Real number

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For the real numbers used in descriptive set theory, see Baire space (set theory). Real number_sentence_0

For the computing datatype, see Floating-point number. Real number_sentence_1

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Real number_sentence_2

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. Real number_sentence_3

The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers. Real number_sentence_4

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. Real number_sentence_5

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. Real number_sentence_6

The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. Real number_sentence_7

All these definitions satisfy the axiomatic definition and are thus equivalent. Real number_sentence_8

History Real number_section_0

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in, c. 600 BC, include what may be the first "use" of irrational numbers. Real number_sentence_9

The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750–690 BC), who were aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Real number_sentence_10

Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Real number_sentence_11

The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Real number_sentence_12

Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. Real number_sentence_13

The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots and fourth roots). Real number_sentence_14

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. Real number_sentence_15

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. Real number_sentence_16

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Real number_sentence_17

Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Real number_sentence_18

Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. Real number_sentence_19

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. Real number_sentence_20

Joseph Liouville (1840) showed that neither e nor e can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof. Real number_sentence_21

Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Real number_sentence_22

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. Real number_sentence_23

The development of calculus in the 18th century used the entire set of real numbers without having defined them rigorously. Real number_sentence_24

The first rigorous definition was published by Georg Cantor in 1871. Real number_sentence_25

In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Real number_sentence_26

Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. Real number_sentence_27

For more, see Cantor's first uncountability proof. Real number_sentence_28

Definition Real number_section_1

Main article: Construction of the real numbers Real number_sentence_29

Axiomatic approach Real number_section_2

Let R denote the set of all real numbers, then: Real number_sentence_30

Real number_unordered_list_0

  • The set R is a field, meaning that addition and multiplication are defined and have the usual properties.Real number_item_0_0
  • The field R is ordered, meaning that there is a total order ≥ such that for all real numbers x, y and z:Real number_item_0_1
    • if x ≥ y, then x + z ≥ y + z;Real number_item_0_2
    • if x ≥ 0 and y ≥ 0, then xy ≥ 0.Real number_item_0_3
  • The order is Dedekind-complete, meaning that every non-empty subset S of R with an upper bound in R has a least upper bound (a.k.a., supremum) in R.Real number_item_0_4

The last property is what differentiates the reals from the rationals (and from other more exotic ordered fields). Real number_sentence_31

For example, the set of rationals with square less than 2 has rational upper bounds (e.g., 1.42), but no rational least upper bound, because the square root of 2 is not rational. Real number_sentence_32

These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers is not upper-bounded in the reals. Real number_sentence_33

In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N. Real number_sentence_34

The real numbers are uniquely specified by the above properties. Real number_sentence_35

More precisely, given any two Dedekind-complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2. Real number_sentence_36

This uniqueness allows us to think of them as essentially the same mathematical object. Real number_sentence_37

For another axiomatization of ℝ, see Tarski's axiomatization of the reals. Real number_sentence_38

Construction from the rational numbers Real number_section_3

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; ...) converges to a unique real number—in this case π. Real number_sentence_39

For details and other constructions of real numbers, see construction of the real numbers. Real number_sentence_40

Properties Real number_section_4

Basic properties Real number_section_5

Real number_unordered_list_1

  • Any non-zero real number is either negative or positive.Real number_item_1_5
  • The sum and the product of two non-negative real numbers is again a non-negative real number, i.e., they are closed under these operations, and form a positive cone, thereby giving rise to a linear order of the real numbers along a number line.Real number_item_1_6
  • The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countably infinite. This establishes that in some sense, there are more real numbers than there are elements in any countable set.Real number_item_1_7
  • There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rationals, the algebraic numbers and the computable numbers, each set being a proper subset of the next in the sequence. The complements of all these sets (irrational, transcendental, and non-computable real numbers) with respect to the reals, are all uncountably infinite sets.Real number_item_1_8
  • Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the ellipsis (three dots) indicates that there would still be more digits to come. This hints to the fact that we can precisely denote only a few, selected real numbers with finitely many symbols.Real number_item_1_9

More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. Real number_sentence_41

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. Real number_sentence_42

The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. Real number_sentence_43

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. Real number_sentence_44

Completeness Real number_section_6

Main article: Completeness of the real numbers Real number_sentence_45

A main reason for using real numbers is that the reals contain all limits. Real number_sentence_46

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. Real number_sentence_47

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). Real number_sentence_48

Real number_sentence_49

A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other. Real number_sentence_50

A sequence (xn) converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − x| is less than ε for n greater than N. Real number_sentence_51

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. Real number_sentence_52

The set of rational numbers is not complete. Real number_sentence_53

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). Real number_sentence_54

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. Real number_sentence_55

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. Real number_sentence_56

For example, the standard series of the exponential function Real number_sentence_57

converges to a real number for every x, because the sums Real number_sentence_58

"The complete ordered field" Real number_section_7

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. Real number_sentence_59

First, an order can be lattice-complete. Real number_sentence_60

It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Real number_sentence_61

Additionally, an order can be Dedekind-complete, as defined in the section Axioms. Real number_sentence_62

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. Real number_sentence_63

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. Real number_sentence_64

These two notions of completeness ignore the field structure. Real number_sentence_65

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; the description in the previous section Completeness is a special case. Real number_sentence_66

(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.) Real number_sentence_67

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". Real number_sentence_68

Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". Real number_sentence_69

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. Real number_sentence_70

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. Real number_sentence_71

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. Real number_sentence_72

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. Real number_sentence_73

Advanced properties Real number_section_8

See also: Real line Real number_sentence_74

The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both sets are infinite. Real number_sentence_75

In fact, the 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 the set of algebraic numbers is countable, almost all real numbers are transcendental. Real number_sentence_76

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. Real number_sentence_77

The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. Real number_sentence_78

As a topological space, the real numbers are separable. Real number_sentence_79

This is because the set of rationals, which is countable, is dense in the real numbers. Real number_sentence_80

The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals. Real number_sentence_81

The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. Real number_sentence_82

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. Real number_sentence_83

The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. Real number_sentence_84

The reals are a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. Real number_sentence_85

The real numbers are locally compact but not compact. Real number_sentence_86

There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. Real number_sentence_87

Every nonnegative real number has a square root in R, although no negative number does. Real number_sentence_88

This shows that the order on R is determined by its algebraic structure. Real number_sentence_89

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. Real number_sentence_90

Proving this is the first half of one proof of the fundamental theorem of algebra. Real number_sentence_91

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. Real number_sentence_92

There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets. Real number_sentence_93

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. Real number_sentence_94

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. Real number_sentence_95

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. Real number_sentence_96

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. Real number_sentence_97

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. Real number_sentence_98

However, this existence theorem is purely theoretical, as such a base has never been explicitly described. Real number_sentence_99

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. Real number_sentence_100

(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.) Real number_sentence_101

Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. Real number_sentence_102

If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula. Real number_sentence_103

A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not. Real number_sentence_104

Applications and connections to other areas Real number_section_9

Real numbers and logic Real number_section_10

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. Real number_sentence_105

In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics. Real number_sentence_106

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others. Real number_sentence_107

Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". Real number_sentence_108

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). Real number_sentence_109

In physics Real number_section_11

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. Real number_sentence_110

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. Real number_sentence_111

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. Real number_sentence_112

In computation Real number_section_12

With some exceptions, most calculators do not operate on real numbers. Real number_sentence_113

Instead, they work with finite-precision approximations called floating-point numbers. Real number_sentence_114

In fact, most scientific computation uses floating-point arithmetic. Real number_sentence_115

Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not. Real number_sentence_116

A real number is called computable if there exists an algorithm that yields its digits. Real number_sentence_117

Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Real number_sentence_118

Moreover, the equality of two computable numbers is an undecidable problem. Real number_sentence_119

Some constructivists accept the existence of only those reals that are computable. Real number_sentence_120

The set of definable numbers is broader, but still only countable. Real number_sentence_121

"Reals" in set theory Real number_section_13

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Real number_sentence_122

Elements of Baire space are referred to as "reals". Real number_sentence_123

Vocabulary and notation Real number_section_14

Mathematicians use the symbol R, or, alternatively, ℝ, the letter "R" in blackboard bold (encoded in Unicode as U+211D ℝ DOUBLE-STRUCK CAPITAL R (HTML ℝ · &reals;, &Ropf;)), to represent the set of all real numbers. Real number_sentence_124

As this set is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration. Real number_sentence_125

The sets of positive real numbers and negative real numbers are often noted R and R, respectively; R+ and R− are also used. Real number_sentence_126

The non-negative real numbers can be noted R≥0 but one often sees this set noted R ∪ {0}. Real number_sentence_127

In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively ℝ+ and ℝ−. Real number_sentence_128

In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted ℝ+* and ℝ−*. Real number_sentence_129

The notation R refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. Real number_sentence_130

For example, a value from R consists of a tuple of three real numbers and specifies the coordinates of a point in 3‑dimensional space. Real number_sentence_131

In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). Real number_sentence_132

For example, real matrix, real polynomial and real Lie algebra. Real number_sentence_133

The word is also used as a noun, meaning a real number (as in "the set of all reals"). Real number_sentence_134

Generalizations and extensions Real number_section_15

The real numbers can be generalized and extended in several different directions: Real number_sentence_135

Real number_unordered_list_2

See also Real number_section_16

Real number_unordered_list_3

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: number.