Algebraic Structure of the Real Numbers - Revision history

Lila at 22:18, 25 October 2021

2021-10-25T22:18:00Z

Lila: /* Vocabulary and notation */

2021-10-25T22:17:02Z

Vocabulary and notation

Lila: /* Real line */

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Real line

Lila: /* Advanced properties */

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Advanced properties

Lila: /* "The complete ordered field" */

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"The complete ordered field"

Lila: /* Completeness */

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Completeness

Lila: /* Construction from the rational numbers */

2021-10-25T22:11:05Z

Construction from the rational numbers

Lila at 22:10, 25 October 2021

2021-10-25T22:10:23Z

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Lila at 20:38, 25 October 2021

2021-10-25T20:38:24Z

Lila at 20:36, 25 October 2021

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 The field <math>\mathbb{R}</math> of real numbers is an extension field of the field <math>\mathbb{Q}</math> of rational numbers, and <math>\mathbb{R}</math> can therefore be seen as a vector space over <math>\mathbb{Q}</math>. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set ''B'' of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of ''B'' is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
-The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on <math>\mathbb{R}</math> with the property that every non-empty subset of <math>\mathbb{R}</math> 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 has not been explicitly described. 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.<ref>{{citation |last=Moschovakis |first=Yiannis N. |title=Descriptive set theory |work=Studies in Logic and the Foundations of Mathematics |volume=100 |publisher=North-Holland Publishing Co. |location=Amsterdam; New York |year=1980 |pages=[https://archive.org/details/descriptivesetth0000mosc/page/ xii, 637] |isbn=978-0-444-85305-9 |url=https://archive.org/details/descriptivesetth0000mosc/page/ }}, chapter V.</ref>
+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 <math>\mathbb{R}</math> with the property that every non-empty subset of <math>\mathbb{R}</math> 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 has not been explicitly described. 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.
 A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.
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 Mathematicians use the symbol '''R''', or, alternatively, <math>\mathbb{R}</math>, the letter "R" in blackboard bold, to represent the set of all real numbers. As this set is naturally endowed with the structure of a field, the expression ''field of real numbers'' is frequently used when its algebraic properties are under consideration.
-The sets of positive real numbers and negative real numbers are often noted <math>\mathbb{R}^+</math> and <math>\mathbb{R}^-</math> respectively; <math>\mathbb{R}_+</math> and <math>\mathbb{R}_-</math> are also used. The non-negative real numbers can be noted <math>\mathbb{R}_{\ge 0}</math> but one often sees this set noted <math>\mathbb{R}^+ \cup \{0\}.</math> In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include zero, and these sets are noted respectively <math>\mathbb{R_+}</math> and <math>\mathbb{R_-}.</math><ref name="nombres-reels-ens-paris"/> In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted <math>\mathbb{R}_{+}*</math> and <math>\mathbb{R}_{-}*.</math><ref name="nombres-reels-ens-paris"/>
+The sets of positive real numbers and negative real numbers are often noted <math>\mathbb{R}^+</math> and <math>\mathbb{R}^-</math> respectively; <math>\mathbb{R}_+</math> and <math>\mathbb{R}_-</math> are also used. The non-negative real numbers can be noted <math>\mathbb{R}_{\ge 0}</math> but one often sees this set noted <math>\mathbb{R}^+ \cup \{0\}.</math> In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include zero, and these sets are noted respectively <math>\mathbb{R_+}</math> and <math>\mathbb{R_-}.</math> In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted <math>\mathbb{R}_{+}*</math> and <math>\mathbb{R}_{-}*.</math>
 The notation <math>\mathbb{R}^n</math> refers to the Cartesian product of {{mvar|n}} copies of <math>\mathbb{R}</math>, which is an {{mvar|n}}-dimensional vector space over the field of the real numbers; this vector space may be identified to the {{mvar|n}}-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from <math>\mathbb{R}^3</math> consists of a tuple of three real numbers and specifies the coordinates of a point in 3‑dimensional space.

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 == Vocabulary and notation ==
-Mathematicians use the symbol '''R''', or, alternatively, <math>\mathbb{R}</math>, the letter "R" in blackboard bold (encoded in Unicode as {{unichar|211D|DOUBLE-STRUCK CAPITAL R|html=}}), to represent the set of all real numbers. 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.
+Mathematicians use the symbol '''R''', or, alternatively, <math>\mathbb{R}</math>, the letter "R" in blackboard bold, to represent the set of all real numbers. As this set is naturally endowed with the structure of a field, the expression ''field of real numbers'' is frequently used when its algebraic properties are under consideration.
 The sets of positive real numbers and negative real numbers are often noted <math>\mathbb{R}^+</math> and <math>\mathbb{R}^-</math> respectively; <math>\mathbb{R}_+</math> and <math>\mathbb{R}_-</math> are also used. The non-negative real numbers can be noted <math>\mathbb{R}_{\ge 0}</math> but one often sees this set noted <math>\mathbb{R}^+ \cup \{0\}.</math> In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include zero, and these sets are noted respectively <math>\mathbb{R_+}</math> and <math>\mathbb{R_-}.</math><ref name="nombres-reels-ens-paris"/> In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted <math>\mathbb{R}_{+}*</math> and <math>\mathbb{R}_{-}*.</math><ref name="nombres-reels-ens-paris"/>
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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'', ''real polynomial'' and ''real Lie algebra''. The word is also used as a noun, meaning a real number (as in "the set of all reals").
 ==Licensing==

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 First, since the real numbers are totally ordered, they carry an order topology.  Second, the real numbers inherit a metric topology from the metric defined above.  The order topology and metric topology on {{math|'''R'''}} are the same.  As a topological space, the real line is homeomorphic to the open interval {{math|(0, 1)}}.
-The real line is trivially a topological manifold of dimension {{Num|1}}.  Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle.  It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.)
+The real line is trivially a topological manifold of dimension 1.  Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle.  It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.)
 The real line is a locally compact space and a paracompact space, as well as second-countable and normal.  It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point.  The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.
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 ===As a vector space===
 [[File:Bijection between vectors and points on number line.svg|thumb|300px|The bijection between points on the real line and vectors]]
-The real line is a vector space over the field {{math|'''R'''}} of real numbers (that is, over itself) of dimension {{Num|1}}. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.
+The real line is a vector space over the field {{math|'''R'''}} of real numbers (that is, over itself) of dimension 1. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.
 ===As a measure space===

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 As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
-The real numbers form a metric space: the distance between ''x'' and ''y'' is defined as the absolute value {{nowrap|{{!}}''x'' − ''y''{{!}}}}. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension&nbsp;1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
+The real numbers form a metric space: the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension&nbsp;1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
 Every nonnegative real number has a square root in <math>\mathbb{R}</math>, although no negative number does. This shows that the order on <math>\mathbb{R}</math> is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make <math>\mathbb{R}</math> 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 real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
-First, an order can be lattice-complete. 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).
+First, an order can be 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'', ''z'' + 1 is larger).
-Additionally, an order can be Dedekind-complete, see {{section link|#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.
+Additionally, an order can be Dedekind-complete.
 These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; 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 for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that <math>\mathbb{R}</math> is the ''only'' uniformly complete ordered field, but it is the only uniformly complete ''Archimedean field'', and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

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 === Completeness ===
 A main reason for using real numbers is so that many sequences have limits.  More formally, 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):
-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''. This definition, originally provided by Cauchy, formalizes the fact that the ''x''<sub>''n''</sub> eventually come and remain arbitrarily close to each other.
+A sequence (''x''<sub>''n''</sub>) 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 |''x<sub>n</sub>'' − ''x<sub>m</sub>''| 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 ''x''<sub>''n''</sub> eventually come and remain arbitrarily close to each other.
-A sequence (''x''<sub>''n''</sub>) ''converges to the limit'' ''x'' if its elements eventually come and remain arbitrarily close to ''x'', that is, 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 for ''n'' greater than ''N''.
+A sequence (''x''<sub>''n''</sub>) ''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 |''x<sub>n</sub>'' − ''x''| is less than ε for ''n'' greater than ''N''.
 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.

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 === Construction from the rational numbers ===
-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 {{pi}}. For details and other constructions of real numbers, see construction of the real numbers.
+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 <math>\pi</math>. For details and other constructions of real numbers, see construction of the real numbers.
 == Properties ==

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−	In mathematics, a '''real number''' is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as {~~{sqrt\|~~2}} (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as <math>\pi</math> (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol '''R''' or <math>\mathbb{R}</math> and is sometimes called "the reals".	+	In mathematics, a '''real number''' is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as <math>\sqrt{2}</math> (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as <math>\pi</math> (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol '''R''' or <math>\mathbb{R}</math> and is sometimes called "the reals".

	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. 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 the real numbers can be thought of as a part of the complex numbers.		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. 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 the real numbers can be thought of as a part of the complex numbers.

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 In mathematics, a '''real number''' is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as {{sqrt|2}} (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as <math>\pi</math> (3.14159265...).  In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol '''R''' or <math>\mathbb{R}</math> and is sometimes called "the reals".
-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 [[integer]]s 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 the real numbers can be thought of as a part of the [[complex number]]s.
+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. 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 the real numbers can be thought of as a part of the complex numbers.
-[[File:Real number line.svg|thumb|upright=1.6|right|Real numbers can be thought of as points on an infinitely long [[number line]]]]
+[[File:Real number line.svg|thumb|upright=1.6|right|Real numbers can be thought of as points on an infinitely long number line]]
-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 current standard axiomatic definition is that real numbers form the unique [[Dedekind-complete]] [[ordered field]] {{nowrap|(<math>\mathbb{R}</math> ; + ; · ; <),}} [[up to]] an [[isomorphism]],{{efn|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.}} 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 infinite [[decimal representation]]s, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.
+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 current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (<math>\mathbb{R}</math> ; + ; · ; <), 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. All these definitions satisfy the axiomatic definition and are thus equivalent.
-The set of all real numbers is [[uncountable set|uncountable]], in the sense that 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. In fact, the [[cardinality]] of the set of all real numbers, denoted by <math>\mathfrak c</math> and called the [[cardinality of the continuum]], is strictly greater than the cardinality of the set of all natural numbers (denoted <math>\aleph_0</math>, [[aleph number#Aleph-nought|'aleph-naught']]).
+The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. In fact, the cardinality of the set of all real numbers, denoted by <math>\mathfrak c</math> and called the cardinality of the continuum, is strictly greater than the cardinality of the set of all natural numbers (denoted <math>\aleph_0</math>, 'aleph-naught').
-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]] (CH). It is neither provable nor refutable using the axioms of [[Zermelo–Fraenkel set theory]] including the [[axiom of choice]] (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.
+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 (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.
 == Definition ==