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id="toc-Axioms-sublist" class="vector-toc-list"> <li id="toc-On_the_least_upper_bound_property" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#On_the_least_upper_bound_property"> <div class="vector-toc-text"> 1.1.1 On the least upper bound property </div> </a> <ul id="toc-On_the_least_upper_bound_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-On_models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#On_models"> <div class="vector-toc-text"> 1.1.2 On models </div> </a> <ul id="toc-On_models-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tarski's_axiomatization_of_the_reals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tarski's_axiomatization_of_the_reals"> <div class="vector-toc-text"> 1.2 Tarski's axiomatization of the reals </div> </a> <ul id="toc-Tarski's_axiomatization_of_the_reals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Explicit_constructions_of_models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Explicit_constructions_of_models"> <div class="vector-toc-text"> 2 Explicit constructions of models </div> </a> <button aria-controls="toc-Explicit_constructions_of_models-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Explicit constructions of models subsection </button> <ul id="toc-Explicit_constructions_of_models-sublist" class="vector-toc-list"> <li id="toc-Construction_from_Cauchy_sequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_from_Cauchy_sequences"> <div class="vector-toc-text"> 2.1 Construction from Cauchy sequences </div> </a> <ul id="toc-Construction_from_Cauchy_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_by_Dedekind_cuts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_by_Dedekind_cuts"> <div class="vector-toc-text"> 2.2 Construction by Dedekind cuts </div> </a> <ul id="toc-Construction_by_Dedekind_cuts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_using_hyperreal_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_using_hyperreal_numbers"> <div class="vector-toc-text"> 2.3 Construction using hyperreal numbers </div> </a> <ul id="toc-Construction_using_hyperreal_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_from_surreal_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_from_surreal_numbers"> <div class="vector-toc-text"> 2.4 Construction from surreal numbers </div> </a> <ul id="toc-Construction_from_surreal_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_from_integers_(Eudoxus_reals)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_from_integers_(Eudoxus_reals)"> <div class="vector-toc-text"> 2.5 Construction from integers (Eudoxus reals) </div> </a> <ul id="toc-Construction_from_integers_(Eudoxus_reals)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_constructions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_constructions"> <div class="vector-toc-text"> 2.6 Other constructions </div> </a> <ul id="toc-Other_constructions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 3 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, there are several equivalent ways of defining the <a href="/wiki/Real_number" title="Real number">real numbers</a>. One of them is that they form a <a href="/wiki/Complete_ordered_field" class="mw-redirect" title="Complete ordered field">complete ordered field</a> that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> that satisfies the definition. The article presents several such constructions.<a href="#cite_note-FOOTNOTEWeiss2015-1">[1]</a> They are equivalent in the sense that, given the result of any two such constructions, there is a unique <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Axiomatic_definitions">Axiomatic definitions</h2>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=1" title="Edit section: Axiomatic definitions">edit</a>]</div> An <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic definition</a> of the real numbers consists of defining them as the elements of a complete ordered field.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> This means the following: The real numbers form a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, containing two distinguished elements denoted 0 and 1, and on which are defined two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> and one <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a>; the operations are called addition and multiplication of real numbers and denoted respectively with + and ×; the binary relation is inequality, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad34d009daf13edc6fd9207e76f6c9f1ed17fb20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \leq .}" /> Moreover, the following properties called <a href="/wiki/Axiom" title="Axiom">axioms</a> must be satisfied. The existence of such a <a href="/wiki/Mathematical_structure" title="Mathematical structure">structure</a> is a <a href="/wiki/Theorem" title="Theorem">theorem</a>, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique <a href="/wiki/Up_to" title="Up to">up to</a> an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction. <div class="mw-heading mw-heading3"><h3 id="Axioms">Axioms</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=2" title="Edit section: Axioms">edit</a>]</div> <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> under addition and multiplication. In other words, <ul><li>For all x, y, and z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x + (y + z) = (x + y) + z and x × (y × z) = (x × y) × z. (<a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> of addition and multiplication)</li> <li>For all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x + y = y + x and x × y = y × x. (<a href="/wiki/Commutative_operation" class="mw-redirect" title="Commutative operation">commutativity</a> of addition and multiplication)</li> <li>For all x, y, and z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x × (y + z) = (x × y) + (x × z). (<a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> of multiplication over addition)</li> <li>For all x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x + 0 = x. (existence of additive <a href="/wiki/Identity_element" title="Identity element">identity</a>)</li> <li>0 is not equal to 1, and for all x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x × 1 = x. (existence of multiplicative identity)</li> <li>For every x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, there exists an element −x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, such that x + (−x) = 0. (existence of additive <a href="/wiki/Inverse_element" title="Inverse element">inverses</a>)</li> <li>For every x ≠ 0 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, there exists an element x−1 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, such that x × x−1 = 1. (existence of multiplicative inverses)</li></ul></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" />. In other words, <ul><li>For all x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x ≤ x. (<a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexivity</a>)</li> <li>For all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, if x ≤ y and y ≤ x, then x = y. (<a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetry</a>)</li> <li>For all x, y, and z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, if x ≤ y and y ≤ z, then x ≤ z. (<a href="/wiki/Transitive_relation" title="Transitive relation">transitivity</a>)</li> <li>For all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, x ≤ y or y ≤ x. (<a href="/wiki/Total_order" title="Total order">totality</a>)</li></ul></li> <li>Addition and multiplication are compatible with the order. In other words, <ul><li>For all x, y and z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)</li> <li>For all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, if 0 ≤ x and 0 ≤ y, then 0 ≤ x × y (preservation of order under multiplication)</li></ul></li> <li>The order ≤ is complete in the following sense: every non-empty subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> that is <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">bounded above</a> has a <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a>. In other words, <ul><li>If A is a non-empty subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, and if A has an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" /> then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.</li></ul></li></ol> <div class="mw-heading mw-heading4"><h4 id="On_the_least_upper_bound_property">On the least upper bound property</h4>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=3" title="Edit section: On the least upper bound property">edit</a>]</div> Axiom 4, which requires the order to be <a href="/wiki/Dedekind-complete" class="mw-redirect" title="Dedekind-complete">Dedekind-complete</a>, implies the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a>. The axiom is crucial in the characterization of the reals. For example, the totally <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms. Note that the axiom is <a href="/wiki/Nonfirstorderizability" title="Nonfirstorderizability">nonfirstorderizable</a>, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a <a href="/wiki/List_of_first-order_theories" title="List of first-order theories">first-order logic theory</a>. <div class="mw-heading mw-heading4"><h4 id="On_models">On models</h4>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=4" title="Edit section: On models">edit</a>]</div> A model of real numbers is a <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> that satisfies the above axioms. Several models are given <a href="#Explicit_constructions_of_models">below</a>. Any two models are isomorphic; so, the real numbers are unique <a href="/wiki/Up_to" title="Up to">up to</a> isomorphisms. Saying that any two models are isomorphic means that for any two models <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,0_{\mathbb {R} },1_{\mathbb {R} },+_{\mathbb {R} },\times _{\mathbb {R} },\leq _{\mathbb {R} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,0_{\mathbb {R} },1_{\mathbb {R} },+_{\mathbb {R} },\times _{\mathbb {R} },\leq _{\mathbb {R} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb0262f2acad4040068856c5cb0bf5715fa6579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.501ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,0_{\mathbb {R} },1_{\mathbb {R} },+_{\mathbb {R} },\times _{\mathbb {R} },\leq _{\mathbb {R} })}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,0_{S},1_{S},+_{S},\times _{S},\leq _{S}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,0_{S},1_{S},+_{S},\times _{S},\leq _{S}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd148e3d4df34f3adb07ce21debae68ff619a461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.336ex; height:2.843ex;" alt="{\displaystyle (S,0_{S},1_{S},+_{S},\times _{S},\leq _{S}),}" /> there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3262d1bff8b7358b8b8752264225f386d14e2840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.104ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to S}" /> that preserves both the field operations and the order. Explicitly, <ul><li>f is both <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> and <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>.</li> <li>f(0ℝ) = 0S and f(1ℝ) = 1S.</li> <li>f(x +ℝ y) = f(x) +S f(y) and f(x ×ℝ y) = f(x) ×S f(y), for all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" /></li> <li>x ≤ℝ y <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> f(x) ≤S f(y), for all x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" /></li></ul> <div class="mw-heading mw-heading3"><h3 id="Tarski's_axiomatization_of_the_reals">Tarski's axiomatization of the reals</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=5" title="Edit section: Tarski's axiomatization of the reals">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization of the reals</a></div> An alternative synthetic <a href="/wiki/Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of the real numbers and their arithmetic was given by <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, consisting of only the 8 <a href="/wiki/Axiom" title="Axiom">axioms</a> shown below and a mere four <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a>: a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> called the real numbers, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> called order, denoted by the <a href="/wiki/Infix_operator" class="mw-redirect" title="Infix operator">infix operator</a> <, a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> called addition, denoted by the infix operator +, and the constant 1. Axioms of order (primitives: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, <): Axiom 1. If x < y, then not y < x. That is, "<" is an <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">asymmetric relation</a>. Axiom 2. If x < z, there exists a y such that x < y and y < z. In other words, "<" is <a href="/wiki/Dense_order" title="Dense order">dense</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />. Axiom 3. "<" is <a href="/wiki/Dedekind-complete" class="mw-redirect" title="Dedekind-complete">Dedekind-complete</a>. More formally, for all X, Y ⊆ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y. To clarify the above statement somewhat, let X ⊆ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> and Y ⊆ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />. We now define two common English verbs in a particular way that suits our purpose: <dl><dd>X precedes Y if and only if for every x ∈ X and every y ∈ Y, x < y.</dd></dl> <dl><dd>The real number z separates X and Y if and only if for every x ∈ X with x ≠ z and every y ∈ Y with y ≠ z, x < z and z < y.</dd></dl> Axiom 3 can then be stated as: <dl><dd>"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."</dd></dl> Axioms of addition (primitives: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, <, +): Axiom 4. x + (y + z) = (x + z) + y. Axiom 5. For all x, y, there exists a z such that x + z = y. Axiom 6. If x + y < z + w, then x < z or y < w. Axioms for one (primitives: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />, <, +, 1): Axiom 7. 1 ∈ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />. Axiom 8. 1 < 1 + 1. These axioms imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is a <a href="/wiki/Linearly_ordered_group" title="Linearly ordered group">linearly ordered</a> <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> under addition with distinguished element 1. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is also <a href="/wiki/Dedekind-complete" class="mw-redirect" title="Dedekind-complete">Dedekind-complete</a> and <a href="/wiki/Divisible_group" title="Divisible group">divisible</a>. <div class="mw-heading mw-heading2"><h2 id="Explicit_constructions_of_models">Explicit constructions of models</h2>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=6" title="Edit section: Explicit constructions of models">edit</a>]</div> We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>/<a href="/wiki/Charles_M%C3%A9ray" title="Charles Méray">Charles Méray</a>, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>/<a href="/wiki/Joseph_Bertrand" title="Joseph Bertrand">Joseph Bertrand</a> and <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> all occurred within a few years of each other. Each has advantages and disadvantages. <div class="mw-heading mw-heading3"><h3 id="Construction_from_Cauchy_sequences">Construction from Cauchy sequences</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=7" title="Edit section: Construction from Cauchy sequences">edit</a>]</div> A standard procedure to force all <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a> in a <a href="/wiki/Metric_space" title="Metric space">metric space</a> to converge is adding new points to the metric space in a process called <a href="/wiki/Completeness_(topology)" class="mw-redirect" title="Completeness (topology)">completion</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is defined as the completion of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> of the rational numbers with respect to the metric |x − y| Normally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.<a href="#cite_note-5">[5]</a> Let R be the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of Cauchy sequences of rational numbers. That is, sequences <dl><dd>(x1, x2, x3,...)</dd></dl> of rational numbers such that for every rational ε > 0, there exists an integer N such that for all natural numbers m, n > N, one has |xm − xn| < ε. Here the vertical bars denote the absolute value. Cauchy sequences (xn) and (yn) can be added and multiplied as follows: <dl><dd>(xn) + (yn) = (xn + yn)</dd> <dd>(xn) × (yn) = (xn × yn).</dd></dl> Two Cauchy sequences (xn) and (yn) are called equivalent if and only if the difference between them tends to zero; that is, for every rational number ε > 0, there exists an integer N such that for all natural numbers n > N, one has |xn − yn| < ε. This defines an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> that is compatible with the operations defined above, and the set R of all <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> can be shown to satisfy <a href="#Axiomatic_definitions">all axioms of the real numbers</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> can be considered as a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> by identifying a rational number r with the equivalence class of the Cauchy sequence (r, r, r, ...). Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences: (xn) ≥ (yn) if and only if x is equivalent to y or there exists an integer N such that xn ≥ yn for all n > N. By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a Cauchy sequence representing x. This reflects the observation that one can often use different sequences to approximate the same real number.<a href="#cite_note-FOOTNOTEKemp2016-6">[6]</a> The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i.e. the <a href="/wiki/Least_upper_bound_property" class="mw-redirect" title="Least upper bound property">least upper bound property</a>. It can be proved as follows: Let S be a non-empty subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} '}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cdef1d27a06f81352c0aa015ac7b8c6de72a19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.363ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} '}" /> and U be an upper bound for S. Substituting a larger value if necessary, we may assume U is rational. Since S is non-empty, we can choose a rational number L such that L < s for some s in S. Now define sequences of rationals (un) and (ln) as follows: Set u0 = U and l0 = L. For each n consider the number mn = (un + ln)/2. If mn is an upper bound for S, set un+1 = mn and ln+1 = ln. Otherwise set ln+1 = mn and un+1 = un. This defines two Cauchy sequences of rationals, and so the real numbers l = (ln) and u = (un). It is easy to prove, by induction on n that un is an upper bound for S for all n and ln is never an upper bound for S for any n Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (un − ln) is 0, and so l = u. Now suppose b < u = l is a smaller upper bound for S. Since (ln) is monotonic increasing it is easy to see that b < ln for some n. But ln is not an upper bound for S and so neither is b. Hence u is a least upper bound for S and ≤ is complete. The usual <a href="/wiki/Decimal_notation" class="mw-redirect" title="Decimal notation">decimal notation</a> can be translated to Cauchy sequences in a natural way. For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). The equation <a href="/wiki/0.999..." title="0.999...">0.999...</a> = 1 states that the sequences (0, 0.9, 0.99, 0.999,...) and (1, 1, 1, 1,...) are equivalent, i.e., their difference converges to 0. An advantage of constructing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> as the completion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> is that this construction can be used for every other metric space. <div class="mw-heading mw-heading3"><h3 id="Construction_by_Dedekind_cuts">Construction by Dedekind cuts</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=8" title="Edit section: Construction by Dedekind cuts">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Dedekind_cut_at_square_root_of_two.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Dedekind_cut_at_square_root_of_two.svg/350px-Dedekind_cut_at_square_root_of_two.svg.png" decoding="async" width="350" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Dedekind_cut_at_square_root_of_two.svg/525px-Dedekind_cut_at_square_root_of_two.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Dedekind_cut_at_square_root_of_two.svg/700px-Dedekind_cut_at_square_root_of_two.svg.png 2x" data-file-width="850" data-file-height="376" /></a><figcaption> Dedekind used his cut to construct the <a href="/wiki/Irrational_number" title="Irrational number">irrational</a>, <a href="/wiki/Real_number" title="Real number">real numbers</a>.</figcaption></figure> A <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cut</a> in an ordered field is a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a>. Real numbers can be constructed as Dedekind cuts of rational numbers.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> For convenience we may take the lower set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> as the representative of any given Dedekind cut <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fa8309a05c1e3d625eeae87ccf5f452e70dd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.737ex; height:2.843ex;" alt="{\displaystyle (A,B)\,}" />, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> completely determines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" />. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /> is any subset of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe2cebb770192c93ef809f66373735e307b0595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle {\textbf {Q}}}" /> of rational numbers that fulfills the following conditions:<a href="#cite_note-FOOTNOTEPugh2002-9">[9]</a> <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /> is not empty</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\neq {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69ab70069c15e74db10ae13da3235fa27dbdac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.155ex; height:2.676ex;" alt="{\displaystyle r\neq {\textbf {Q}}}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /> is closed downwards. In other words, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f1e6640c8ad45b49265625959e34d82d4432de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.368ex; height:2.509ex;" alt="{\displaystyle x,y\in {\textbf {Q}}}" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" />, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff293f5871b6477970fb1499cfa6b8b6dadcb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.045ex; height:2.176ex;" alt="{\displaystyle y\in r}" /> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656cc65bdd3b8c63f070da9acb19b9e9def8f3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.219ex; height:1.843ex;" alt="{\displaystyle x\in r}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /> contains no greatest element. In other words, there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656cc65bdd3b8c63f070da9acb19b9e9def8f3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.219ex; height:1.843ex;" alt="{\displaystyle x\in r}" /> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff293f5871b6477970fb1499cfa6b8b6dadcb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.045ex; height:2.176ex;" alt="{\displaystyle y\in r}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de6a6e4f44d9dfcbfaadbdcf388d4b8a6fed109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle y\leq x}" /></li></ol> <ul><li>We form the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">R</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110532b0727e9c4bc5b8e0ac19bb8fe581939162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle {\textbf {R}}}" /> of real numbers as the set of all Dedekind cuts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe2cebb770192c93ef809f66373735e307b0595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle {\textbf {Q}}}" />, and define a <a href="/wiki/Total_ordering" class="mw-redirect" title="Total ordering">total ordering</a> on the real numbers as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y\Leftrightarrow x\subseteq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo stretchy="false">⇔</mo> <mi>x</mi> <mo>⊆</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y\Leftrightarrow x\subseteq y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f4f37413b587896d2f7406e2fb163a9850aadf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.781ex; height:2.343ex;" alt="{\displaystyle x\leq y\Leftrightarrow x\subseteq y}" /></li> <li>We <a href="/wiki/Embedding" title="Embedding">embed</a> the rational numbers into the reals by identifying the rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /> with the set of all smaller rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in {\textbf {Q}}:x<q\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo>:</mo> <mi>x</mi> <mo><</mo> <mi>q</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in {\textbf {Q}}:x<q\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80dbaaae982559db0ae3f63728d230379d8b56c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.938ex; height:2.843ex;" alt="{\displaystyle \{x\in {\textbf {Q}}:x<q\}}" />.<a href="#cite_note-FOOTNOTEPugh2002-9">[9]</a> Since the rational numbers are <a href="/wiki/Dense_order" title="Dense order">dense</a>, such a set can have no greatest element and thus fulfills the conditions for being a real number laid out above.</li> <li><a href="/wiki/Addition" title="Addition">Addition</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B:=\{a+b:a\in A\land b\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B:=\{a+b:a\in A\land b\in B\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea4f9e637833df82d89bbfdea390f89aee6fbb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.421ex; height:2.843ex;" alt="{\displaystyle A+B:=\{a+b:a\in A\land b\in B\}}" /><a href="#cite_note-FOOTNOTEPugh2002-9">[9]</a></li> <li><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-B:=\{a-b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>b</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-B:=\{a-b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e177dd99ddc30fd982cba6ed502599a9cd2cff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.433ex; height:2.843ex;" alt="{\displaystyle A-B:=\{a-b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Q}}\setminus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo class="MJX-variant">∖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Q}}\setminus B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc7037b5ba6aa51258259b53dc80f95dda34ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.967ex; height:2.843ex;" alt="{\displaystyle {\textbf {Q}}\setminus B}" /> denotes the <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">relative complement</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe2cebb770192c93ef809f66373735e307b0595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle {\textbf {Q}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x:x\in {\textbf {Q}}\land x\notin B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo>∧</mo> <mi>x</mi> <mo>∉</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x:x\in {\textbf {Q}}\land x\notin B\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166f1b4ef654de75dd5c194b20deb7837e43e946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.287ex; height:2.843ex;" alt="{\displaystyle \{x:x\in {\textbf {Q}}\land x\notin B\}}" /></li> <li><a href="/wiki/Negation_of_a_number" class="mw-redirect" title="Negation of a number">Negation</a> is a special case of subtraction: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -B:=\{a-b:a<0\land b\in ({\textbf {Q}}\setminus B)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>B</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo><</mo> <mn>0</mn> <mo>∧</mo> <mi>b</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -B:=\{a-b:a<0\land b\in ({\textbf {Q}}\setminus B)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4cb97d4846de4ee736d352ebf07bb2ce94749e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.335ex; height:2.843ex;" alt="{\displaystyle -B:=\{a-b:a<0\land b\in ({\textbf {Q}}\setminus B)\}}" /></li> <li>Defining <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> is less straightforward.<a href="#cite_note-FOOTNOTEPugh2002-9">[9]</a> <ul><li>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa663f3ae395e714dc6df465c3204174f85abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\displaystyle A,B\geq 0}" /> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B:=\{a\times b:a\geq 0\land a\in A\land b\geq 0\land b\in B\}\cup \{x\in \mathrm {Q} :x<0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>≥</mo> <mn>0</mn> <mo>∧</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> <mo>∧</mo> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B:=\{a\times b:a\geq 0\land a\in A\land b\geq 0\land b\in B\}\cup \{x\in \mathrm {Q} :x<0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24807ee10d9e6d8374df07e1a49cbc8ea2d917f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.749ex; height:2.843ex;" alt="{\displaystyle A\times B:=\{a\times b:a\geq 0\land a\in A\land b\geq 0\land b\in B\}\cup \{x\in \mathrm {Q} :x<0\}}" /></li> <li>if either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a72cbbfdbb8b9d0dad053538c330994b308bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,}" /> is negative, we use the identities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>×</mo> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <mo>×</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <mo>×</mo> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5faf52d7cca19df2f5925e01c825d356e574af42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.349ex; height:2.843ex;" alt="{\displaystyle A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)\,}" /> to convert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> and/or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a72cbbfdbb8b9d0dad053538c330994b308bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,}" /> to positive numbers and then apply the definition above.</li></ul></li> <li>We define <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> in a similar manner: <ul><li>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\geq 0{\mbox{ and }}B>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>B</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\geq 0{\mbox{ and }}B>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94903b16765e2015703f2b4dae907f5594124ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.938ex; height:2.343ex;" alt="{\displaystyle A\geq 0{\mbox{ and }}B>0}" /> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A/B:=\{a/b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>b</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/B:=\{a/b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be23430ebb1de221541a48d8a6a9786c1d4223f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.077ex; height:2.843ex;" alt="{\displaystyle A/B:=\{a/b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}" /></li> <li>if either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a72cbbfdbb8b9d0dad053538c330994b308bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,}" /> is negative, we use the identities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A/B=-(A/{-B})=-(-A/B)=-A/{-B}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>B</mi> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/B=-(A/{-B})=-(-A/B)=-A/{-B}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6864b99952e8ff65fc371cda33e62b3d070d7f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.828ex; height:2.843ex;" alt="{\displaystyle A/B=-(A/{-B})=-(-A/B)=-A/{-B}\,}" /> to convert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> to a non-negative number and/or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a72cbbfdbb8b9d0dad053538c330994b308bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,}" /> to a positive number and then apply the definition above.</li></ul></li> <li><a href="/wiki/Supremum" class="mw-redirect" title="Supremum">Supremum</a>. If a nonempty set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> of real numbers has any upper bound in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">R</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110532b0727e9c4bc5b8e0ac19bb8fe581939162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle {\textbf {R}}}" />, then it has a least upper bound in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">R</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110532b0727e9c4bc5b8e0ac19bb8fe581939162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle {\textbf {R}}}" /> that is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee5de6dfb89da11866e7f6d4a1af6fb403dfeac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.468ex; height:3.843ex;" alt="{\displaystyle \bigcup S}" />.<a href="#cite_note-FOOTNOTEPugh2002-9">[9]</a></li></ul> As an example of a Dedekind cut representing an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, we may take the <a href="/wiki/Square_root_of_2" title="Square root of 2">positive square root of 2</a>. This can be defined by the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{x\in {\textbf {Q}}:x<0\lor x\times x<2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> <mo>:</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo>∨</mo> <mi>x</mi> <mo>×</mo> <mi>x</mi> <mo><</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{x\in {\textbf {Q}}:x<0\lor x\times x<2\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf3157dc5e967e42b31d15768328b6323db9e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.216ex; height:2.843ex;" alt="{\displaystyle A=\{x\in {\textbf {Q}}:x<0\lor x\times x<2\}}" />.<a href="#cite_note-FOOTNOTEHersh1997-10">[10]</a> It can be seen from the definitions above that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> is a real number, and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times A=2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times A=2\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf3685b8ba87a001cdbad6bafa638762261b962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.975ex; height:2.176ex;" alt="{\displaystyle A\times A=2\,}" />. However, neither claim is immediate. Showing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}" /> is real requires showing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> has no greatest element, i.e. that for any positive rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times x<2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mi>x</mi> <mo><</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times x<2\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4c7a8e2024948f11b1adb31a1945e4c882b50c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.148ex; height:2.176ex;" alt="{\displaystyle x\times x<2\,}" />, there is a rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c233e7cc39fac816991250d86e09b515d02e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.543ex; height:2.009ex;" alt="{\displaystyle y\,}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ce4ac40139ac6343aec4897ccfc9606dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.176ex;" alt="{\displaystyle x<y\,}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\times y<2\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>×</mo> <mi>y</mi> <mo><</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\times y<2\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43391e9bb678d08cc1388350221aee1078b2a876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.446ex; height:2.509ex;" alt="{\displaystyle y\times y<2\,.}" /> The choice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {2x+2}{x+2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {2x+2}{x+2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/172837d44574c114f29a7a760f6c806df0059768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.972ex; height:5.343ex;" alt="{\displaystyle y={\frac {2x+2}{x+2}}\,}" /> works. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times A\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times A\leq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7fc57b2ab3614d41c06995e3babf1fe612f1fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.588ex; height:2.343ex;" alt="{\displaystyle A\times A\leq 2}" /> but to show equality requires showing that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08ce4d4c86c5b43f36c8435fb598da6471047c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,}" /> is any rational number with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r<2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<2\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811e751050b8fd8ef08f8feddd72a4f49c33f808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.697ex; height:2.176ex;" alt="{\displaystyle r<2\,}" />, then there is positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r<x\times x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <mi>x</mi> <mo>×</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<x\times x\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101adb922e38964f4a8341fc8c76368fc6b799e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.034ex; height:1.843ex;" alt="{\displaystyle r<x\times x\,}" />. An advantage of this construction is that each real number corresponds to a unique cut. Furthermore, by relaxing the first two requirements of the definition of a cut, the <a href="/wiki/Extended_real_number" class="mw-redirect" title="Extended real number">extended real number</a> system may be obtained by associating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /> with the empty set and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" /> with all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Q</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe2cebb770192c93ef809f66373735e307b0595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle {\textbf {Q}}}" />. <div class="mw-heading mw-heading3"><h3 id="Construction_using_hyperreal_numbers">Construction using hyperreal numbers</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=9" title="Edit section: Construction using hyperreal numbers">edit</a>]</div> As in the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a>, one constructs the hyperrationals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{*}\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{*}\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5937515b3d1e4fda68bd1b932d15e0f718dee650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle ^{*}\mathbb {Q} }" /> from the rational numbers by means of an <a href="/wiki/Ultrafilter" title="Ultrafilter">ultrafilter</a>.<a href="#cite_note-11">[11]</a> Here a hyperrational is by definition a ratio of two <a href="/wiki/Hyperinteger" title="Hyperinteger">hyperintegers</a>. Consider the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> of all limited (i.e. finite) elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{*}\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{*}\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5937515b3d1e4fda68bd1b932d15e0f718dee650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle ^{*}\mathbb {Q} }" />. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> has a unique <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" />, the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> hyperrational numbers. The quotient ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e247d9e33f9921ba42fd3219101cd637e0097b6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle B/I}" /> gives the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> of real numbers.<a href="#cite_note-12">[12]</a> This construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. It turns out that the maximal ideal respects the order on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{*}\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{*}\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5937515b3d1e4fda68bd1b932d15e0f718dee650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle ^{*}\mathbb {Q} }" />. Hence the resulting field is an ordered field. Completeness can be proved in a similar way to the construction from the Cauchy sequences. <div class="mw-heading mw-heading3"><h3 id="Construction_from_surreal_numbers">Construction from surreal numbers</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=10" title="Edit section: Construction from surreal numbers">edit</a>]</div> Every ordered field can be embedded in the <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a>. The real numbers form a maximal subfield that is <a href="/wiki/Archimedean_group" title="Archimedean group">Archimedean</a> (meaning that no real number is infinitely large or infinitely small). This embedding is not unique, though it can be chosen in a canonical way. <div class="mw-heading mw-heading3"><h3 id="Construction_from_integers_(Eudoxus_reals)">Construction from integers (Eudoxus reals)</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=11" title="Edit section: Construction from integers (Eudoxus reals)">edit</a>]</div> A relatively less known construction allows to define real numbers using only the additive group of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /> with different versions.<a href="#cite_note-FOOTNOTEArthan2004-13">[13]</a><a href="#cite_note-FOOTNOTEA'Campo2003-14">[14]</a><a href="#cite_note-FOOTNOTEStreet2003-15">[15]</a> <a href="#CITEREFArthan2004">Arthan (2004)</a>, who attributes this construction to unpublished work by <a href="/wiki/Stephen_Schanuel" title="Stephen Schanuel">Stephen Schanuel</a>, refers to this construction as the Eudoxus reals, naming them after ancient Greek astronomer and mathematician <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>. As noted by <a href="#CITEREFShenitzer1987">Shenitzer (1987)</a> and <a href="#CITEREFArthan2004">Arthan (2004)</a>, Eudoxus's treatment of quantity using the behavior of <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportions</a> became the basis for this construction. This construction has been <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">formally verified</a> to give a Dedekind-complete ordered field by the IsarMathLib project.<a href="#cite_note-FOOTNOTEIsarMathLib-16">[16]</a> Let an almost homomorphism be a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {Z} \to \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {Z} \to \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08222a32dbc48971408c22a1019a8141d34c33d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.93ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {Z} \to \mathbb {Z} }" /> such that the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f(n+m)-f(m)-f(n):n,m\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f(n+m)-f(m)-f(n):n,m\in \mathbb {Z} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a914e1f6bd2f640dc77332ad8f84894df6d2d341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.777ex; height:2.843ex;" alt="{\displaystyle \{f(n+m)-f(m)-f(n):n,m\in \mathbb {Z} \}}" /> is finite. (Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=\lfloor \alpha n\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>α</mi> <mi>n</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=\lfloor \alpha n\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb1410adf6a02e2fd7addf728847038dadbd555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.528ex; height:2.843ex;" alt="{\displaystyle f(n)=\lfloor \alpha n\rfloor }" /> is an almost homomorphism for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7988141e89a37e7f4deb883dbd74d9bbd6d11317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} }" />.) Almost homomorphisms form an abelian group under pointwise addition. We say that two almost homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}" /> are almost equal if the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f(n)-g(n):n\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f(n)-g(n):n\in \mathbb {Z} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2d409a4a83f82264e4d0f9e23bd41c77b0489c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.691ex; height:2.843ex;" alt="{\displaystyle \{f(n)-g(n):n\in \mathbb {Z} \}}" /> is finite. This defines an equivalence relation on the set of almost homomorphisms. Real numbers are defined as the equivalence classes of this relation. Alternatively, the almost homomorphisms taking only finitely many values form a subgroup, and the underlying additive group of the real number is the quotient group. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to functional composition of almost homomorphisms. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [f]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7414a730e8157655ff770265ec3a03ec9f09dd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle [f]}" /> denotes the real number represented by an almost homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> we say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq [f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq [f]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a619cad56c3597fafd65950aa99007a14efaa385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.833ex; height:2.843ex;" alt="{\displaystyle 0\leq [f]}" /> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> is bounded or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> takes an infinite number of positive values on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628778fcf14bd3629e9b9ebacffa172b0ad6ce41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{+}}" />. This defines the <a href="/wiki/Total_order" title="Total order">linear order</a> relation on the set of real numbers constructed this way. <div class="mw-heading mw-heading3"><h3 id="Other_constructions">Other constructions</h3>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=12" title="Edit section: Other constructions">edit</a>]</div> <a href="#CITEREFFaltinMetropolisRossRota1975">Faltin et al. (1975)</a> write: "Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives."<a href="#cite_note-FOOTNOTEFaltinMetropolisRossRota1975-17">[17]</a> A number of other constructions have been given, by: <ul><li><a href="#CITEREFde_Bruijn1976">de Bruijn (1976)</a>, <a href="#CITEREFde_Bruijn1977">de Bruijn (1977)</a></li> <li><a href="#CITEREFRieger1982">Rieger (1982)</a></li> <li><a href="#CITEREFKnopfmacherKnopfmacher1987">Knopfmacher & Knopfmacher (1987)</a>, <a href="#CITEREFKnopfmacherKnopfmacher1988">Knopfmacher & Knopfmacher (1988)</a></li></ul> For an overview, see <a href="#CITEREFWeiss2015">Weiss (2015)</a>. As a reviewer of one noted: "The details are all included, but as usual they are tedious and not too instructive."<a href="#cite_note-18">[18]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=13" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Constructivism_(mathematics)#Example_from_real_analysis" class="mw-redirect" title="Constructivism (mathematics)">Constructivism (mathematics)#Example from real analysis</a> – Mathematical viewpoint that existence proofs must be constructivePages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Decidability_of_first-order_theories_of_the_real_numbers" title="Decidability of first-order theories of the real numbers">Decidability of first-order theories of the real numbers</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=14" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEWeiss2015-1"><a href="#cite_ref-FOOTNOTEWeiss2015_1-0">^</a> <a href="#CITEREFWeiss2015">Weiss 2015</a>. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://math.colorado.edu/~nita/RealNumbers.pdf">"Real Numbers"</a> (PDF). <a href="/wiki/University_of_Colorado_Boulder" title="University of Colorado Boulder">University of Colorado Boulder</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+Colorado+Boulder&rft.atitle=Real+Numbers&rft_id=http%3A%2F%2Fmath.colorado.edu%2F~nita%2FRealNumbers.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSaunders2015" class="citation web cs1">Saunders, Bonnie (August 21, 2015). <a rel="nofollow" class="external text" href="http://homepages.math.uic.edu/~saunders/MATH313/INRA/INRA_chapters0and1.pdf">"Interactive Notes for Real Analysis"</a> (PDF). <a href="/wiki/University_of_Illinois_at_Chicago" class="mw-redirect" title="University of Illinois at Chicago">University of Illinois at Chicago</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+Illinois+at+Chicago&rft.atitle=Interactive+Notes+for+Real+Analysis&rft.date=2015-08-21&rft.aulast=Saunders&rft.aufirst=Bonnie&rft_id=http%3A%2F%2Fhomepages.math.uic.edu%2F~saunders%2FMATH313%2FINRA%2FINRA_chapters0and1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101226005339/http://math.uci.edu/~mfinkels/140A/Introduction%20and%20Logic%20Notes.pdf">"Axioms of the Real Number System"</a> (PDF). <a href="/wiki/University_of_California,_Irvine" title="University of California, Irvine">University of California, Irvine</a>. Archived from <a rel="nofollow" class="external text" href="https://www.math.uci.edu/~mfinkels/140A/Introduction%2520and%2520Logic%2520Notes.pdf">the original</a> (PDF) on December 26, 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+California%2C+Irvine&rft.atitle=Axioms+of+the+Real+Number+System&rft_id=https%3A%2F%2Fwww.math.uci.edu%2F~mfinkels%2F140A%2FIntroduction%252520and%252520Logic%252520Notes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> For completions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> with respect to other metrics, see <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">p-adic numbers</a>). </li> <li id="cite_note-FOOTNOTEKemp2016-6"><a href="#cite_ref-FOOTNOTEKemp2016_6-0">^</a> <a href="#CITEREFKemp2016">Kemp 2016</a>. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a rel="nofollow" class="external text" href="https://www.math.ucdavis.edu/~temple/MAT25/HomeworkProblems.pdf">Math 25 Exercises</a> ucdavis.edu </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a rel="nofollow" class="external text" href="http://math.furman.edu/~tlewis/math41/Pugh/chap1/sec2.pdf">1.2–Cuts</a> furman.edu </li> <li id="cite_note-FOOTNOTEPugh2002-9">^ <a href="#cite_ref-FOOTNOTEPugh2002_9-0">a</a> <a href="#cite_ref-FOOTNOTEPugh2002_9-1">b</a> <a href="#cite_ref-FOOTNOTEPugh2002_9-2">c</a> <a href="#cite_ref-FOOTNOTEPugh2002_9-3">d</a> <a href="#cite_ref-FOOTNOTEPugh2002_9-4">e</a> <a href="#CITEREFPugh2002">Pugh 2002</a>. </li> <li id="cite_note-FOOTNOTEHersh1997-10"><a href="#cite_ref-FOOTNOTEHersh1997_10-0">^</a> <a href="#CITEREFHersh1997">Hersh 1997</a>. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKrakoff2015" class="citation web cs1">Krakoff, Gianni (June 8, 2015). <a rel="nofollow" class="external text" href="https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf">"Hyperreals and a Brief Introduction to Non-Standard Analysis"</a> (PDF). Department of Mathematics, <a href="/wiki/University_of_Washington" title="University of Washington">University of Washington</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Department+of+Mathematics%2C+University+of+Washington&rft.atitle=Hyperreals+and+a+Brief+Introduction+to+Non-Standard+Analysis&rft.date=2015-06-08&rft.aulast=Krakoff&rft.aufirst=Gianni&rft_id=https%3A%2F%2Fsites.math.washington.edu%2F~morrow%2F336_15%2Fpapers%2Fgianni.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoldblatt1998" class="citation book cs1">Goldblatt, Robert (1998). "Exercise 5.7 (4)". Lectures on the Hyperreals: An introduction to nonstandard analysis. Graduate Texts in Mathematics. Vol. 188. New York: Springer-Verlag. p. 54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0615-6">10.1007/978-1-4612-0615-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98464-X" title="Special:BookSources/0-387-98464-X"><bdi>0-387-98464-X</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1643950">1643950</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exercise+5.7+%284%29&rft.btitle=Lectures+on+the+Hyperreals%3A+An+introduction+to+nonstandard+analysis&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pages=54&rft.pub=Springer-Verlag&rft.date=1998&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1643950%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0615-6&rft.isbn=0-387-98464-X&rft.aulast=Goldblatt&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArthan2004-13"><a href="#cite_ref-FOOTNOTEArthan2004_13-0">^</a> <a href="#CITEREFArthan2004">Arthan 2004</a>. </li> <li id="cite_note-FOOTNOTEA'Campo2003-14"><a href="#cite_ref-FOOTNOTEA'Campo2003_14-0">^</a> <a href="#CITEREFA'Campo2003">A'Campo 2003</a>. </li> <li id="cite_note-FOOTNOTEStreet2003-15"><a href="#cite_ref-FOOTNOTEStreet2003_15-0">^</a> <a href="#CITEREFStreet2003">Street 2003</a>. </li> <li id="cite_note-FOOTNOTEIsarMathLib-16"><a href="#cite_ref-FOOTNOTEIsarMathLib_16-0">^</a> <a href="#CITEREFIsarMathLib">IsarMathLib</a>. </li> <li id="cite_note-FOOTNOTEFaltinMetropolisRossRota1975-17"><a href="#cite_ref-FOOTNOTEFaltinMetropolisRossRota1975_17-0">^</a> <a href="#CITEREFFaltinMetropolisRossRota1975">Faltin et al. 1975</a>. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=693180">693180</a> (84j:26002) review of <a href="#CITEREFRieger1982">Rieger1982</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Construction_of_the_real_numbers&action=edit&section=15" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFA'Campo2003" class="citation arxiv cs1"><a href="/wiki/Norbert_A%27Campo" title="Norbert A'Campo">A'Campo, Norbert</a> (2003). "A natural construction for the real numbers". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0301015">math/0301015</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+natural+construction+for+the+real+numbers&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0301015&rft.aulast=A%27Campo&rft.aufirst=Norbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArthan2004" class="citation arxiv cs1">Arthan, R.D. (2004). "The Eudoxus Real Numbers". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0405454">math/0405454</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Eudoxus+Real+Numbers&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0405454&rft.aulast=Arthan&rft.aufirst=R.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFde_Bruijn1976" class="citation journal cs1">de Bruijn, N.G. (1976). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/138572587690055X/pdf">"Defining reals without the use of rationals"</a>. Indagationes Mathematicae (Proceedings). 79 (2): 100–108. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F1385-7258%2876%2990055-X">10.1016/1385-7258(76)90055-X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indagationes+Mathematicae+%28Proceedings%29&rft.atitle=Defining+reals+without+the+use+of+rationals&rft.volume=79&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E100-%3C%2Fspan%3E108&rft.date=1976&rft_id=info%3Adoi%2F10.1016%2F1385-7258%2876%2990055-X&rft.aulast=de+Bruijn&rft.aufirst=N.G.&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2F138572587690055X%2Fpdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"> also at <a rel="nofollow" class="external free" href="http://alexandria.tue.nl/repository/freearticles/597556.pdf">http://alexandria.tue.nl/repository/freearticles/597556.pdf</a></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFde_Bruijn1977" class="citation journal cs1">de Bruijn, N.G. (1977). "Construction of the system of real numbers". Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 86 (9): 121–125.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nederl.+Akad.+Wetensch.+Verslag+Afd.+Natuurk.&rft.atitle=Construction+of+the+system+of+real+numbers&rft.volume=86&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E121-%3C%2Fspan%3E125&rft.date=1977&rft.aulast=de+Bruijn&rft.aufirst=N.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFaltinMetropolisRossRota1975" class="citation journal cs1">Faltin, F.; Metropolis, M.; Ross, B.; Rota, G.-C. (1975). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2875%2990115-2">"The real numbers as a wreath product"</a>. <a href="/wiki/Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>. 16 (3): 278–304. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2875%2990115-2">10.1016/0001-8708(75)90115-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=The+real+numbers+as+a+wreath+product&rft.volume=16&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E278-%3C%2Fspan%3E304&rft.date=1975&rft_id=info%3Adoi%2F10.1016%2F0001-8708%2875%2990115-2&rft.aulast=Faltin&rft.aufirst=F.&rft.au=Metropolis%2C+M.&rft.au=Ross%2C+B.&rft.au=Rota%2C+G.-C.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0001-8708%252875%252990115-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHersh1997" class="citation book cs1">Hersh, Reuben (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R-qgdx2A5b0C">What is Mathematics, Really?</a>. 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The Mathematical Intelligencer. 9 (3): 44–52. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf03023955">10.1007/bf03023955</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122199850">122199850</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=A+topics+course+in+mathematics&rft.volume=9&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E44-%3C%2Fspan%3E52&rft.date=1987&rft_id=info%3Adoi%2F10.1007%2Fbf03023955&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122199850%23id-name%3DS2CID&rft.aulast=Shenitzer&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConstruction+of+the+real+numbers" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStreet2003" class="citation web cs1">Street, Ross (September 2003). <a rel="nofollow" class="external text" href="http://www.maths.mq.edu.au/~street/reals.pdf">"Update on the efficient reals"</a> (PDF). 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