictly greater thanℵ0{\displaystyle \aleph _{0}} and strictly smaller thanc{\displaystyle {\mathfrak {c}}} is known as thecontinuum hypothesis (CH). It is neither provable nor refutable using the axioms ofZermelo–Fraenkel set theory including theaxiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.[5]

Other properties

[edit]
See also:Real line

As a topological space, the real numbers areseparable. This is because the set of rationals, which is countable, isdense 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 ametric space: the distance betweenx andy is defined as theabsolute value|x −y|. By virtue of being a totally ordered set, they also carry anorder topology; thetopology 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 acontractible (henceconnected andsimply connected),separable andcomplete metric space ofHausdorff dimension 1. The real numbers arelocally compact but notcompact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separableorder topologies are necessarilyhomeomorphic to the reals.

Every nonnegative real number has asquare root inR{\displaystyle \mathbb {R} }, although no negative number does. This shows that the order onR{\displaystyle \mathbb {R} } is determined by its algebraic structure. Also, everypolynomial of odd degree admits at least one real root: these two properties makeR{\displaystyle \mathbb {R} } the premier example of areal closed field. Proving this is the first half of one proof of thefundamental theorem of algebra.

The reals carry a canonicalmeasure, theLebesgue measure, which is theHaar measure on their structure as atopological group normalized such that theunit interval [0;1] 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 withfirst-order logic alone: theLöwenheim–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 ofhyperreal numbers satisfies the same first order sentences asR{\displaystyle \mathbb {R} }. Ordered fields that satisfy the same first-order sentences asR{\displaystyle \mathbb {R} } are callednonstandard models ofR{\displaystyle \mathbb {R} }. This is what makesnonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it inR{\displaystyle \mathbb {R} }), we know that the same statement must also be true ofR{\displaystyle \mathbb {R} }.

ThefieldR{\displaystyle \mathbb {R} } of real numbers is anextension field of the fieldQ{\displaystyle \mathbb {Q} } of rational numbers, andR{\displaystyle \mathbb {R} } can therefore be seen as avector space overQ{\displaystyle \mathbb {Q} }.Zermelo–Fraenkel set theory with theaxiom of choice guarantees the existence of abasis of this vector space: there exists a setB of real numbers such that every real number can be written uniquely as a finitelinear combination of elements of this set, using rational coefficients only, and such that no element ofB is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

Thewell-ordering theorem implies that the real numbers can bewell-ordered if the axiom of choice is assumed: there exists a total order onR{\displaystyle \mathbb {R} } with the property that every nonemptysubset ofR{\displaystyle \mathbb {R} } has aleast element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. anopen 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. IfV=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.[6]

A real number may be eithercomputable or uncomputable; eitheralgorithmically random or not; and eitherarithmetically random or not.

History

[edit]
Real numbers(R){\displaystyle (\mathbb {R} )} include the rational numbers(Q){\displaystyle (\mathbb {Q} )}, which include the integers(Z){\displaystyle (\mathbb {Z} )}, which in turn include the natural numbers(N){\displaystyle (\mathbb {N} )}

Simple fractions were used by theEgyptians around 1000 BC; theVedic "Shulba Sutras" ("The rules of chords") inc. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by earlyIndian mathematicians such asManava(c. 750–690 BC), who was aware that thesquare roots of certain numbers, such as 2 and 61, could not be exactly determined.[7]

Around 500 BC, theGreek mathematicians led byPythagoras also realized that thesquare root of 2 is irrational.

For Greek mathematicians, numbers were only thenatural numbers. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length. Two lengths are "commensurable", if there is a unit in which they are both measured by integers, that is, in modern terminology, if their ratio is arational number.Eudoxus of Cnidus (c. 390−340 BC) provided a definition of the equality of two irrational proportions in a way that is similar toDedekind cuts (introduced more than 2,000 years later), except that he did not use anyarithmetic operation other than multiplication of a length by a natural number (seeEudoxus of Cnidus). This may be viewed as the first definition of the real numbers.

TheMiddle Ages brought about the acceptance ofzero,negative numbers, integers, andfractional numbers, first byIndian andChinese mathematicians, and then byArabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra).[8] Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.[9] The Egyptian mathematicianAbū Kāmil Shujā ibn Aslam(c. 850–930) was the first to accept irrational numbers as solutions toquadratic equations, or ascoefficients in anequation (often in the form of square roots,cube roots, andfourth roots).[10] In Europe, such numbers, not commensurable with the numerical unit, were calledirrational orsurd ("deaf").

In the 16th century,Simon Stevin created the basis for moderndecimal 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 apolynomial, distinguishing them from "imaginary" numbers.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers.Lambert (1761) gave a flawed proof thatπ cannot be rational;Legendre (1794) completed the proof[11] and showed thatπ is not the square root of a rational number.[12]Liouville (1840) showed that neithere nore2 can be a root of an integerquadratic equation, and then established the existence of transcendental numbers;Cantor (1873) extended and greatly simplified this proof.[13]Hermite (1873) proved thate is transcendental, andLindemann (1882), showed thatπ is transcendental. Lindemann's proof was much simplified by Weierstrass (1885),Hilbert (1893),Hurwitz,[14] andGordan.[15]

The concept that many points existed between rational numbers, such as the square root of 2, was well known to the ancient Greeks. The existence of a continuous number line was considered self-evident, but the nature of this continuity, presently calledcompleteness, was not understood. The rigor developed for geometry did not cross over to the concept of numbers until the 1800s.[16]

Modern analysis

[edit]

The developers ofcalculus used real numbers andlimits without defining them rigorously. In hisCours d'Analyse (1821),Cauchy made calculus rigorous, but he used the real numbers without defining them, and assumed without proof that everyCauchy sequence has a limit and that this limit is a real number.

In 1854Bernhard Riemann highlighted the limitations of calculus in the method ofFourier series, showing the need for a rigorous definition of the real numbers.[17]: 672 

Beginning withRichard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, includingHermann Hankel,Charles Méray, andEduard Heine, leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, asDedekind cuts, and the other one byGeorg Cantor, as equivalence classes of Cauchy sequences.[18] Several problems were left open by these definitions, which contributed to thefoundational crisis of mathematics. Firstly both definitions suppose thatrational numbers and thusnatural numbers are rigorously defined; this was done a few years later withPeano axioms. Secondly, both definitions involveinfinite sets (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor'sset theory was published several years later. Thirdly, these definitions implyquantification on infinite sets, and this cannot be formalized in the classicallogic offirst-order predicates. This is one of the reasons for whichhigher-order logics were developed in the first half of the 20th century.

In 1874 Cantor showed that the set of all real numbers isuncountably infinite, but the set of all algebraic numbers iscountably infinite.Cantor's first uncountability proof was different from his famousdiagonal argument published in 1891.

Formal definitions

[edit]
Main article:Construction of the real numbers

The real number system(R;+;⋅;<){\displaystyle (\mathbb {R} ;{}+{};{}\cdot {};{}<{})} can be definedaxiomatically up to anisomorphism, which is described hereinafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of theirCauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers.[19] Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or ofTarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems areisomorphic.

Axiomatic approach

[edit]

LetR{\displaystyle \mathbb {R} } denote theset of all real numbers. Then:

The last property applies to the real numbers but not to the rational numbers (or toother more exotic ordered fields). For example,{x∈Q:x2<2}{\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}} has a rational upper bound (e.g., 1.42), but noleast rational upper bound, because2{\displaystyle {\sqrt {2}}} is not rational.

These properties imply theArchimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper boundN; then,N – 1 would not be an upper bound, and there would be an integern such thatn >N – 1, and thusn + 1 >N, which is a contradiction with the upper-bound property ofN.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fieldsR1{\displaystyle \mathbb {R} _{1}} andR2{\displaystyle \mathbb {R} _{2}}, there exists a unique fieldisomorphism fromR1{\displaystyle \mathbb {R} _{1}} toR2{\displaystyle \mathbb {R_{2}} }. This uniqueness allows us to think of them as essentially the same mathematical object.

For another axiomatization ofR{\displaystyle \mathbb {R} } seeTarski's axiomatization of the reals.

Construction from the rational numbers

[edit]

The real numbers can be constructed as acompletion 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—in this caseπ. For details and other constructions of real numbers, seeConstruction of the real numbers.

Applications and connections

[edit]

Physics

[edit]

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 asclassical mechanics,electromagnetism,quantum mechanics,general relativity, and thestandard model are described using mathematical structures, typicallysmooth manifolds orHilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finiteaccuracy 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.[20]

Logic

[edit]

The real numbers are most often formalized using theZermelo–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 inreverse mathematics and inconstructive mathematics.[21]

Thehyperreal numbers as developed byEdwin Hewitt,Abraham Robinson, and others extend the set of the real numbers by introducinginfinitesimal and infinite numbers, allowing for buildinginfinitesimal calculus in a way closer to the original intuitions ofLeibniz,Euler,Cauchy, and others.

Edward Nelson'sinternal set theory enriches theZermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).

Thecontinuum hypothesis posits that the cardinality of the set of the real numbers isℵ1{\displaystyle \aleph _{1}}; i.e. the smallest infinitecardinal number afterℵ0{\displaystyle \aleph _{0}}, the cardinality of the integers.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.

Computation

[edit]

Electronic calculators andcomputers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrarydefinable real numbers, which are inconvenient to manipulate.

Instead, computers typically work with finite-precision approximations calledfloating-point numbers, a representation similar toscientific notation. The achievable precision is limited by thedata storage space allocated for each number, whether asfixed-point, floating-point, orarbitrary-precision numbers, or some other representation. Mostscientific computation usesbinary floating-point arithmetic, often a64-bit representation with around 16 decimaldigits of precision. Real numbers satisfy theusual rules of arithmetic, butfloating-point numbers do not. The field ofnumerical analysis studies thestability andaccuracy of numericalalgorithms implemented with approximate arithmetic.

Alternately,computer algebra systems can operate on irrational quantities exactly bymanipulating symbolic formulas for them (such as2,{\textstyle {\sqrt {2}},}arctan⁡5,{\textstyle \arctan 5,} or∫01xxdx{\textstyle \int _{0}^{1}x^{x}\,dx}) rather than their rational or decimal approximation.[22] But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal (theconstant problem); and arithmetic operations can causeexponential explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring apolynomial roughly doubles its number of terms), overwhelming finite computer storage.[23]

A real number is calledcomputable if there exists an algorithm that yields its digits. Because there are onlycountably many algorithms,[24] but an uncountable number of reals,almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is anundecidable problem. Someconstructivists accept the existence of only those reals that are computable. The set ofdefinable numbers is broader, but still only countable.

Set theory

[edit]

Inset theory, specificallydescriptive set theory, theBaire 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".

Vocabulary and notation

[edit]

Theset of all real numbers is denotedR{\displaystyle \mathbb {R} } (blackboard bold) orR (upright bold). As it is naturally endowed with the structure of afield, the expressionfield of real numbers is frequently used when its algebraic properties are under consideration.

The sets of positive real numbers and negative real numbers are often notedR+{\displaystyle \mathbb {R} ^{+}} andR−{\displaystyle \mathbb {R} ^{-}},[25] respectively;R+{\displaystyle \mathbb {R} _{+}} andR−{\displaystyle \mathbb {R} _{-}} are also used.[26] The non-negative real numbers can be notedR≥0{\displaystyle \mathbb {R} _{\geq 0}} but one often sees this set notedR+∪{0}.{\displaystyle \mathbb {R} ^{+}\cup \{0\}.}[25] In French mathematics, thepositive real numbers andnegative real numbers commonly includezero, and these sets are noted respectivelyR+{\displaystyle \mathbb {R_{+}} } andR−.{\displaystyle \mathbb {R} _{-}.}[26] In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are notedR+∗{\displaystyle \mathbb {R} _{+}^{*}} andR−∗.{\displaystyle \mathbb {R} _{-}^{*}.}[26]

The notationRn{\displaystyle \mathbb {R} ^{n}} refers to the set of then-tuples of elements ofR{\displaystyle \mathbb {R} } (real coordinate space), which can be identified to theCartesian product ofn copies ofR.{\displaystyle \mathbb {R} .} It is ann-dimensionalvector space over the field of the real numbers, often called thecoordinate space of dimensionn; this space may be identified to then-dimensionalEuclidean space as soon as aCartesian coordinate system has been chosen in the latter. In this identification, apoint of the Euclidean space is identified with the tuple of itsCartesian coordinates.

In mathematicsreal is used as an adjective, meaning that the underlying field is the field of the real numbers (orthe real field). For example,realmatrix,real polynomial andrealLie algebra. The word is also used as anoun, meaning a real number (as in "the set of all reals").

Generalizations and extensions

[edit]

The real numbers can be generalized and extended in several different directions:

See also

[edit]
Number systems
Complex:C{\displaystyle :\;\mathbb {C} }
Real:R{\displaystyle :\;\mathbb {R} }
Rational:Q{\displaystyle :\;\mathbb {Q} }
Integer:Z{\displaystyle :\;\mathbb {Z} }
Natural:N{\displaystyle :\;\mathbb {N} }
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Irrational period
Transcendental
Imaginary

Notes

[edit]
  1. ^This is not sufficient for distinguishing the real numbers from therational numbers; a property ofcompleteness is also required.
  2. ^Rational numbers with a "terminating" decimal expansion have two decimal expansions (see0.999...); the other real numbers have one decimal expansion.
  3. ^Limits and continuity can be defined ingeneral topology without reference to real numbers, but these generalizations are relatively recent, and used only in very specific cases.
  4. ^More precisely, given two complete totally ordered fields, there is aunique isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering. In fact, the identity is the unique field automorphism of the reals, sincex>y{\displaystyle x>y} is equivalent to∃z∣x−y=z2,{\displaystyle \exists z\mid x-y=z^{2},} and the second formula is stable under field automorphisms.

References

[edit]

Citations

[edit]
  1. ^"Real number".Oxford Reference. 2011-08-03.
  2. ^"real".Oxford English Dictionary (3rd ed.). 2008. 'real',n.2, B.4.Mathematics. A real number. Usually inplural
  3. ^Webb, Stephen (2018)."Set of Natural Numbers ℕ".Clash Of Symbols: A Ride Through The Riches Of Glyphs. Springer. pp. 198–199.
  4. ^ab"Real number".Encyclopedia Britannica.
  5. ^Koellner, Peter (2013)."The Continuum Hypothesis". InZalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy. Stanford University.
  6. ^Moschovakis, Yiannis N. (1980),"5. The Constructible Universe",Descriptive Set Theory, North-Holland, pp. 274–285,ISBN 978-0-444-85305-9
  7. ^T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 410–11. In:Selin, Helaine;D'Ambrosio, Ubiratan, eds. (2000),Mathematics Across Cultures: The History of Non-western Mathematics,Springer,ISBN 978-1-4020-0260-1.
  8. ^O'Connor, John J.;Robertson, Edmund F. (1999),"Arabic mathematics: forgotten brilliance?",MacTutor History of Mathematics Archive,University of St Andrews
  9. ^Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics",Annals of the New York Academy of Sciences,500 (1): 253–77 [254],Bibcode:1987NYASA.500..253M,doi:10.1111/j.1749-6632.1987.tb37206.x,S2CID 121416910
  10. ^Jacques Sesiano, "Islamic mathematics", p. 148, inSelin, Helaine; D'Ambrosio, Ubiratan (2000),Mathematics Across Cultures: The History of Non-western Mathematics,Springer,ISBN 978-1-4020-0260-1
  11. ^Beckmann, Petr (1971).A History ofπ (PI). St. Martin's Press. p. 170.ISBN 9780312381851.
  12. ^Arndt, Jörg; Haenel, Christoph (2001),Pi Unleashed, Springer, p. 192,ISBN 978-3-540-66572-4, retrieved2015-11-15.
  13. ^Dunham, William (2015),The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, p. 127,ISBN 978-1-4008-6679-3, retrieved2015-02-17,Cantor found a remarkable shortcut to reach Liouville's conclusion with a fraction of the work
  14. ^Hurwitz, Adolf (1893). "Beweis der Transendenz der Zahl e".Mathematische Annalen (43):134–35.
  15. ^Gordan, Paul (1893)."Transcendenz vone und π".Mathematische Annalen.43 (2–3):222–224.doi:10.1007/bf01443647.S2CID 123203471.
  16. ^Stefan Drobot "Real numbers". Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 vii+102 pp.
  17. ^Robson, Eleanor; Stedall, Jacqueline A., eds. (2009).The Oxford handbook of the history of mathematics. Oxford handbooks. Oxford; New York: Oxford University Press.ISBN 978-0-19-921312-2.OCLC 229023665.
  18. ^O'Connor, John J.;Robertson, Edmund F. (October 2005),"The real numbers: Stevin to Hilbert",MacTutor History of Mathematics Archive,University of St Andrews
  19. ^"Lecture #1"(PDF).18.095 Lecture Series in Mathematics. 2015-01-05.
  20. ^Wheeler, John Archibald (1986). "Hermann Weyl and the Unity of Knowledge: In the linkage of four mysteries—the "how come" of existence, time, the mathematical continuum, and the discontinuous yes-or-no of quantum physics—may lie the key to deep new insight".American Scientist.74 (4):366–75.Bibcode:1986AmSci..74..366W.JSTOR 27854250.
    Bengtsson, Ingemar (2017). "The Number Behind the Simplest SIC-POVM".Foundations of Physics.47 (8):1031–41.arXiv:1611.09087.Bibcode:2017FoPh...47.1031B.doi:10.1007/s10701-017-0078-3.S2CID 118954904.
  21. ^Bishop, Errett; Bridges, Douglas (1985),Constructive analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279, Berlin, New York:Springer-Verlag,ISBN 978-3-540-15066-4, chapter 2.
  22. ^Cohen, Joel S. (2002),Computer algebra and symbolic computation: elementary algorithms, vol. 1, A K Peters, p. 32,ISBN 978-1-56881-158-1
  23. ^Trefethen, Lloyd N. (2007)."Computing numerically with functions instead of numbers"(PDF).Mathematics in Computer Science.1 (1):9–19.doi:10.1007/s11786-007-0001-y.
  24. ^Hein, James L. (2010), "14.1.1",Discrete Structures, Logic, and Computability (3 ed.), Sudbury, MA: Jones and Bartlett Publishers,ISBN 97-80763772062, retrieved2015-11-15
  25. ^abSchumacher, Carol (1996).Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley. pp. 114–115.ISBN 9780201826531.
  26. ^abcÉcole Normale Supérieure ofParis,"Nombres réels" ("Real numbers")Archived 2014-05-08 at theWayback Machine, p. 6

Sources

[edit]

External links

[edit]
Number systems
Sets ofdefinable numbers
Composition algebras
Split
types
Otherhypercomplex
Infinities andinfinitesimals
Other types
Authority control databases: NationalEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Real_number&oldid=1280948651"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp