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aria-controls="toc-Reaalarvude_konstruktsioonid-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Lülita ümber alaosa "Reaalarvude konstruktsioonid"</span> </button> <ul id="toc-Reaalarvude_konstruktsioonid-sublist" class="vector-toc-list"> <li id="toc-Cantori_fundamentaaljadade_teooria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cantori_fundamentaaljadade_teooria"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Cantori fundamentaaljadade teooria</span> </div> </a> <ul id="toc-Cantori_fundamentaaljadade_teooria-sublist" class="vector-toc-list"> <li id="toc-Kokkuvõte" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Kokkuvõte"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Kokkuvõte</span> </div> </a> <ul id="toc-Kokkuvõte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cauchy_koonduvuskriteerium_ja_selle_kasutamine_Cantoril" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cauchy_koonduvuskriteerium_ja_selle_kasutamine_Cantoril"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Cauchy koonduvuskriteerium ja selle kasutamine Cantoril</span> </div> </a> <ul id="toc-Cauchy_koonduvuskriteerium_ja_selle_kasutamine_Cantoril-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reaalarvude_konstruktsioon_Cantori_järgi" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reaalarvude_konstruktsioon_Cantori_järgi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Reaalarvude konstruktsioon Cantori järgi</span> </div> </a> <ul id="toc-Reaalarvude_konstruktsioon_Cantori_järgi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reaalarvude_hulga_täielikkus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reaalarvude_hulga_täielikkus"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.4</span> <span>Reaalarvude hulga täielikkus</span> </div> </a> <ul id="toc-Reaalarvude_hulga_täielikkus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lõpmatute_kümnendmurdude_teooria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lõpmatute_kümnendmurdude_teooria"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Lõpmatute kümnendmurdude teooria</span> </div> </a> <ul id="toc-Lõpmatute_kümnendmurdude_teooria-sublist" class="vector-toc-list"> <li id="toc-Ratsionaalarvud_ja_kümnendmurrud" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ratsionaalarvud_ja_kümnendmurrud"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Ratsionaalarvud ja kümnendmurrud</span> </div> </a> <ul id="toc-Ratsionaalarvud_ja_kümnendmurrud-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lõpmatute_kümnendmurdude_teooria_konstruktsioon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lõpmatute_kümnendmurdude_teooria_konstruktsioon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Lõpmatute kümnendmurdude teooria konstruktsioon</span> </div> </a> <ul id="toc-Lõpmatute_kümnendmurdude_teooria_konstruktsioon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ajalooline_kommentaar" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ajalooline_kommentaar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.3</span> <span>Ajalooline kommentaar</span> </div> </a> <ul id="toc-Ajalooline_kommentaar-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lõigete_teooria_ratsionaalarvude_vallas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lõigete_teooria_ratsionaalarvude_vallas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Lõigete teooria ratsionaalarvude vallas</span> </div> </a> <ul id="toc-Lõigete_teooria_ratsionaalarvude_vallas-sublist" class="vector-toc-list"> <li id="toc-Küsimuseasetus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Küsimuseasetus"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>Küsimuseasetus</span> </div> </a> <ul id="toc-Küsimuseasetus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ratsionaalarvude_võrdlus_sirge_punktidega" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ratsionaalarvude_võrdlus_sirge_punktidega"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.2</span> <span>Ratsionaalarvude võrdlus sirge punktidega</span> </div> </a> <ul id="toc-Ratsionaalarvude_võrdlus_sirge_punktidega-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sirge_pidevus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sirge_pidevus"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.3</span> <span>Sirge pidevus</span> </div> </a> <ul id="toc-Sirge_pidevus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pidevus_Dedekindi_järgi" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pidevus_Dedekindi_järgi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.4</span> <span>Pidevus Dedekindi järgi</span> </div> </a> <ul id="toc-Pidevus_Dedekindi_järgi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irratsionaalarvude_konstrueerimine" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Irratsionaalarvude_konstrueerimine"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.5</span> <span>Irratsionaalarvude konstrueerimine</span> </div> </a> <ul id="toc-Irratsionaalarvude_konstrueerimine-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Reaalarvude_aksiomaatikad" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reaalarvude_aksiomaatikad"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Reaalarvude aksiomaatikad</span> </div> </a> <button aria-controls="toc-Reaalarvude_aksiomaatikad-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Lülita ümber alaosa "Reaalarvude aksiomaatikad"</span> </button> <ul id="toc-Reaalarvude_aksiomaatikad-sublist" class="vector-toc-list"> <li id="toc-Reaalarvude_hulk_kui_pidev_järjestatud_korpus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reaalarvude_hulk_kui_pidev_järjestatud_korpus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Reaalarvude hulk kui pidev järjestatud korpus</span> </div> </a> <ul id="toc-Reaalarvude_hulk_kui_pidev_järjestatud_korpus-sublist" class="vector-toc-list"> <li id="toc-Korpuse_aksioomid" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Korpuse_aksioomid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Korpuse aksioomid</span> </div> </a> <ul id="toc-Korpuse_aksioomid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Järjestuse_aksioomid" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Järjestuse_aksioomid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Järjestuse aksioomid</span> </div> </a> <ul id="toc-Järjestuse_aksioomid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pidevuse_aksioomid" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pidevuse_aksioomid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Pidevuse aksioomid</span> </div> </a> <ul id="toc-Pidevuse_aksioomid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kokkuvõte_ja_definitsioon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Kokkuvõte_ja_definitsioon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Kokkuvõte ja definitsioon</span> </div> </a> <ul id="toc-Kokkuvõte_ja_definitsioon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Reaalarvude_hulk_kui_maksimaalne_arhimeediline_järjestatud_korpus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reaalarvude_hulk_kui_maksimaalne_arhimeediline_järjestatud_korpus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Reaalarvude hulk kui maksimaalne arhimeediline järjestatud korpus</span> </div> </a> <ul id="toc-Reaalarvude_hulk_kui_maksimaalne_arhimeediline_järjestatud_korpus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tarski_reaalarvude_aksiomaatika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tarski_reaalarvude_aksiomaatika"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Tarski reaalarvude aksiomaatika</span> </div> </a> <ul id="toc-Tarski_reaalarvude_aksiomaatika-sublist" class="vector-toc-list"> <li id="toc-Aksioomid" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aksioomid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Aksioomid</span> </div> </a> <ul id="toc-Aksioomid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kuidas_need_aksioomid_toovad_kaasa_korpuse" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Kuidas_need_aksioomid_toovad_kaasa_korpuse"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Kuidas need aksioomid toovad kaasa korpuse</span> </div> </a> <ul id="toc-Kuidas_need_aksioomid_toovad_kaasa_korpuse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kirjandus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Kirjandus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Kirjandus</span> </div> </a> <ul id="toc-Kirjandus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Lõpmatu_kümnendarendus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lõpmatu_kümnendarendus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lõpmatu kümnendarendus</span> </div> </a> <ul id="toc-Lõpmatu_kümnendarendus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reaalarvude_korpus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reaalarvude_korpus"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Reaalarvude korpus</span> </div> </a> <ul id="toc-Reaalarvude_korpus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ajalugu" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ajalugu"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ajalugu</span> </div> </a> <button aria-controls="toc-Ajalugu-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Lülita ümber alaosa "Ajalugu"</span> </button> <ul id="toc-Ajalugu-sublist" class="vector-toc-list"> <li id="toc-Naiivne_reaalarvude_teooria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Naiivne_reaalarvude_teooria"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Naiivne reaalarvude teooria</span> </div> </a> <ul id="toc-Naiivne_reaalarvude_teooria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Range_teooria_loomine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Range_teooria_loomine"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Range teooria loomine</span> </div> </a> <ul id="toc-Range_teooria_loomine-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vaata_ka" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vaata_ka"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Vaata ka</span> </div> </a> <ul id="toc-Vaata_ka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Viited" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Viited"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Viited</span> </div> </a> <ul id="toc-Viited-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sisukord" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Lülita sisukord ümber" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Lülita sisukord ümber</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Reaalarv</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Mine teises keeles artiklisse. Saadaval 118 keeles" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-118" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">118 keelt</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Re%C3%ABle_getal" title="Reële getal – afrikaani" lang="af" hreflang="af" data-title="Reële getal" data-language-autonym="Afrikaans" data-language-local-name="afrikaani" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – šveitsisaksa" lang="gsw" hreflang="gsw" data-title="Reelle Zahl" data-language-autonym="Alemannisch" data-language-local-name="šveitsisaksa" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Reaalloho" title="Reaalloho – Inari saami" lang="smn" hreflang="smn" data-title="Reaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="Inari saami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%D9%8A%D9%82%D9%8A" title="عدد حقيقي – araabia" lang="ar" hreflang="ar" data-title="عدد حقيقي" data-language-autonym="العربية" data-language-local-name="araabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_real" title="Númberu real – astuuria" lang="ast" hreflang="ast" data-title="Númberu real" data-language-autonym="Asturianu" data-language-local-name="astuuria" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99qiqi_%C9%99d%C9%99dl%C9%99r" title="Həqiqi ədədlər – aserbaidžaani" lang="az" hreflang="az" data-title="Həqiqi ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="aserbaidžaani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="حقیقی ساییلار – lõunaaserbaidžaani" lang="azb" hreflang="azb" data-title="حقیقی ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="lõunaaserbaidžaani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_riil" title="Bilangan riil – indoneesia" lang="id" hreflang="id" data-title="Bilangan riil" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_nyata" title="Nombor nyata – malai" lang="ms" hreflang="ms" data-title="Nombor nyata" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A6%AC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বাস্তব সংখ্যা – bengali" lang="bn" hreflang="bn" data-title="বাস্তব সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%CC%8Dt-s%C3%B2%CD%98" title="Si̍t-sò͘ – lõunamini" lang="nan" hreflang="nan" data-title="Si̍t-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="lõunamini" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD" title="Ысын һан – baškiiri" lang="ba" hreflang="ba" data-title="Ысын һан" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – valgevene" lang="be" hreflang="be" data-title="Рэчаісны лік" data-language-autonym="Беларуская" data-language-local-name="valgevene" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – valgevene (taraškievitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Рэчаісны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="valgevene (taraškievitsa)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – bihaari" lang="bh" hreflang="bh" data-title="वास्तविक संख्या" data-language-autonym="भोजपुरी" data-language-local-name="bihaari" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – keskbikoli" lang="bcl" hreflang="bcl" data-title="Tunay na bilang" data-language-autonym="Bikol Central" data-language-local-name="keskbikoli" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Realan_broj" title="Realan broj – bosnia" lang="bs" hreflang="bs" data-title="Realan broj" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Реално число – bulgaaria" lang="bg" hreflang="bg" data-title="Реално число" data-language-autonym="Български" data-language-local-name="bulgaaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%91%D0%BE%D0%B4%D0%BE%D1%82%D0%BE_%D1%82%D0%BE%D0%BE" title="Бодото тоо – burjaadi" lang="bxr" hreflang="bxr" data-title="Бодото тоо" data-language-autonym="Буряад" data-language-local-name="burjaadi" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="hea artikkel"><a href="https://ca.wikipedia.org/wiki/Nombre_real" title="Nombre real – katalaani" lang="ca" hreflang="ca" data-title="Nombre real" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A7%C4%83%D0%BD_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Чăн хисеп – tšuvaši" lang="cv" hreflang="cv" data-title="Чăн хисеп" data-language-autonym="Чӑвашла" data-language-local-name="tšuvaši" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo – tšehhi" lang="cs" hreflang="cs" data-title="Reálné číslo" data-language-autonym="Čeština" data-language-local-name="tšehhi" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_real" title="Rhif real – kõmri" lang="cy" hreflang="cy" data-title="Rhif real" data-language-autonym="Cymraeg" data-language-local-name="kõmri" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reelle_tal" title="Reelle tal – taani" lang="da" hreflang="da" data-title="Reelle tal" data-language-autonym="Dansk" data-language-local-name="taani" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – saksa" lang="de" hreflang="de" data-title="Reelle Zahl" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πραγματικός αριθμός – kreeka" lang="el" hreflang="el" data-title="Πραγματικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="kreeka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_re%C3%A8l" title="Nómmer reèl – emiilia-romanja" lang="egl" hreflang="egl" data-title="Nómmer reèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="emiilia-romanja" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Real_number" title="Real number – inglise" lang="en" hreflang="en" data-title="Real number" data-language-autonym="English" data-language-local-name="inglise" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – hispaania" lang="es" hreflang="es" data-title="Número real" data-language-autonym="Español" data-language-local-name="hispaania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Reelo" title="Reelo – esperanto" lang="eo" hreflang="eo" data-title="Reelo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_erreal" title="Zenbaki erreal – baski" lang="eu" hreflang="eu" data-title="Zenbaki erreal" data-language-autonym="Euskara" data-language-local-name="baski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%DB%8C%D9%82%DB%8C" title="عدد حقیقی – pärsia" lang="fa" hreflang="fa" data-title="عدد حقیقی" data-language-autonym="فارسی" data-language-local-name="pärsia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Reelt_tal" title="Reelt tal – fääri" lang="fo" hreflang="fo" data-title="Reelt tal" data-language-autonym="Føroyskt" data-language-local-name="fääri" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_r%C3%A9el" title="Nombre réel – prantsuse" lang="fr" hreflang="fr" data-title="Nombre réel" data-language-autonym="Français" data-language-local-name="prantsuse" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Numars_re%C3%A2i" title="Numars reâi – friuuli" lang="fur" hreflang="fur" data-title="Numars reâi" data-language-autonym="Furlan" data-language-local-name="friuuli" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9aduimhir" title="Réaduimhir – iiri" lang="ga" hreflang="ga" data-title="Réaduimhir" data-language-autonym="Gaeilge" data-language-local-name="iiri" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Feer_earroo" title="Feer earroo – mänksi" lang="gv" hreflang="gv" data-title="Feer earroo" data-language-autonym="Gaelg" data-language-local-name="mänksi" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – galeegi" lang="gl" hreflang="gl" data-title="Número real" data-language-autonym="Galego" data-language-local-name="galeegi" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – kani" lang="gan" hreflang="gan" data-title="實數" data-language-autonym="贛語" data-language-local-name="kani" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D3%99%D3%99%D0%BB%D2%BB%D0%B0%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Бәәлһан тойг – kalmõki" lang="xal" hreflang="xal" data-title="Бәәлһан тойг" data-language-autonym="Хальмг" data-language-local-name="kalmõki" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%A4%EC%88%98" title="실수 – korea" lang="ko" hreflang="ko" data-title="실수" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Իրական թիվ – armeenia" lang="hy" hreflang="hy" data-title="Իրական թիվ" data-language-autonym="Հայերեն" data-language-local-name="armeenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – hindi" lang="hi" hreflang="hi" data-title="वास्तविक संख्या" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Realni_broj" title="Realni broj – horvaadi" lang="hr" hreflang="hr" data-title="Realni broj" data-language-autonym="Hrvatski" data-language-local-name="horvaadi" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Reala_nombro" title="Reala nombro – ido" lang="io" hreflang="io" data-title="Reala nombro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_real" title="Numero real – interlingua" lang="ia" hreflang="ia" data-title="Numero real" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%91%C3%A6%D0%BB%D0%B2%D1%8B%D1%80%D0%B4_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Бæлвырд нымæц – osseedi" lang="os" hreflang="os" data-title="Бæлвырд нымæц" data-language-autonym="Ирон" data-language-local-name="osseedi" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Rauntala" title="Rauntala – islandi" lang="is" hreflang="is" data-title="Rauntala" data-language-autonym="Íslenska" data-language-local-name="islandi" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_reale" title="Numero reale – itaalia" lang="it" hreflang="it" data-title="Numero reale" data-language-autonym="Italiano" data-language-local-name="itaalia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי – heebrea" lang="he" hreflang="he" data-title="מספר ממשי" data-language-autonym="עברית" data-language-local-name="heebrea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Si%C5%8B%C5%8B_%C3%B1%CA%8A%C5%8B_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Siŋŋ ñʊŋ (tʊʊzʊʊ) – kabije" lang="kbp" hreflang="kbp" data-title="Siŋŋ ñʊŋ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="kabije" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B3%88%E0%B2%9C_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ನೈಜ ಸಂಖ್ಯೆ – kannada" lang="kn" hreflang="kn" data-title="ನೈಜ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%9B%E1%83%93%E1%83%95%E1%83%98%E1%83%9A%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ნამდვილი რიცხვი – gruusia" lang="ka" hreflang="ka" data-title="ნამდვილი რიცხვი" data-language-autonym="ქართული" data-language-local-name="gruusia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D2%9B%D1%82%D1%8B_%D1%81%D0%B0%D0%BD" title="Нақты сан – kasahhi" lang="kk" hreflang="kk" data-title="Нақты сан" data-language-autonym="Қазақша" data-language-local-name="kasahhi" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Анык сан – kirgiisi" lang="ky" hreflang="ky" data-title="Анык сан" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_halisi" title="Namba halisi – suahiili" lang="sw" hreflang="sw" data-title="Namba halisi" data-language-autonym="Kiswahili" data-language-local-name="suahiili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_r%C3%A9y%C3%A8l" title="Nonm réyèl – Guajaana kreoolkeel" lang="gcr" hreflang="gcr" data-title="Nonm réyèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guajaana kreoolkeel" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_rast%C3%AEn" title="Hejmarên rastîn – kurdi" lang="ku" hreflang="ku" data-title="Hejmarên rastîn" data-language-autonym="Kurdî" data-language-local-name="kurdi" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%88%E0%BA%B4%E0%BA%87" title="ຈຳນວນຈິງ – lao" lang="lo" hreflang="lo" data-title="ຈຳນວນຈິງ" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_realis" title="Numerus realis – ladina" lang="la" hreflang="la" data-title="Numerus realis" data-language-autonym="Latina" data-language-local-name="ladina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Re%C4%81ls_skaitlis" title="Reāls skaitlis – läti" lang="lv" hreflang="lv" data-title="Reāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="läti" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Realusis_skai%C4%8Dius" title="Realusis skaičius – leedu" lang="lt" hreflang="lt" data-title="Realusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="leedu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Numeri_re%C3%A6" title="Numeri reæ – liguuri" lang="lij" hreflang="lij" data-title="Numeri reæ" data-language-autonym="Ligure" data-language-local-name="liguuri" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – limburgi" lang="li" hreflang="li" data-title="Reëel getal" data-language-autonym="Limburgs" data-language-local-name="limburgi" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_real" title="Numero real – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero real" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/pavycimdyna%27u" title="pavycimdyna'u – ložban" lang="jbo" hreflang="jbo" data-title="pavycimdyna'u" data-language-autonym="La .lojban." data-language-local-name="ložban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_real" title="Numer real – lombardi" lang="lmo" hreflang="lmo" data-title="Numer real" data-language-autonym="Lombard" data-language-local-name="lombardi" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok – ungari" lang="hu" hreflang="hu" data-title="Valós számok" data-language-autonym="Magyar" data-language-local-name="ungari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реален број – makedoonia" lang="mk" hreflang="mk" data-title="Реален број" data-language-autonym="Македонски" data-language-local-name="makedoonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_voatsapa" title="Isa voatsapa – malagassi" lang="mg" hreflang="mg" data-title="Isa voatsapa" data-language-autonym="Malagasy" data-language-local-name="malagassi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B4%B5%E0%B4%BF%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="വാസ്തവികസംഖ്യ – malajalami" lang="ml" hreflang="ml" data-title="വാസ്തവികസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="malajalami" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – marathi" lang="mr" hreflang="mr" data-title="वास्तविक संख्या" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%85%E1%80%85%E1%80%BA" title="ကိန်းစစ် – birma" lang="my" hreflang="my" data-title="ကိန်းစစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – hollandi" lang="nl" hreflang="nl" data-title="Reëel getal" data-language-autonym="Nederlands" data-language-local-name="hollandi" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक सङ्ख्या – nepali" lang="ne" hreflang="ne" data-title="वास्तविक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%9F%E6%95%B0" title="実数 – jaapani" lang="ja" hreflang="ja" data-title="実数" data-language-autonym="日本語" data-language-local-name="jaapani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Re%27el_taal" title="Re'el taal – põhjafriisi" lang="frr" hreflang="frr" data-title="Re'el taal" data-language-autonym="Nordfriisk" data-language-local-name="põhjafriisi" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Reelt_tall" title="Reelt tall – norra bokmål" lang="nb" hreflang="nb" data-title="Reelt tall" data-language-autonym="Norsk bokmål" data-language-local-name="norra bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Reelle_tal" title="Reelle tal – uusnorra" lang="nn" hreflang="nn" data-title="Reelle tal" data-language-autonym="Norsk nynorsk" data-language-local-name="uusnorra" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_real" title="Nombre real – oksitaani" lang="oc" hreflang="oc" data-title="Nombre real" data-language-autonym="Occitan" data-language-local-name="oksitaani" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Haqiqiy_sonlar" title="Haqiqiy sonlar – usbeki" lang="uz" hreflang="uz" data-title="Haqiqiy sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B5%E0%A8%BF%E0%A8%95_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਵਾਸਤਵਿਕ ਅੰਕ – pandžabi" lang="pa" hreflang="pa" data-title="ਵਾਸਤਵਿਕ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Riil_nomba" title="Riil nomba – Jamaica kreoolkeel" lang="jam" hreflang="jam" data-title="Riil nomba" data-language-autonym="Patois" data-language-local-name="Jamaica kreoolkeel" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%96%E1%9E%B7%E1%9E%8F" title="ចំនួនពិត – khmeeri" lang="km" hreflang="km" data-title="ចំនួនពិត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmeeri" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_real" title="Nùmer real – piemonte" lang="pms" hreflang="pms" data-title="Nùmer real" data-language-autonym="Piemontèis" data-language-local-name="piemonte" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_rzeczywiste" title="Liczby rzeczywiste – poola" lang="pl" hreflang="pl" data-title="Liczby rzeczywiste" data-language-autonym="Polski" data-language-local-name="poola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – portugali" lang="pt" hreflang="pt" data-title="Número real" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Aqiqiy_say%C4%B1" title="Aqiqiy sayı – krimmitatari" lang="crh" hreflang="crh" data-title="Aqiqiy sayı" data-language-autonym="Qırımtatarca" data-language-local-name="krimmitatari" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_real" title="Număr real – rumeenia" lang="ro" hreflang="ro" data-title="Număr real" data-language-autonym="Română" data-language-local-name="rumeenia" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вещественное число – vene" lang="ru" hreflang="ru" data-title="Вещественное число" data-language-autonym="Русский" data-language-local-name="vene" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D1%8C%D0%B8%D2%A5%D0%BD%D1%8D%D1%8D%D1%85_%D1%87%D1%8B%D1%8B%D2%BB%D1%8B%D0%BB%D0%B0%D0%BB%D0%B0%D1%80" title="Дьиҥнээх чыыһылалар – jakuudi" lang="sah" hreflang="sah" data-title="Дьиҥнээх чыыһылалар" data-language-autonym="Саха тыла" data-language-local-name="jakuudi" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_real%C3%AB" title="Numrat realë – albaania" lang="sq" hreflang="sq" data-title="Numrat realë" data-language-autonym="Shqip" data-language-local-name="albaania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_riali" title="Nùmmuru riali – sitsiilia" lang="scn" hreflang="scn" data-title="Nùmmuru riali" data-language-autonym="Sicilianu" data-language-local-name="sitsiilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="තාත්වික සංඛ්‍යා – singali" lang="si" hreflang="si" data-title="තාත්වික සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="singali" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Real_number" title="Real number – lihtsustatud inglise" lang="en-simple" hreflang="en-simple" data-title="Real number" data-language-autonym="Simple English" data-language-local-name="lihtsustatud inglise" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Re%C3%A1lne_%C4%8D%C3%ADslo" title="Reálne číslo – slovaki" lang="sk" hreflang="sk" data-title="Reálne číslo" data-language-autonym="Slovenčina" data-language-local-name="slovaki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Realno_%C5%A1tevilo" title="Realno število – sloveeni" lang="sl" hreflang="sl" data-title="Realno število" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D9%82%DB%8C%D9%86%DB%95" title="ژمارەی ڕاستەقینە – sorani" lang="ckb" hreflang="ckb" data-title="ژمارەی ڕاستەقینە" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реалан број – serbia" lang="sr" hreflang="sr" data-title="Реалан број" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Realan_broj" title="Realan broj – serbia-horvaadi" lang="sh" hreflang="sh" data-title="Realan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbia-horvaadi" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reaaliluku" title="Reaaliluku – soome" lang="fi" hreflang="fi" data-title="Reaaliluku" data-language-autonym="Suomi" data-language-local-name="soome" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reella_tal" title="Reella tal – rootsi" lang="sv" hreflang="sv" data-title="Reella tal" data-language-autonym="Svenska" data-language-local-name="rootsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – tagalogi" lang="tl" hreflang="tl" data-title="Tunay na bilang" data-language-autonym="Tagalog" data-language-local-name="tagalogi" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%AF%E0%AF%8D%E0%AE%AF%E0%AF%86%E0%AE%A3%E0%AF%8D" title="மெய்யெண் – tamili" lang="ta" hreflang="ta" data-title="மெய்யெண்" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%88%E0%B8%A3%E0%B8%B4%E0%B8%87" title="จำนวนจริง – tai" lang="th" hreflang="th" data-title="จำนวนจริง" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_th%E1%BB%B1c" title="Số thực – vietnami" lang="vi" hreflang="vi" data-title="Số thực" data-language-autonym="Tiếng Việt" data-language-local-name="vietnami" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Reel_say%C4%B1lar" title="Reel sayılar – türgi" lang="tr" hreflang="tr" data-title="Reel sayılar" data-language-autonym="Türkçe" data-language-local-name="türgi" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B9%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Дійсне число – ukraina" lang="uk" hreflang="uk" data-title="Дійсне число" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B9%D8%AF%D8%AF" title="حقیقی عدد – urdu" lang="ur" hreflang="ur" data-title="حقیقی عدد" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Reaalarv" title="Reaalarv – võru" lang="vro" hreflang="vro" data-title="Reaalarv" data-language-autonym="Võro" data-language-local-name="võru" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – klassikaline hiina" lang="lzh" hreflang="lzh" data-title="實數" data-language-autonym="文言" data-language-local-name="klassikaline hiina" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – uu" lang="wuu" hreflang="wuu" data-title="实数" data-language-autonym="吴语" data-language-local-name="uu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A8%D7%A2%D7%90%D7%9C%D7%A2_%D7%A6%D7%90%D7%9C" title="רעאלע צאל – jidiši" lang="yi" hreflang="yi" data-title="רעאלע צאל" data-language-autonym="ייִדיש" data-language-local-name="jidiši" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="hea artikkel"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_gidi" title="Nọ́mbà gidi – joruba" lang="yo" hreflang="yo" data-title="Nọ́mbà gidi" data-language-autonym="Yorùbá" data-language-local-name="joruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – kantoni" lang="yue" hreflang="yue" data-title="實數" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_reel" title="Amaro reel – dõmli" lang="diq" hreflang="diq" data-title="Amaro reel" data-language-autonym="Zazaki" data-language-local-name="dõmli" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – hiina" lang="zh" hreflang="zh" data-title="实数" data-language-autonym="中文" data-language-local-name="hiina" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_bendar" title="Lumur bendar – ibani" lang="iba" hreflang="iba" data-title="Lumur bendar" data-language-autonym="Jaku Iban" data-language-local-name="ibani" class="interlanguage-link-target"><span>Jaku Iban</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12916#sitelinks-wikipedia" title="Muuda keelelinke" class="wbc-editpage">Muuda linke</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Nimeruumid"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Reaalarv" title="Vaata sisulehekülge [c]" accesskey="c"><span>Artikkel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Arutelu:Reaalarv" rel="discussion" title="Arutelu selle lehekülje sisu kohta [t]" accesskey="t"><span>Arutelu</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown 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width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Number-systems.svg/330px-Number-systems.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Number-systems.svg/440px-Number-systems.svg.png 2x" data-file-width="800" data-file-height="400" /></a><figcaption><a class="mw-selflink selflink">Reaalarvude</a> hulk ℝ sisaldab kõigi ratsionaalarvude hulka ℚ, mis omakorda sisaldab kõigi <a href="/wiki/T%C3%A4isarv" title="Täisarv">täisarvude</a> hulka ℤ, mis sisaldab kõigi <a href="/wiki/Naturaalarv" title="Naturaalarv">naturaalarvude</a> hulka ℕ</figcaption></figure> <p><b>Reaalarvud</b> on kõik <a href="/wiki/Ratsionaalarvud" class="mw-redirect" title="Ratsionaalarvud">ratsionaal-</a> ja <a href="/wiki/Irratsionaalarvud" title="Irratsionaalarvud">irratsionaalarvud</a> ehk kõik <a href="/wiki/Positiivne_arv" title="Positiivne arv">positiivsed</a> ja <a href="/wiki/Negatiivne_arv" title="Negatiivne arv">negatiivsed arvud</a> ja <a href="/wiki/Null" title="Null">null</a> ehk kõik <a href="/w/index.php?title=Algebraline_arv&action=edit&redlink=1" class="new" title="Algebraline arv (pole veel kirjutatud)">algebralised arvud</a> ja <a href="/wiki/Transtsendentne_arv" title="Transtsendentne arv">transtsendentsed arvud</a>. </p><p>Reaalarvud moodustavad <a href="/wiki/Reaalarvude_hulk" title="Reaalarvude hulk">reaalarvude hulga</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> ehk <b>R</b> ning tähtsaima <a href="/w/index.php?title=Arvuvald&action=edit&redlink=1" class="new" title="Arvuvald (pole veel kirjutatud)">arvuvalla</a> <a href="/wiki/Matemaatika" title="Matemaatika">matemaatikas</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fail:Real_number_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/220px-Real_number_line.svg.png" decoding="async" width="220" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/330px-Real_number_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Real_number_line.svg/440px-Real_number_line.svg.png 2x" data-file-width="689" data-file-height="225" /></a><figcaption><a href="/wiki/Arvsirge" class="mw-redirect" title="Arvsirge">Arvsirge</a>, millel on näidatud arvude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> (<a href="/wiki/Ruutjuur_kahest" title="Ruutjuur kahest">ruutjuur kahest</a>), <i><a href="/wiki/E_(arv)" title="E (arv)">e</a></i> ja <i><a href="/wiki/Pii" title="Pii">π</a></i> asukoht</figcaption></figure> <p>Reaalarvud on konstrueeritud nii, et oleks võimalik loomulik <a href="/w/index.php?title=%C3%9Cks%C3%BChene_vastavus&action=edit&redlink=1" class="new" title="Üksühene vastavus (pole veel kirjutatud)">üksühene vastavus</a> reaalarvude hulga ja <a href="/wiki/Sirge" title="Sirge">sirge</a> (<a href="/wiki/Arvsirge" class="mw-redirect" title="Arvsirge">arvsirge</a>) <a href="/wiki/Punkt_(matemaatika)" title="Punkt (matemaatika)">punktide</a> hulga vahel. Sellepärast samastatakse reaalarvude hulk mõnikord arvsirgega. </p><p>Nimetus "reaalarv" ('tegelik arv') iseloomustab erinevust <a href="/wiki/Imaginaararv" title="Imaginaararv">imaginaararvudest</a>. </p><p>Reaalarvu mõiste väljakujunemine võttis palju aega. <a href="/w/index.php?title=Vana-Kreeka_matemaatika&action=edit&redlink=1" class="new" title="Vana-Kreeka matemaatika (pole veel kirjutatud)">Vana-Kreeka matemaatikas</a> <a href="/wiki/Pythagoras" title="Pythagoras">Pythagorase</a> koolkonnas, kus kõige aluseks peeti <a href="/wiki/Naturaalarv" title="Naturaalarv">naturaalarve</a> ja nende suhteid, avastati, et on olemas ühismõõdutud suurused (ruudu külje ja diagonaali <a href="/w/index.php?title=%C3%9Chism%C3%B5%C3%B5dutus&action=edit&redlink=1" class="new" title="Ühismõõdutus (pole veel kirjutatud)">ühismõõdutus</a>), tänapäeva mõistes avastati <a href="/wiki/Arv" title="Arv">arvud</a>, mis ei ole <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvud</a>. <a href="/wiki/Eudoxos" title="Eudoxos">Eudoxos</a> püüdis välja töötada ühismõõdutute suurustega opereerivat teooriat. 19. sajandi 2. poolel formuleeriti <a href="/wiki/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs">matemaatiline analüüs</a> kõrgemal <a href="/w/index.php?title=Rangus&action=edit&redlink=1" class="new" title="Rangus (pole veel kirjutatud)">rangusastmel</a> ning selle käigus töötasid <a href="/w/index.php?title=Karl_Weierstra%C3%9F&action=edit&redlink=1" class="new" title="Karl Weierstraß (pole veel kirjutatud)">Karl Weierstraß</a>, <a href="/w/index.php?title=Richard_Dedekind&action=edit&redlink=1" class="new" title="Richard Dedekind (pole veel kirjutatud)">Richard Dedekind</a>, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, <a href="/w/index.php?title=Eduard_Heine&action=edit&redlink=1" class="new" title="Eduard Heine (pole veel kirjutatud)">Eduard Heine</a> ja <a href="/w/index.php?title=Charles_M%C3%A9ray&action=edit&redlink=1" class="new" title="Charles Méray (pole veel kirjutatud)">Charles Méray</a> välja reaalarvude range teooria. </p><p>Tänapäeva matemaatika seisukohast moodustab reaalarvude hulk <a href="/w/index.php?title=Reaalarvude_hulga_pidevus&action=edit&redlink=1" class="new" title="Reaalarvude hulga pidevus (pole veel kirjutatud)">pideva</a> <a href="/w/index.php?title=J%C3%A4rjestatud_korpus&action=edit&redlink=1" class="new" title="Järjestatud korpus (pole veel kirjutatud)">järjestatud korpuse</a>. See definitsioon või sellega samaväärne <a href="/w/index.php?title=Aksiomaatika&action=edit&redlink=1" class="new" title="Aksiomaatika (pole veel kirjutatud)">aksiomaatika</a> määratleb reaalarvud üheselt: <a href="/wiki/Isomorfism" title="Isomorfism">isomorfismi</a> täpsusega leidub ainult üks pidev järjestatud <a href="/wiki/Korpus_(matemaatika)" title="Korpus (matemaatika)">korpus</a>. </p><p><a href="/wiki/Reaalarvude_hulk" title="Reaalarvude hulk">Kõikide reaalarvude hulga</a> tavaline tähis on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> (ℝ) või ka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" 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 \mathbf {R} }"></span> või <big><b>R</b></big>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Reaalarvude_konstruktsioonid">Reaalarvude konstruktsioonid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=1" title="Muuda alaosa "Reaalarvude konstruktsioonid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=1" title="Muuda alaosa "Reaalarvude konstruktsioonid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Reaalarvude konstruktsioonid ehk konstruktiivsed definitsioonid on sellised reaalarvude defineerimise viisid, mille korral võetakse aluseks <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvud</a> või muud matemaatilised objektid, mis loetakse antuks. Neist konstrueeritakse uued objektid, mis vastavad meie intuitiivsele arusaamale <a href="/wiki/Irratsionaalarv" class="mw-redirect" title="Irratsionaalarv">irratsionaalarvudest</a> ja mida nimetatakse irratsionaalarvudeks, ja lisatakse need ratsionaalarvudele. Erinevalt nendest konstruktsioonidest mõistetakse reaalarve ainult intuitiivselt ja need ei ole esialgu rangelt defineeritud matemaatiline mõiste. Ratsionaalarvud ja irratsionaalarvud kokku moodustavad reaalarvud. Nendel defineeritakse põhitehted ja järjestus ning tõestatakse nende omadused. </p><p>Ajalooliselt esimesed reaalarvude ranged definitsioonid olidki konstruktiivsed. Aastal <a href="/wiki/1872" title="1872">1872</a> avaldati üheaegselt kolm tööd – <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantori</a> <a href="/wiki/Fundamentaaljada" class="mw-redirect" title="Fundamentaaljada">fundamentaaljadade</a> teooria, <a href="/w/index.php?title=Karl_Weierstrass&action=edit&redlink=1" class="new" title="Karl Weierstrass (pole veel kirjutatud)">Karl Weierstrassi</a> teooria (tänapäevases variandis <a href="/w/index.php?title=L%C3%B5pmatu_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Lõpmatu kümnendmurd (pole veel kirjutatud)">lõpmatute kümnendmurdude</a> teooria) ning <a href="/w/index.php?title=Richard_Dedekind&action=edit&redlink=1" class="new" title="Richard Dedekind (pole veel kirjutatud)">Richard Dedekindi</a> lõigete teooria ratsionaalarvude vallas. </p> <div class="mw-heading mw-heading3"><h3 id="Cantori_fundamentaaljadade_teooria">Cantori fundamentaaljadade teooria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=2" title="Muuda alaosa "Cantori fundamentaaljadade teooria"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=2" title="Muuda alaosa "Cantori fundamentaaljadade teooria" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Allpool esitatud lähenemise reaalarvude defineerimisele esitas <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> <a href="/wiki/1872" title="1872">1872</a>. aastal avaldatud artiklis<sup id="cite_ref-Кантор_1-0" class="reference"><a href="#cite_note-Кантор-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. Sarnaseid ideid väljendasid <a href="/w/index.php?title=Eduard_Heine&action=edit&redlink=1" class="new" title="Eduard Heine (pole veel kirjutatud)">Eduard Heine</a> ja <a href="/w/index.php?title=Charles_M%C3%A9rais&action=edit&redlink=1" class="new" title="Charles Mérais (pole veel kirjutatud)">Charles Mérais</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Kokkuvõte"><span id="Kokkuv.C3.B5te"></span>Kokkuvõte</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=3" title="Muuda alaosa "Kokkuvõte"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=3" title="Muuda alaosa "Kokkuvõte" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Selles lähenemises defineeritakse arv ratsionaalarvude <a href="/wiki/Jada_piirv%C3%A4%C3%A4rtus" title="Jada piirväärtus">jada piirväärtusena</a>. Et ratsionaalarvude jada <a href="/wiki/Koonduv_jada" title="Koonduv jada">koonduks</a>, asetatakse sellele <a href="/w/index.php?title=Cauchy_tingimus&action=edit&redlink=1" class="new" title="Cauchy tingimus (pole veel kirjutatud)">Cauchy tingimus</a>: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon ):\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon ):\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9cd8e1244bd39a39eed80f71a40be02c0555ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.674ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon ):\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }"></span></div> <p>Selle tingimuse mõte seisneb selles, et jada liikmed on alates teatud numbrist üksteisele kui tahes lähedal. Jadasid, mis seda tingimust rahuldavad, nimetatakse <a href="/wiki/Fundamentaaljada" class="mw-redirect" title="Fundamentaaljada">fundamentaaljadadeks</a> ehk Cauchy jadadeks. </p><p>Reaalarvu, mille defineerib ratsionaalarvude fundamentaaljada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span>, tähistame <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac819208e1146ac08706a43030b597eb7b7a803f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.742ex; height:2.843ex;" alt="{\displaystyle [a_{n}]}"></span>. </p><p>Kaht reaalarvu </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =[a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =[a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222a30a9a92c2e3eef762e7c2c43dceb2bb4d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle \alpha =[a_{n}]}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =[b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =[b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26985ffa366493195dac10ceda47058558cd721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\displaystyle \beta =[b_{n}]}"></span>, </p> </div> <p>mis on defineeritud vastavalt fundamentaaljadadele <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span>, nimetatakse <a href="/w/index.php?title=V%C3%B5rdsus_(matemaatika)&action=edit&redlink=1" class="new" title="Võrdsus (matemaatika) (pole veel kirjutatud)">võrdseteks</a>, kui </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left(a_{n}-b_{n}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left(a_{n}-b_{n}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef91da6be0c8e66ea3699244f878c65848c797e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.235ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }\left(a_{n}-b_{n}\right)=0}"></span></div> <p>Kui on antud kaks reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =[a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =[a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222a30a9a92c2e3eef762e7c2c43dceb2bb4d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle \alpha =[a_{n}]}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =[b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =[b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26985ffa366493195dac10ceda47058558cd721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\displaystyle \beta =[b_{n}]}"></span>, siis nende summaks ja korrutiseks nimetatakse arve, mis on defineeritud vastavalt jadade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span>: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]\qquad \alpha \cdot \beta {\overset {\text{def}}{=}}[a_{n}\cdot b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mspace width="2em" /> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]\qquad \alpha \cdot \beta {\overset {\text{def}}{=}}[a_{n}\cdot b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce999eb318c4febc0f482ca4c85c6559b94ae9ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.534ex; height:3.843ex;" alt="{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]\qquad \alpha \cdot \beta {\overset {\text{def}}{=}}[a_{n}\cdot b_{n}]}"></span></div> <p><a href="/wiki/Summa" class="mw-redirect" title="Summa">summa</a> ja <a href="/wiki/Korrutis" class="mw-redirect" title="Korrutis">korrutisena</a>. </p><p><a href="/wiki/J%C3%A4rjestusseos" class="mw-redirect" title="Järjestusseos">Järjestusseos</a> reaalarvude hulgal kehtestatakse kokkuleppega, mille järgi arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =[a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =[a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222a30a9a92c2e3eef762e7c2c43dceb2bb4d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle \alpha =[a_{n}]}"></span> on definitsiooni kohaselt suurem arvust <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =[b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =[b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26985ffa366493195dac10ceda47058558cd721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\displaystyle \beta =[b_{n}]}"></span> ehk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878aa4ba937a08258b46361229c9c35d970672ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha >\beta }"></span>, kui </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \varepsilon >0\;\exists N:\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>N</mi> <mo>:</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩾</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \varepsilon >0\;\exists N:\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bdcfdc772a0099192d76c6cf5a7b8d4694117e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.402ex; height:2.509ex;" alt="{\displaystyle \exists \varepsilon >0\;\exists N:\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }"></span></div> <p>Reaalarvude hulga konstrueerimine fundamentaaljadade kaudu on mis tahes <a href="/wiki/Meetriline_ruum" title="Meetriline ruum">meetrilise ruumi</a> <a href="/w/index.php?title=T%C3%A4ielikustamine&action=edit&redlink=1" class="new" title="Täielikustamine (pole veel kirjutatud)">täielikustamise</a> konstruktsioon. Nagu ka üldjuhul, on täielikustamise tulemuseks saadud reaalarvude hulk ise juba <a href="/w/index.php?title=T%C3%A4ielik_meetriline_ruum&action=edit&redlink=1" class="new" title="Täielik meetriline ruum (pole veel kirjutatud)">täielik</a>, st temasse kuuluvad kõigi tema elementide fundamentaaljadade piirväärtused. </p> <div class="mw-heading mw-heading4"><h4 id="Cauchy_koonduvuskriteerium_ja_selle_kasutamine_Cantoril">Cauchy koonduvuskriteerium ja selle kasutamine Cantoril</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=4" title="Muuda alaosa "Cauchy koonduvuskriteerium ja selle kasutamine Cantoril"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=4" title="Muuda alaosa "Cauchy koonduvuskriteerium ja selle kasutamine Cantoril" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cantori teooria lähtekoht oli järgmine idee<sup id="cite_ref-Арнольд_2-0" class="reference"><a href="#cite_note-Арнольд-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. Iga reaalarvu saab esitada ratsionaalarvude <a href="/wiki/L%C3%B5pmatu_jada" class="mw-redirect" title="Lõpmatu jada">lõpmatu jada</a> </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c9df9a9127bb614a7f44403ec97e534adb7f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.986ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }"></span></div> <p>abil, mille liikmed on selle reaalarvu <a href="/w/index.php?title=L%C3%A4hend&action=edit&redlink=1" class="new" title="Lähend (pole veel kirjutatud)">lähendid</a> kasvava täpsusastmega, nii et see jada <a href="/wiki/Koonduv_jada" title="Koonduv jada">koondub</a> selleks arvuks. </p><p>Mõistame nüüd reaalarvu all mingit objekti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, mille defineerib ratsionaalarvude koonduv jada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1902f224782e1a7d0af83cb15aef4f9ea983479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.359ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots }"></span>. </p><p>Ent siin peitub <a href="/wiki/Vigane_ring" class="mw-redirect" title="Vigane ring">vigane ring</a>. Koonduva jada definitsioonis figureerib reaalarv, mis on selle <a href="/wiki/Jada_piirv%C3%A4%C3%A4rtus" title="Jada piirväärtus">jada piirväärtus</a> – seesama mõiste, mille me koonduvate jadade abil tahame defineerida: </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> koondub <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⟺</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a3a1a17366e695966bae38466f8466653a43f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.317ex; height:1.843ex;" alt="{\displaystyle \Longleftrightarrow }"></span> leidub <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>, nõnda et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adea9f214dd631da7e7416d7ae44ebdd46d5bdd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.694ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=\alpha }"></span> </p> </div> <p>Et vigast ringi ei tekiks, on tarvis leida mingi tunnus, mis võimaldab väljendada jada koonduvuse tingimust selle jada liikmete kaudu, see tähendab mainimata <a href="/wiki/Jada_piirv%C3%A4%C3%A4rtus" title="Jada piirväärtus">jada piirväärtuse</a> väärtust ennast. </p><p>Cantori ajaks oli niisugune tingimus juba leitud. Selle tegi üldisel kujul kindlaks prantsuse matemaatik <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a><sup id="cite_ref-0TL20_3-0" class="reference"><a href="#cite_note-0TL20-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. <a href="/w/index.php?title=Cauchy_kriteerium&action=edit&redlink=1" class="new" title="Cauchy kriteerium (pole veel kirjutatud)">Cauchy kriteeriumi</a> järgi koondub jada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1902f224782e1a7d0af83cb15aef4f9ea983479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.359ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots }"></span> <a href="/wiki/Siis_ja_ainult_siis,_kui" class="mw-redirect" title="Siis ja ainult siis, kui">siis ja ainult siis, kui</a> </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bd2f5e14be81f21b514b68f78fb10fbb0312c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.736ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;\forall m>0\;|a_{n+m}-a_{n}|<\varepsilon }"></span></div> <p>Kujundlikult öeldes seisneb jada koonduvuse tingimus Cauchy kriteeriumis selles, et jada liikmed on mingist numbrist alates üksteisele kui tahes lähedal. </p><p>Cauchy ei saanud seda kriteeriumi kuigivõrd rangelt põhjendada, sest puudus reaalarvu teooria. </p><p>Cantor pööras tähelepanu sellele, et see kriteerium iseenesest iseloomustab koonduva jada sisemisi omadusi: teda saab formuleerida ja kontrollida, ilma et oleks juttu selle jada piirväärtuseks olevast reaalarvust endast. Ja sellepärast saab seda tunnust kasutada selleks, et tuua välja jadade klass, mille abil saab reaalarve defineerida. </p><p>Nii et põhiline samm, mille tegi Cantor reaalarvu teooria rajamisel, seisnes selles, et ta vaatles iga ratsionaalarvude jada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1902f224782e1a7d0af83cb15aef4f9ea983479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.359ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots }"></span>, mis rahuldab Cauchy tingimust, mingit (ratsionaalset või irratsionaalset) reaalarvu defineerivana. "Kui ma räägin arvsuurusest üldistatud mõttes, toimub see eelkõige juhtumil, mil on ette pandud ratsionaalarvude lõpmatu jada </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c9df9a9127bb614a7f44403ec97e534adb7f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.986ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{n},\ldots }"></span>,</div> <p>mis on antud mingi seaduse abil ning millel on see omadus, et <a href="/wiki/Vahe" class="mw-redirect" title="Vahe">vahe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n+m}-a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n+m}-a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2ca215868168baae0521b7a2dd875122986c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.458ex; height:2.343ex;" alt="{\displaystyle a_{n+m}-a_{n}}"></span> saab <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-i kasvades <a href="/w/index.php?title=L%C3%B5pmata_v%C3%A4ike_suurus&action=edit&redlink=1" class="new" title="Lõpmata väike suurus (pole veel kirjutatud)">lõpmata väikeseks</a>, olgu <a href="/w/index.php?title=Positiivne_t%C3%A4isarv&action=edit&redlink=1" class="new" title="Positiivne täisarv (pole veel kirjutatud)">positiivne täisarv</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> milline tahes, ehk teiste sõnadega, suvaliselt valitud (<a href="/w/index.php?title=Positiivne_ratsionaalarv&action=edit&redlink=1" class="new" title="Positiivne ratsionaalarv (pole veel kirjutatud)">positiivse ratsionaalarvu</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> korral leidub selline <a href="/wiki/T%C3%A4isarv" title="Täisarv">täisarv</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{n+m}-a_{n}|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{n+m}-a_{n}|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3e3778c19701e6f9d02ddc54f407839ea2da6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.934ex; height:2.843ex;" alt="{\displaystyle |a_{n+m}-a_{n}|<\varepsilon }"></span>, ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> on mis tahes positiivne täisarv."<sup id="cite_ref-Кантор_1-1" class="reference"><a href="#cite_note-Кантор-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Tänapäeval nimetatakse Cauchy tingimust rahuldavat jada Cauchy jadaks ehk <a href="/wiki/Fundamentaaljada" class="mw-redirect" title="Fundamentaaljada">fundamentaaljadaks</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Reaalarvude_konstruktsioon_Cantori_järgi"><span id="Reaalarvude_konstruktsioon_Cantori_j.C3.A4rgi"></span>Reaalarvude konstruktsioon Cantori järgi</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=5" title="Muuda alaosa "Reaalarvude konstruktsioon Cantori järgi"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=5" title="Muuda alaosa "Reaalarvude konstruktsioon Cantori järgi" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kaks fundamentaaljada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span> võivad defineerida üht ja sedasama reaalarvu. See on nii tingimusel </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;|a_{n}-b_{n}|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;|a_{n}-b_{n}|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aeb9f35d771d07033b71b9697a9a037e85f371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.544ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\;\exists N(\varepsilon )\;\forall n>N(\varepsilon )\;|a_{n}-b_{n}|<\varepsilon }"></span></div> <p>Niisiis tekib ratsionaalarvude kõigi fundamentaaljadade hulgal <a href="/wiki/Ekvivalentsusseos" class="mw-redirect" title="Ekvivalentsusseos">ekvivalentsusseos</a>, ja üldise printsiibi järgi jagunevad kõik fundamentaaljadad <a href="/w/index.php?title=Ekvivalentsusklass&action=edit&redlink=1" class="new" title="Ekvivalentsusklass (pole veel kirjutatud)">ekvivalentsusklassideks</a>. Selle <a href="/wiki/Klassijaotus" class="mw-redirect" title="Klassijaotus">klassijaotuse</a> mõte seisneb selles, et ühte klassi kuuluvad jadad defineerivad ühe ja sellesama reaalarvu ning eri klassidesse kuuluvad jadad erinevad. Niisiis on olemas <a href="/w/index.php?title=%C3%9Cks%C3%BChene_vastavus&action=edit&redlink=1" class="new" title="Üksühene vastavus (pole veel kirjutatud)">üksühene vastavus</a> reaalarvude ning ratsionaalarvude fundamentaaljadade klasside vahel. </p><p>Nüüd saame sõnastada Cantori reaalarvude teooria põhidefinitsiooni. </p><p>Definitsioon. Reaalarv on ratsionaalarvude fundamentaaljadade ekvivalentsusklass. </p><p>Reaalarvu (ekvivalentsusklassi), mille defineerib ratsionaalarvude fundamentaaljada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span>, tähistame <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac819208e1146ac08706a43030b597eb7b7a803f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.742ex; height:2.843ex;" alt="{\displaystyle [a_{n}]}"></span>. </p><p>Aritmeetilised tehted reaalarvudega defineeritakse nii. Kui on antud kaks reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, mis on defineeritud fundamentaaljadadega <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span>, nii et </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =[a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =[a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222a30a9a92c2e3eef762e7c2c43dceb2bb4d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle \alpha =[a_{n}]}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =[b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =[b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26985ffa366493195dac10ceda47058558cd721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\displaystyle \beta =[b_{n}]}"></span>, </p> </div> <p>siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> <a href="/wiki/Summa" class="mw-redirect" title="Summa">summaks</a> nimetatakse reaalarvu, mis on defineeritud jadaga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}+b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}+b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d62f5b89a287f42f4b9788758a348f4b6bcddfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.83ex; height:2.843ex;" alt="{\displaystyle \{a_{n}+b_{n}\}}"></span>, see on ekvivalentsusklassi, millesse see jada kuulub: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a6a24fe3fde45ad8904f2b2d99267279f3fdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.606ex; height:3.843ex;" alt="{\displaystyle \alpha +\beta {\overset {\text{def}}{=}}[a_{n}+b_{n}]}"></span></div> <p>Pole raske kontrollida, et see definitsioon on korrektne, st ei sõltu konkreetsete jadade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> klassist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span> klassist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> valikust. </p><p>Analoogselt defineeritakse reaalarvude <a href="/wiki/Vahe" class="mw-redirect" title="Vahe">vahe</a>, <a href="/wiki/Korrutis" class="mw-redirect" title="Korrutis">korrutis</a> ja <a href="/wiki/Jagatis" title="Jagatis">jagatis</a>. </p><p>Reaalarv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =[a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =[a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222a30a9a92c2e3eef762e7c2c43dceb2bb4d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle \alpha =[a_{n}]}"></span> on definitsiooni järgi suurem kui arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =[b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =[b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26985ffa366493195dac10ceda47058558cd721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\displaystyle \beta =[b_{n}]}"></span>, see tähendab <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878aa4ba937a08258b46361229c9c35d970672ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha >\beta }"></span>, kui </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \varepsilon >0\;\exists N\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>N</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩾</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \varepsilon >0\;\exists N\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a16d5d63fe6b5fc466f0ea4b6d697a7b7e29d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.465ex; height:2.509ex;" alt="{\displaystyle \exists \varepsilon >0\;\exists N\;\forall n>N\;a_{n}\geqslant b_{n}+\varepsilon }"></span></div> <p>See definitsioon ei sõltu jadade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339d461b497fb8f8670bb29308fe09f0e7bfd34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}"></span> klassist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \{b_{n}\}}"></span> klassist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> valikust. </p><p>Ratsionaalarvude süsteem sisestatakse reaalarvude süsteemi täiendava kokkuleppe abil, mille järgi jada </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,a,\ldots a,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mo>…</mo> <mi>a</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,a,\ldots a,\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a580fbf41d501eebee0d7c7d5aa5d5b5feb24c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.659ex; height:2.009ex;" alt="{\displaystyle a,a,\ldots a,\ldots ,}"></span></div> <p>mille kõik liikmed on võrdsed ühe ja sellesama ratsionaalarvuga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, defineerib sellesama arvu, nii et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a]{\overset {\text{def}}{=}}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a]{\overset {\text{def}}{=}}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ddfa221b62aca4b05e2aeae88624720468b4679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.901ex; height:3.843ex;" alt="{\displaystyle [a]{\overset {\text{def}}{=}}a}"></span>. Teiste sõnadega, iga klass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea82bc70a8e322f13a3c4e5b9d5d69e8ef097ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.523ex; height:2.843ex;" alt="{\displaystyle [a]}"></span>, mis sisaldab statsionaarset jada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,a,\ldots a,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mo>…</mo> <mi>a</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,a,\ldots a,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11feb8b1a7097aa6dc2267c19906420b844b17d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.625ex; height:2.009ex;" alt="{\displaystyle a,a,\ldots a,\ldots }"></span>, samastatakse arvuga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Niisiis, reaalarvude hulk, mille konstrueerisime, on ratsionaalarvude hulga <a href="/w/index.php?title=Hulga_laiend&action=edit&redlink=1" class="new" title="Hulga laiend (pole veel kirjutatud)">laiend</a>. </p><p>Sellega on reaalarvude hulga konstrueerimine lõpule viidud. </p><p>Edasi saab toodud definitsioonide abil tõestada reaalarvude teadaolevad omadused. </p> <div class="mw-heading mw-heading4"><h4 id="Reaalarvude_hulga_täielikkus"><span id="Reaalarvude_hulga_t.C3.A4ielikkus"></span>Reaalarvude hulga täielikkus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=6" title="Muuda alaosa "Reaalarvude hulga täielikkus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=6" title="Muuda alaosa "Reaalarvude hulga täielikkus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definitsioonist järeldub, et iga ratsionaalarvude fundamentaaljada koondub mingiks reaalarvuks. See printsiip oli reaalarvu definitsiooni aluseks. Tänu sellele lisandusid ratsionaalarvudele irratsionaalarvud – ratsionaalarvude nende fundamentaaljadade piirväärtused, millel ratsionaalarvude seas piirväärtust ei olnud. </p><p>Tekib õigustatud küsimus, kas analoogset täielikustamise protseduuri ei või läbi viia veel kord, juba konstrueeritud reaalarvude hulgal. Osutub, et see protseduur ei anna uut tulemust, sest igal reaalarvude fundamentaaljadal on piirväärtus reaalarvude seas. Seda reaalarvude hulga omadust nimetatakse <a href="/w/index.php?title=T%C3%A4ielik_meetriline_ruum&action=edit&redlink=1" class="new" title="Täielik meetriline ruum (pole veel kirjutatud)">täielikkuseks</a>. Ja väide, et iga reaalarvude fundamentaaljada koondub, ongi Cauchy koonduvuskriteeriumi põhisisu. </p><p>Sama ideed kasutas hiljem <a href="/w/index.php?title=Felix_Hausdorff&action=edit&redlink=1" class="new" title="Felix Hausdorff (pole veel kirjutatud)">Felix Hausdorff</a>, kui ta tõestas <a href="/w/index.php?title=Hausdorffi_teoreem&action=edit&redlink=1" class="new" title="Hausdorffi teoreem (pole veel kirjutatud)">Hausdorffi teoreemi</a> meetrilise ruumi täielikustamisest. </p> <div class="mw-heading mw-heading3"><h3 id="Lõpmatute_kümnendmurdude_teooria"><span id="L.C3.B5pmatute_k.C3.BCmnendmurdude_teooria"></span>Lõpmatute kümnendmurdude teooria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=7" title="Muuda alaosa "Lõpmatute kümnendmurdude teooria"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=7" title="Muuda alaosa "Lõpmatute kümnendmurdude teooria" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/12px-Next.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/18px-Next.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/24px-Next.svg.png 2x" data-file-width="160" data-file-height="160" /></span></span> <i>Pikemalt artiklis <a href="/wiki/K%C3%BCmnendmurd" title="Kümnendmurd">Kümnendmurd</a></i></dd></dl> <p><a href="/w/index.php?title=L%C3%B5pmatu_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Lõpmatu kümnendmurd (pole veel kirjutatud)">Lõpmatute kümnendmurdude</a> teooria pärineb <a href="/w/index.php?title=Karl_Weierstra%C3%9F&action=edit&redlink=1" class="new" title="Karl Weierstraß (pole veel kirjutatud)">Karl Weierstraßilt</a>. 1863. aasta paiku töötas ta välja reaalarvude teooria, mis avaldati tema loengumärkmetena <a href="/wiki/1872" title="1872">1872</a><sup id="cite_ref-Пути_4-0" class="reference"><a href="#cite_note-Пути-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>. Muide, Weierstraßi teooria algversioon erineb mõnevõrra lõpmatute kümnendmurdude teooriast, mida esitatakse tänapäeva <a href="/wiki/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs">matemaatilise analüüsi</a> õpikutes. </p> <div class="mw-heading mw-heading4"><h4 id="Ratsionaalarvud_ja_kümnendmurrud"><span id="Ratsionaalarvud_ja_k.C3.BCmnendmurrud"></span>Ratsionaalarvud ja kümnendmurrud</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=8" title="Muuda alaosa "Ratsionaalarvud ja kümnendmurrud"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=8" title="Muuda alaosa "Ratsionaalarvud ja kümnendmurrud" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/12px-Next.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/18px-Next.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/24px-Next.svg.png 2x" data-file-width="160" data-file-height="160" /></span></span> <i>Pikemalt artiklis <a href="/wiki/K%C3%BCmnendmurd" title="Kümnendmurd">Kümnendmurd</a></i></dd></dl> <p>Loeme antuks <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvude</a> hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>. On teada, et iga ratsionaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953f2689b703cc4c998e48999db91aeadac070a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.14ex; height:2.843ex;" alt="{\displaystyle p/q\in \mathbb {Q} }"></span> saab lahutada kümnendmurruks, mille paneme kirja kujul: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q\sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>∼</mo> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q\sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9aa88ee9086dac350237631874068268dae6121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:25.34ex; height:2.843ex;" alt="{\displaystyle p/q\sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"></span></div> <p>Kui kümnendmurruks lahutamise protsess lõpeb lõpliku arvu sammude järel, siis kümnendmurd on <a href="/w/index.php?title=L%C3%B5plik_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Lõplik kümnendmurd (pole veel kirjutatud)">lõplik</a>, vastasel juhul <a href="/w/index.php?title=L%C3%B5pmatu_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Lõpmatu kümnendmurd (pole veel kirjutatud)">lõpmatu</a>. </p><p>Iga lõplikku või lõpmatut kümnendmurdu võib vaadelda formaalse reana kujul </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b51b4ce4b7ea4eac9d18bed32582eb0a55c64667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.274ex; height:5.509ex;" alt="{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k},}"></span></div> <p>kus indeksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> väärtused on kas esimesed <a href="/wiki/Naturaalarv" title="Naturaalarv">naturaalarvud</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff4508f2c3fb59e9c506dd1944390b07f184a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.932ex; height:2.509ex;" alt="{\displaystyle 0,1,\ldots ,n}"></span> või vastavalt kõik naturaalarvud <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,2,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1da8ed7e74b31b6314f23f122a1198c104fcaad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.312ex; height:2.509ex;" alt="{\displaystyle 0,1,2,\ldots }"></span> . Saab näidata, et rida, mis saadakse ratsionaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa5bd4cf049744deac0ac4a04c07998bd6befa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.491ex; height:2.843ex;" alt="{\displaystyle p/q}"></span> lahutamisel kümnendmurruks, alati <a href="/w/index.php?title=Koonduv_rida&action=edit&redlink=1" class="new" title="Koonduv rida (pole veel kirjutatud)">koondub</a> ning tema <a href="/w/index.php?title=Rea_summa&action=edit&redlink=1" class="new" title="Rea summa (pole veel kirjutatud)">summa</a> võrdub algse ratsionaalarvuga. </p><p>Kui ratsionaalarvu lahutamisel kümnendmurruks saadakse lõpmatu kümnendmurd, siis see on alati <a href="/w/index.php?title=Perioodiline_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Perioodiline kümnendmurd (pole veel kirjutatud)">perioodiline</a>. </p><p>Seega vastab igale ratsionaalarvule üksainus kümnendmurd, kuid mõni kümnendmurd (näiteks lõpmatud mitteperioodilised) ei vasta ühelegi ratsionaalarvule. Loomulik on oletada, et ka nendele murdudele vastavad mingid hüpoteetilised arvud, mis ei ole ratsionaalarvud. Võttes vaatluse alla need hüpoteetilised arvud, mida hakkame nimetama <a href="/wiki/Irratsionaalarv" class="mw-redirect" title="Irratsionaalarv">irratsionaalarvudeks</a>, just nagu täidame lüngad kõigi kümnendmurdude kogumis. </p><p>Seega võtame reaalarvude teooria aluseks oletuse (idee), et iga kümnendmurd on mõne ratsionaal- või irratsionaalarvu (reaalarvu) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> lahutus kümnendmurruks: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∼</mo> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf4ee4d8d80299a6dfe4a092f733db86ee59073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.336ex; height:2.509ex;" alt="{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"></span></div> <p>Seejuures tõlgendame seda lahutust samamoodi nagu ratsionaalarvude puhul, nimelt peame reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> rea </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ba24dfccff0da4fae3dc09f79bda96079ab5b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.627ex; height:5.509ex;" alt="{\displaystyle \pm \sum _{k}a_{k}\cdot 10^{-k}}"></span></div> <p>summaks. </p> <div class="mw-heading mw-heading4"><h4 id="Lõpmatute_kümnendmurdude_teooria_konstruktsioon"><span id="L.C3.B5pmatute_k.C3.BCmnendmurdude_teooria_konstruktsioon"></span>Lõpmatute kümnendmurdude teooria konstruktsioon</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=9" title="Muuda alaosa "Lõpmatute kümnendmurdude teooria konstruktsioon"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=9" title="Muuda alaosa "Lõpmatute kümnendmurdude teooria konstruktsioon" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definitsioon. Reaalarv on lõpmatu kümnendmurd, s.o on avaldis kujul </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258468ddd86f0944d01443c8462385dafda5b046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.75ex; height:2.509ex;" alt="{\displaystyle \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"></span>,</div> <p>kus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869e366caf596564de4de06cb0ba124056d4064b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }"></span> on üks sümbolitest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" 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 +}"></span> või <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" 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 -}"></span>, mida nimetame arvu <a href="/w/index.php?title=M%C3%A4rk_(matemaatika)&action=edit&redlink=1" class="new" title="Märk (matemaatika) (pole veel kirjutatud)">märgiks</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></span> on <a href="/w/index.php?title=Mittenegatiivne_t%C3%A4isarv&action=edit&redlink=1" class="new" title="Mittenegatiivne täisarv (pole veel kirjutatud)">mittenegatiivne täisarv</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots a_{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots a_{n},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcce35066d06cba4f83890fab5d04eed96ddb64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.952ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots a_{n},\ldots }"></span> on <a href="/w/index.php?title=K%C3%BCmnendnumbrim%C3%A4rk&action=edit&redlink=1" class="new" title="Kümnendnumbrimärk (pole veel kirjutatud)">kümnendnumbrimärkide</a> (mida võib tõlgendada arvuhulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,\ldots 9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,\ldots 9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf462c5f5c24e4542aee24dfdfd89dfc33d30e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.99ex; height:2.843ex;" alt="{\displaystyle \{0,1,\ldots 9\}}"></span> elementidena) jada. </p><p>Seejuures samastame definitsiooni kohaselt kümnendmurrud <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +0,00\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>0</mn> <mo>,</mo> <mn>00</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +0,00\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94778fadecb40419a3d8e9f8dc1f9becbe731705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.44ex; height:2.509ex;" alt="{\displaystyle +0,00\ldots }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0,00\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>00</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0,00\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd18b8b39cb5d2d0419c07425044883ddf6c0bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.44ex; height:2.509ex;" alt="{\displaystyle -0,00\ldots }"></span>, samuti kümnendmurrud kujul <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm a_{0},a_{1}\ldots a_{n}999\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mn>999</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm a_{0},a_{1}\ldots a_{n}999\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc51f13c359ef18f0fb92272c21027bb66d5275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.954ex; height:2.509ex;" alt="{\displaystyle \pm a_{0},a_{1}\ldots a_{n}999\ldots }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm a_{0},a_{1}\ldots (a_{n}+1)000\ldots ,\;(a_{n}\neq 9)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mn>000</mn> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm a_{0},a_{1}\ldots (a_{n}+1)000\ldots ,\;(a_{n}\neq 9)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d96e02f697d790a74d47ed05a6f19e9787661d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.35ex; height:2.843ex;" alt="{\displaystyle \pm a_{0},a_{1}\ldots (a_{n}+1)000\ldots ,\;(a_{n}\neq 9)}"></span>. Selle kokkuleppe mõte on ilmne, sest neile kümnendmurdudele vastavad ratsionaalarvud langevad kokku.<sup id="cite_ref-Nsa5G_5-0" class="reference"><a href="#cite_note-Nsa5G-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Loomulik on kohe kokku leppida, et perioodilised lõpmatud kümnendmurrud kujutavad neile vastavaid ratsionaalarve. Teiste sõnadega, me samastame perioodilised kümnendmurrud ratsionaalarvudega. Sellise kokkuleppe puhul on <a href="/wiki/Ratsionaalarvude_hulk" class="mw-redirect" title="Ratsionaalarvude hulk">ratsionaalarvude hulk</a> <a href="/wiki/Reaalarvude_hulk" title="Reaalarvude hulk">kõikide reaalarvude hulga</a> <a href="/wiki/Alamhulk" title="Alamhulk">alamhulk</a>. </p><p>Järgneb lõpmatute kümnendmurdude teooria konstruktsiooni visand. </p><p>Kõigepealt defineeritakse <a href="/wiki/J%C3%A4rjestus" title="Järjestus">järjestus</a> kõigi lõpmatute kümnendmurdude hulgal. Aluseks võetakse arvude kümnendjärkude järjestikune võrdlemine suurematest väiksemateni. Olgu näiteks antud kaks mittenegatiivset arvu </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\alpha &=+a_{0},a_{1}a_{2}\ldots a_{n}\ldots \\\beta &=+b_{0},b_{1}b_{2}\ldots b_{n}\ldots \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mo>=</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> </mtd> <mtd> <mo>=</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\alpha &=+a_{0},a_{1}a_{2}\ldots a_{n}\ldots \\\beta &=+b_{0},b_{1}b_{2}\ldots b_{n}\ldots \end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3f3df86eeaa754c85d20bfe06ae38e0e544de3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.765ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}\alpha &=+a_{0},a_{1}a_{2}\ldots a_{n}\ldots \\\beta &=+b_{0},b_{1}b_{2}\ldots b_{n}\ldots \end{matrix}}}"></span></div> <p>Olgu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span> esimesed mittekokkulangevad numbrimärgid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> üleskirjutustes. Kui nüüd <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}<b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}<b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8e8dce32064580a5c50a9a1569bfb3988bbd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.763ex; height:2.509ex;" alt="{\displaystyle a_{n}<b_{n}}"></span>, siis definitsiooni kohaselt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4250c976810fd3abc5916c47ee4984a7e5882b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha <\beta }"></span>, ja kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}>b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}>b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4047244d69264c4e3662e2a2b7ff2d127da8600d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.763ex; height:2.509ex;" alt="{\displaystyle a_{n}>b_{n}}"></span>, siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878aa4ba937a08258b46361229c9c35d970672ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha >\beta }"></span>. Kahe mittenegatiivse arvu võrdluse alusel defineeritakse mis tahes kahe reaalarvu võrreldavus. </p><p>Saab näidata, et defineeritud võrdlusseos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> annab lõpmatute kümnendmurdude hulgal <a href="/wiki/Lineaarselt_j%C3%A4rjestatud_hulk" class="mw-redirect" title="Lineaarselt järjestatud hulk">lineaarselt järjestatud hulga</a> struktuuri. Samuti saab näidata, et perioodiliste kümnendmurdude korral langeb kehtestatud järjestussuhe kokku juba olemasoleva võrreldavusseosega ratsionaalarvude seas. </p><p>Pärast lõpmatute kümnendmurdude hulgal järjestusseose defineerimist tõestatakse reaalarvu teooria jaoks põhimõttelise tähtsusega teoreem täpsest ülemrajast. See teoreem väljendab asjaolu, et reaalarvude järjestatud kogumil on pidevuse (<a href="/w/index.php?title=Dedekindi_t%C3%A4ielikkus&action=edit&redlink=1" class="new" title="Dedekindi täielikkus (pole veel kirjutatud)">Dedekindi täielikkuse</a>) omadus. </p><p>Nüüd laiendatakse aritmeetilised tehted, mis on juba defineeritud ratsionaalarvude seas, pidevuse alusel kõigile reaalarvudele. </p><p>Nimelt, olgu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> kaks reaalarvu. Nende summaks nimetatakse reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b80a3fdecb9cf75091789bb4335a1bb3561b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }"></span>, mis rahuldab järgmist tingimust: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'+b'\leqslant \alpha +\beta \leqslant a''+b'')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>α</mi> <mo>⩽</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>β</mi> <mo>⩽</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>⩽</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo>+</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'+b'\leqslant \alpha +\beta \leqslant a''+b'')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000e80d785e96dd07c40cebda6ef93326a1e2886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.36ex; height:3.009ex;" alt="{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'+b'\leqslant \alpha +\beta \leqslant a''+b'')}"></span></div> <p>Saab näidata, et seda tingimust rahuldav reaalarv eksisteerib ja on ainuke. </p><p>Analoogselt defineeritakse reaalarvude korrutamine. Kahe <i>positiivse</i> reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>korrutiseks nimetatakse reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \cdot \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \cdot \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753c9592c42595168e4a85ce28a9c338c5cff3ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.499ex; height:2.509ex;" alt="{\displaystyle \alpha \cdot \beta }"></span>, mis rahuldab järgmist tingimust: </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'>0)\land (b'>0)\land (a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'\cdot b'\leqslant \alpha \cdot \beta \leqslant a''\cdot b'')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>α</mi> <mo>⩽</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>β</mi> <mo>⩽</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mo>⩽</mo> <msup> <mi>a</mi> <mo>″</mo> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'>0)\land (b'>0)\land (a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'\cdot b'\leqslant \alpha \cdot \beta \leqslant a''\cdot b'')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4f24f318e3a5b4d1105000a2fa074540160634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:95.778ex; height:3.009ex;" alt="{\displaystyle \forall a',a'',b',b''\in \mathbb {Q} \;(a'>0)\land (b'>0)\land (a'\leqslant \alpha \leqslant a'')\land (b'\leqslant \beta \leqslant b'')\Rightarrow (a'\cdot b'\leqslant \alpha \cdot \beta \leqslant a''\cdot b'')}"></span></div> <p>Nagu ka liitmise puhul, seda tingimust rahuldav arv eksisteerib ja on ainuke. Pärast seda on lihtne defineerida kahe suvalise märgiga reaalarvu korrutis. </p><p>Saab kontrollida, et reaalarvude hulgal defineeritud liitmise ja korrutamise tehe langevad kokku ratsionaalarvude liitmise ja korrutamise tehtega. </p><p>Sellega on lõpmatute kümnendmurdude teooria konstruktsioon lõpule viidud. Edasi saab antud definitsioonide põhjal tõestada reaalarvude teadaolevad omadused, mis puudutavad aritmeetilisi tehteid ja järjestust. </p><p>Lõpuks olgu märgitud, et pärast reaalarvude jada piirväärtuse ja rea summa defineerimist saab tõestada, et iga reaalarv on oma kümnendlahutuse rea summa. See tähendab, kui </p> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∼</mo> <mo>±</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf4ee4d8d80299a6dfe4a092f733db86ee59073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.336ex; height:2.509ex;" alt="{\displaystyle \alpha \sim \pm a_{0},a_{1}a_{2}\ldots a_{n}\ldots }"></span>,</div> <p>siis </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo>±</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c935bb66b3f1922f524900f964240c1d70820979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.213ex; height:7.009ex;" alt="{\displaystyle \alpha =\pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}"></span>. </p> </div> <div class="mw-heading mw-heading4"><h4 id="Ajalooline_kommentaar">Ajalooline kommentaar</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=10" title="Muuda alaosa "Ajalooline kommentaar"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=10" title="Muuda alaosa "Ajalooline kommentaar" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weierstrass ise vaatles pisut teistsugust konstruktsiooni<sup id="cite_ref-Пути_4-1" class="reference"><a href="#cite_note-Пути-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-f5k4l_6-0" class="reference"><a href="#cite_note-f5k4l-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>. </p><p>Ülalesitatud teooriat võib lühidalt määratleda teooriana <a href="/w/index.php?title=Formaalne_rida&action=edit&redlink=1" class="new" title="Formaalne rida (pole veel kirjutatud)">formaalsetest ridadest</a> kujul </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cbb0d68bfa5080b71129310bb43ccefa333b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.627ex; height:7.009ex;" alt="{\displaystyle \pm \sum _{k=0}^{\infty }a_{k}\cdot 10^{-k}}"></span>, </p> </div> <p>kus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></span> on mittenegatiivne täisarv ning <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k},k=1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k},k=1,2,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a723ee22818696a83b5563583675aad6a4ae864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.778ex; height:2.509ex;" alt="{\displaystyle a_{k},k=1,2,\ldots }"></span> on – kümnendnumbrimärgid. </p><p>Weierstrass aga vaatles formaalseid ridasid üldisemal kujul: </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sum _{n=0}^{\infty }a_{n}\cdot 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sum _{n=0}^{\infty }a_{n}\cdot 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf449286daa257213d5f7ec821980737e68141f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.784ex; height:6.843ex;" alt="{\displaystyle \pm \sum _{n=0}^{\infty }a_{n}\cdot 1/n}"></span>, </p> </div> <p>kus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n},n=1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n},n=1,2,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3761e25a26f2c9c03eef9b25b34d58de902df16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.091ex; height:2.509ex;" alt="{\displaystyle a_{n},n=1,2,\ldots }"></span> on suvalised mittenegatiivsed täisarvud. </p><p>On ilmne, et niisuguses konstruktsioonis saab reaalarvu kujutada lõpmata paljudel viisidel. Peale selle on selge, et kaugeltki mitte kõigile niisugustele ridadele ei saa omistada arvväärtust. Näiteks rida </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af384e90b895564fe0c1efa82f9e63008fa0dca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:7.462ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }1/n}"></span> </p> </div> <p><a href="/w/index.php?title=Hajuv_rida&action=edit&redlink=1" class="new" title="Hajuv rida (pole veel kirjutatud)">hajub</a>. </p><p>Sellepärast Weierstrass esiteks vaatleb ainult koonduvaid ridasid (ta defineerib need read <a href="/w/index.php?title=T%C3%B5kestatud_osasummadega_rida&action=edit&redlink=1" class="new" title="Tõkestatud osasummadega rida (pole veel kirjutatud)">tõkestatud osasummadega ridadena</a> (vaata <a href="/w/index.php?title=Mittenegatiivsete_liikmetega_rea_koonduvuse_kriteerium&action=edit&redlink=1" class="new" title="Mittenegatiivsete liikmetega rea koonduvuse kriteerium (pole veel kirjutatud)">mittenegatiivsete liikmetega rea koonduvuse kriteeriumi</a>)) ning teiseks defineerib sellel hulgal ekvivalentsusseose. Reaalarv on defineeritud ekvivalentsete koonduvate ridade klassina. </p><p>Reaalarvude defineerimisviis kümnendmurdude abil, see tähendab lahutuse abil mitte kõikide alikvootsete murdude (see on murdude kujul <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span>) kaupa, vaid ainult kümne astmete <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/10^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/10^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d55a9ae7c1ba21eeb52bf77021e826f6df5537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.739ex; height:3.176ex;" alt="{\displaystyle 1/10^{k}}"></span> kaupa on mugavam, sest sellega saavutatakse reaalarvu rea kujul kujutamise ühesus. Kui aga tulla tagasi Weierstrassi üldise viisi juurde, siis saab ilmsiks analoogia Weierstrassi lähenemise ja Cantori lähenemise vahel. Cantor defineeris reaalarvu ratsionaalarvude koonduvate jadade ekvivalentsusklassina, kusjuures jada koonduvuse määratlemiseks ta kasutas <a href="/w/index.php?title=Cauchy_kriteerium&action=edit&redlink=1" class="new" title="Cauchy kriteerium (pole veel kirjutatud)">Cauchy kriteeriumi</a>. Weierstrass tegi sedasama, ainult et koonduvate jadade asemel vaatles ta koonduvaid ridasid ning jada koonduvuse Cauchy kriteeriumi asemel kasutas ta mittenegatiivsete liikmetega rea koonduvuse tunnust (muide, ekvivalentne teoreem monotoonse jada piirväärtusest kannab Weierstrassi nime). </p> <div class="mw-heading mw-heading3"><h3 id="Lõigete_teooria_ratsionaalarvude_vallas"><span id="L.C3.B5igete_teooria_ratsionaalarvude_vallas"></span>Lõigete teooria ratsionaalarvude vallas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=11" title="Muuda alaosa "Lõigete teooria ratsionaalarvude vallas"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=11" title="Muuda alaosa "Lõigete teooria ratsionaalarvude vallas" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/12px-Next.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/18px-Next.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/24px-Next.svg.png 2x" data-file-width="160" data-file-height="160" /></span></span> <i>Pikemalt artiklis <a href="/wiki/Dedekindi_l%C3%B5ige" title="Dedekindi lõige">Dedekindi lõige</a></i></dd></dl> <p><a href="/w/index.php?title=Richard_Dedekind&action=edit&redlink=1" class="new" title="Richard Dedekind (pole veel kirjutatud)">Richard Dedekindi</a> teooria on kõige lihtsam ja ajalooliselt esimene range reaalarvuteooria. Erinevalt Cantori ja Weierstrassi analüütilisest lähenemisest on Dedekindi teooria aluseks geomeetrilised kaalutlused, millest tuleneb selle näitlikkus. </p><p>Dedekindi teooria väärtus seisneb selles, et peale reaalarvude konstrueerimise toodi seal esimest korda välja <a href="/w/index.php?title=Pidevus_(filosoofia)&action=edit&redlink=1" class="new" title="Pidevus (filosoofia) (pole veel kirjutatud)">pidevuse</a> mõiste matemaatiline olemus. See mõiste on <a href="/wiki/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs">matemaatilise analüüsi</a> aluseks ning seda oli sajandeid kasutatud, viidates selle ilmsusele või geomeetrilistele kaalutlustele. </p><p>Dedekind lõi oma teooria <a href="/wiki/1858" title="1858">1858</a>, kuid see avaldati esmakordselt <a href="/wiki/1872" title="1872">1872</a> väikeses brošüüris "<a href="/w/index.php?title=Stetigkeit_und_irrationale_Zahlen&action=edit&redlink=1" class="new" title="Stetigkeit und irrationale Zahlen (pole veel kirjutatud)">Stetigkeit und irrationale Zahlen</a>"<sup id="cite_ref-QMGcp_7-0" class="reference"><a href="#cite_note-QMGcp-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> ('Pidevus ja irratsionaalarvud'). See raamat on tänini üks paremaid ja arusaadavaid aine esitusi. Siinne esitus järgib põhilises Dedekindi mõttekäiku. </p> <div class="mw-heading mw-heading4"><h4 id="Küsimuseasetus"><span id="K.C3.BCsimuseasetus"></span>Küsimuseasetus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=12" title="Muuda alaosa "Küsimuseasetus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=12" title="Muuda alaosa "Küsimuseasetus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dedekindi ajal tuli <a href="/wiki/Diferentsiaalarvutus" title="Diferentsiaalarvutus">diferentsiaalarvutuse</a> kursuse esituses, mis enamasti kasutas rangeid meetodeid, mõne väite tõestamisel siiski tugineda geomeetrilisele näitlikkusele. </p><p>Näiteks kui tõestati teoreemi <a href="/w/index.php?title=Monotoonne_jada&action=edit&redlink=1" class="new" title="Monotoonne jada (pole veel kirjutatud)">monotoonse jada</a> piirväärtusest, siis joonestati <a href="/wiki/Sirgjoon" class="mw-redirect" title="Sirgjoon">sirgjoon</a>, millele märgiti punktid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span>, mis kujutasid jada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> liikmeid. Edasi öeldi näiteks, et "ilmselt" eksisteerib <a href="/wiki/Punkt_(matemaatika)" title="Punkt (matemaatika)">punkt</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, millele punktid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span> piiramatult lähenevad, või et niisugune punkt "peab" eksisteerima, sest <a href="/wiki/Arvsirge" class="mw-redirect" title="Arvsirge">arvsirge</a> on "punktidega pidevalt täidetud". Edasi, kuna igale punktile sirgel vastab mingi ratsionaal- või irratsionaalarv, siis punktile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> vastava arvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> korral kehtib: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim a_{n}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">lim</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim a_{n}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d390caa436925bb71ae9d3ca80dc87ab333e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.393ex; height:2.509ex;" alt="{\displaystyle \lim a_{n}=a}"></span>. Dedekind ütleb: "Sageli öeldakse, et diferentsiaalarvutus tegeleb <a href="/w/index.php?title=Pidev_suurus&action=edit&redlink=1" class="new" title="Pidev suurus (pole veel kirjutatud)">pidevate suurustega</a>, ent mitte kuskil ei anta selle <a href="/w/index.php?title=Pidevus_(filosoofia)&action=edit&redlink=1" class="new" title="Pidevus (filosoofia) (pole veel kirjutatud)">pidevuse</a> <a href="/wiki/Definitsioon" title="Definitsioon">definitsiooni</a>, ja isegi diferentsiaalarvutuse kõige rangemal esitamisel ei rajata <a href="/wiki/T%C3%B5estus" title="Tõestus">tõestusi</a> pidevusele, vaid apelleeritakse enam-vähem teadlikult kas geomeetrilistele ettekujutustele või ettekujutustele, mis saavad alguse <a href="/wiki/Geomeetria" title="Geomeetria">geomeetriast</a>, või lõpuks rajatakse tõestus teesidele, mida endid pole mitte kunagi tõestatud puhtaritmeetilisel teel." </p><p>Vajadust kasutada puhtaritmeetilise (<a href="/wiki/Arv" title="Arv">arvude</a> kohta käiva) väite tõestamiseks kasutada geomeetrilisi kaalutlusi tekitab teatava rahulolematuse tunde ning annab tunnistust "aritmeetika puudulikust põhjendatusest", arvu range ja täieliku teooria puudumisest. Ent kui isegi pidada lubatavaks geomeetrilist argumentatsiooni, tekib küsimus pidevusest sirge enese punktide suhtes. Osutub, et sirge pidevuse mõistel puudub siin loogiline definitsioon. </p><p>Sellest lähtudes püstitas Dedekind järgmised kaks ülesannet: </p> <ol><li>Leida loogiline formuleering sirge põhiomadusele, mis kätkeb meie näitlikes ettekujutustes "sirge pidevast täidetusest punktidega".</li> <li>Konstrueerida arvu range puhtaritmeetiline teooria, nii et need arvude süsteemi omadused, mille põhjendamiseks varem oli kasutatud näitlikke geomeetrilisi ettekujutusi, nüüd järelduksid arvu üldisest definitsioonist.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Ratsionaalarvude_võrdlus_sirge_punktidega"><span id="Ratsionaalarvude_v.C3.B5rdlus_sirge_punktidega"></span>Ratsionaalarvude võrdlus sirge punktidega</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=13" title="Muuda alaosa "Ratsionaalarvude võrdlus sirge punktidega"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=13" title="Muuda alaosa "Ratsionaalarvude võrdlus sirge punktidega" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dedekind lähtub <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvudest</a>, mille omadused ta loeb teadaolevateks. Ratsionaalarvude süsteemi kõrvutab ta sirge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" 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 {L} }"></span> punktide kogumiga, et tuua välja viimase omadused. </p><p>Ratsionaalarvud <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> moodustavad kogumi, millel on antud aritmeetilised tehted liitmine ja korrutamine, millel on teatud omadused. Kogum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on <a href="/wiki/Lineaarselt_j%C3%A4rjestatud_hulk" class="mw-redirect" title="Lineaarselt järjestatud hulk">lineaarselt järjestatud</a>: mis tahes kahe erineva arvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> korral võib öelda, et üks neist on teisest väiksem. </p><p>Sirge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" 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 {L} }"></span> punktide kogum on samuti lineaarselt järjestatud hulk. Kahe punkti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> vaheline järjestusseos väljendub siin selles, et üks punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> asetseb teisest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> vasakul. </p><p>Selle sarnasuse ratsionaalarvude ja punktide vahel saab välja arendada, seades nende vahele vastavuse. Nii saadakse <a href="/wiki/Arvsirge" class="mw-redirect" title="Arvsirge">arvsirge</a>. Selleks valitakse sirgel kindel <a href="/w/index.php?title=Algpunkt&action=edit&redlink=1" class="new" title="Algpunkt (pole veel kirjutatud)">algpunkt</a>, kindel pikkusühik ehk <a href="/w/index.php?title=%C3%9Chikl%C3%B5ik&action=edit&redlink=1" class="new" title="Ühiklõik (pole veel kirjutatud)">ühiklõik</a> ehk skaala <a href="/wiki/L%C3%B5ik" title="Lõik">lõikude</a> mõõtmiseks ning <a href="/w/index.php?title=Positiivne_suund&action=edit&redlink=1" class="new" title="Positiivne suund (pole veel kirjutatud)">positiivne suund</a>. Iga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb03a87dc03fc305e7fe382cc673c048b6bd599e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.879ex; height:2.509ex;" alt="{\displaystyle a\in \mathbb {Q} }"></span> jaoks saab konstrueerida vastava pikkuse, ning asetades selle algpunktist paremale või vasakule olenevalt sellest, kas arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> on positiivne või mitte, saame kindla punkti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a7b393da9bb4b10142aee4130abcac27764aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.65ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {L} }"></span>, mis vastab ratsionaalarvule <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. </p><p>Seega saab igale ratsionaalarvule <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb03a87dc03fc305e7fe382cc673c048b6bd599e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.879ex; height:2.509ex;" alt="{\displaystyle a\in \mathbb {Q} }"></span> seada vastavusse kindla punkti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a7b393da9bb4b10142aee4130abcac27764aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.65ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {L} }"></span>. Seejuures vastavad eri arvudele eri punktid. Kui arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> on väiksem kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, siis punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, mis vastab arvule <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, asetseb vasakul punktist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, mis vastab arvule <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Teiste sõnadega, kehtestatud vastavus säilitab järjestuse. </p> <div class="mw-heading mw-heading4"><h4 id="Sirge_pidevus">Sirge pidevus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=14" title="Muuda alaosa "Sirge pidevus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=14" title="Muuda alaosa "Sirge pidevus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ühtlasi osutub, et sirgel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" 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 {L} }"></span> on lõpmata palju arve, mis ei vasta mitte ühelegi ratsionaalarvule. See järeldub <a href="/w/index.php?title=%C3%9Chism%C3%B5%C3%B5dutus&action=edit&redlink=1" class="new" title="Ühismõõdutus (pole veel kirjutatud)">ühismõõdutute</a> lõikude olemasolust, mis oli teada juba antiikajal (näiteks <a href="/wiki/Ruut" title="Ruut">ruudu</a> <a href="/wiki/Diagonaal" title="Diagonaal">diagonaali</a> ja <a href="/w/index.php?title=K%C3%BClg&action=edit&redlink=1" class="new" title="Külg (pole veel kirjutatud)">külje</a> ühismõõdutus, see tähendab arvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> (<a href="/wiki/Ruutjuur_kahest" title="Ruutjuur kahest">ruutjuur kahest</a>) irratsionaalsus). </p><p>Piltlikult öeldes on sirge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" 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 {L} }"></span> täidetud punktidega tihedamalt kui ratsionaalarvude kogum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> arvudega. Me näeme, et ratsionaalarvude hulga sees on tühjad kohad, mis vastavad neile sirge punktidele, millele ei leidunud vastavat ratsionaalarvu, sellal kui sirge kohta ütleme, et ta on "punktidega pidevalt täidetud". Dedekind ütleb: "Eelnev ratsionaalarvude valla võrdlus sirgega viis selleni, et esimeses avastati lünklikkus, mittetäielikkus ehk katkelisus, sellal kui sirgele me omistame täielikkuse, lünkade puudumise, pidevuse." </p><p>Milles see pidevus õigupoolest seisneb? Kuidas seda sirge omadust matemaatiliselt väljendada? </p><p>Dedekind teeb järgmise tähelepaneku. Kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> on sirge kindel punkt, siis sirge kõik punktid jagunevad kaheks klassiks: need, mis asetsevad punktist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> vasakul, ja need, mis asetsevad punktist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> paremal; punkti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> võib aga suvaliselt arvata kas esimesse või teise klassi. Ühtlasi kehtib sirge punktide kohta pöördprintsiip: "Kui sirge punktid jagunevad kaheks niisuguseks klassiks, et iga esimese klassi punkt asetseb igast teise klassi punktist vasakul, siis eksisteerib üks ja ainult üks punkt, mis tekitab sirge niisuguse jagunemise kaheks klassiks, sirge niisuguse lõigatuse kaheks tükiks." </p><p>Geomeetriliselt tundub see väide ilmsena, kuid tõestada me seda ei suuda. Dedekind osutab, et tegelikult ei ole see printsiip midagi muud kui postulaat, milles väljendub sirge pidevuse olemus. Kui võtame selle postulaadi omaks, siis omistame sirgele selle omaduse, mida me nimetame tema pidevuseks. "Selle omaduse omaksvõtmine pole midagi muud kui aksioom, mille abil me alles mööname sirgele pidevuse, paigutame mõttes sirge sisse pidevuse." </p><p>Selgitame Dedekindi printsiibi sisu ja geomeetrilist tõlgendust. Kujutame ette, et kõik sirge punktid on värvitud kahte värvi – roheliseks ja punaseks, nii et iga rohelist värvi punkt asetseb vasakul igast punast värvi punktist. </p><p>On geomeetriselt ilmne, et peab eksisteerima sirge niisugune punkt, milles värvid puutuvad kokku. See punkt jagabki sirge kaheks klassiks: kõik rohelist värvi punktid asetsevad temast vasakul ja kõik punast värvi punktid paremal. Selles seisnebki Dedekindi printsiip. </p><p>Sealjuures peab ka kokkupuutepunkt ise olema kindlat värvi, sest tingimuse kohaselt on värvitud eranditult kõik sirge punktid. See punkt peab olema kas roheline, olles sel juhul viimane roheline punkt, või punane, olles sel juhul esimene punane punkt. Nagu on kerge näha, need kaks varianti välistavad teineteist: esimesel juhtumil ei eksisteeri esimest punast punkti – eksisteerivad kokkupuutepunktile kui tahes lähedased punased punktid, kuid esimest nende seas ei ole, teisel juhtumil aga puudub analoogsetel põhjustel viimane roheline punkt. </p><p>Nüüd pöörame tähelepanu sellele, millised loogilised võimalused me välistasime, apelleerides geomeetrilisele näitlikkusele. Kerge on näha, et neid on ainult kaks: esiteks võiks juhtuda, et üheaegselt eksisteerivad nii viimane roheline kui ka esimene punane punkt; teiseks võiks juhtuda, et pole ei viimast rohelist ega esimest punast punkti. </p><p>Esimese olukorra kohta öeldakse, et leiab aset hüpe. Selline pilt on võimalik sirge korral, millest on välja visatud terve vahemik vahepealseid punkte. </p><p>Teise olukorra kirjeldamiseks kasutatakse terminit "lünk". See pilt võib aset leida sirge korral, millest on eemaldatud terve lõik, kaasa arvatud selle otsad – sealhulgas juhtum, kui on eemaldatud üksainus punkt. </p><p>Seega tähendab sirge pidevus, et temas ei ole ei hüppeid ega lünki – lühidalt, ei ole tühje kohti. </p><p>On märkimisväärne, et ülaltoodud pidevuse definitsioon on rakendatav mis tahes elementide järjestatud kogumile. </p> <div class="mw-heading mw-heading4"><h4 id="Pidevus_Dedekindi_järgi"><span id="Pidevus_Dedekindi_j.C3.A4rgi"></span>Pidevus Dedekindi järgi</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=15" title="Muuda alaosa "Pidevus Dedekindi järgi"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=15" title="Muuda alaosa "Pidevus Dedekindi järgi" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Anname nüüd pidevuse täpse definitsiooni Dedekindi järgi, mis on rakendatav suvalisele lineaarselt järjestatud hulgale. </p><p>Definitsioon. Olgu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25704a85c37e68d78dd9f549587912bf314b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.176ex;" alt="{\displaystyle {\mathsf {L}}}"></span> lineaarselt järjestatud hulk. Hulkade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span> järjestatud paari nimetatakse lõikeks järjestatud hulgal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25704a85c37e68d78dd9f549587912bf314b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.176ex;" alt="{\displaystyle {\mathsf {L}}}"></span> ning hulki <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span> endid selle lõike vastavalt alumiseks ja ülemiseks klassiks, kui on täidetud järgmised tingimused: </p><p>1. Klassid on mittetühjad:<i></i> </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\neq \varnothing ,A'\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq \varnothing ,A'\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b8929c995ccfb515819458cfab8b85390f1749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.018ex; height:3.009ex;" alt="{\displaystyle A\neq \varnothing ,A'\neq \varnothing }"></span> </p> </div> <p>2. Järjestatud hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25704a85c37e68d78dd9f549587912bf314b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.176ex;" alt="{\displaystyle {\mathsf {L}}}"></span> iga element kuulub vähemalt ühesse klassidest': </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup A'={\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup A'={\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3880d4df2fb564e4fbdd7088188af7fd52baf226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.112ex; height:2.509ex;" alt="{\displaystyle A\cup A'={\mathsf {L}}}"></span> </p> </div> <p>3. Alumise klassi iga element on väiksem ülemise klassi suvalisest elemendist: </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a\in A,\forall a'\in A'\;(a<a')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mi mathvariant="normal">∀</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>∈</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo><</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A,\forall a'\in A'\;(a<a')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f93f65b3a695fa94fab31896989baf59a615e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.313ex; height:3.009ex;" alt="{\displaystyle \forall a\in A,\forall a'\in A'\;(a<a')}"></span> </p> </div> <p>Lõiget tähistame <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A|A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d677f90895eb598b46e0a31b67c3776b9b231c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:3.009ex;" alt="{\displaystyle A|A'}"></span>. </p><p>Definitsioon. Lineaarselt järjestatud hulka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25704a85c37e68d78dd9f549587912bf314b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.26ex; height:2.176ex;" alt="{\displaystyle {\mathsf {L}}}"></span> nimetatakse <a href="/w/index.php?title=Pidev_lineaarselt_j%C3%A4rjestatud_hulk&action=edit&redlink=1" class="new" title="Pidev lineaarselt järjestatud hulk (pole veel kirjutatud)">pidevaks</a> (Dedekindi järgi), kui tema mis tahes lõike puhul kas alumises klassis leidub suurim element ja ülemises klassis ei leidu vähimat või ülemises klassis leidub vähim element ja alumises klassis ei leidu suurimat (niisuguseid lõikeid nimetatakse <a href="/wiki/Dedekindi_l%C3%B5ige" title="Dedekindi lõige">Dedekindi lõigeteks</a>). </p><p>Näitena vaatleme ratsionaalarvude hulka. Kerge on näha, et selles ei saa olla hüppeid: kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> on alumise klassi maksimaalne element, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> on ülemise klassi minimaalne element, siis arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)/2}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3045bc796115de4496f1b5474aa35178ad9c139a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.202ex; height:2.843ex;" alt="{\displaystyle (a+b)/2}"></span>, mis asetseb <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> vahel keskel, ei saa kuuluda ei alumisse ega ülemisse klassi, mis on vastuolus lõike definitsiooniga. </p><p>Samal ajal on ratsionaalarvude hulga sees tühikud – just neis kohtades, kus peavad asuma irratsionaalarvud. Vaatleme näiteks lõiget <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A|A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d677f90895eb598b46e0a31b67c3776b9b231c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:3.009ex;" alt="{\displaystyle A|A'}"></span>, mille defineerivad hulgad </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{x\in \mathbb {Q} :x\leq 0\lor (x>0\land x^{2}<2)\},A'=\{x\in \mathbb {Q} :x>0\land x^{2}>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"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>≤</mo> <mn>0</mn> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>∧</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>∧</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{x\in \mathbb {Q} :x\leq 0\lor (x>0\land x^{2}<2)\},A'=\{x\in \mathbb {Q} :x>0\land x^{2}>2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d407ef3024c0acb7d3fffdc77999e966a8ac08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.501ex; height:3.176ex;" alt="{\displaystyle A=\{x\in \mathbb {Q} :x\leq 0\lor (x>0\land x^{2}<2)\},A'=\{x\in \mathbb {Q} :x>0\land x^{2}>2\}}"></span> </p> </div> <p>Kerge on näha, et see on tõesti lõige, kuid alumises klassis ei ole maksimaalset elementi ja ülemises klassis ei ole minimaalset. Seega on tegu lüngaga. </p> <div class="mw-heading mw-heading4"><h4 id="Irratsionaalarvude_konstrueerimine">Irratsionaalarvude konstrueerimine</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=16" title="Muuda alaosa "Irratsionaalarvude konstrueerimine"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=16" title="Muuda alaosa "Irratsionaalarvude konstrueerimine" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seega ratsionaalarvude kogum ei ole pidev: selles on lüngad. Et konstrueerida reaalarvude hulk, mille elemendid assotsieeruvad reaalarvudega, tuleb täita kõik lüngad ratsionaalarvude kogumis. </p><p>Ratsionaalarvude hulga iga lüngatüüpi lõike <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A|A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d677f90895eb598b46e0a31b67c3776b9b231c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:3.009ex;" alt="{\displaystyle A|A'}"></span> korral lisame kogumile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> uue elemendi (irratsionaalarvu) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, mis <i>definitsiooni kohaselt</i> on suurem igast alumisse klassi kuuluvast arvust <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ning väiksem igast ülemisse klassi kuuluvast arvust <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb64c0f02687512818f37839ce23ee049c37743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.915ex; height:2.509ex;" alt="{\displaystyle a'}"></span>. Sellega täidamegi lõike klasside vahelise tühja koha. Me ütleme, et lõige <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A|A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d677f90895eb598b46e0a31b67c3776b9b231c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:3.009ex;" alt="{\displaystyle A|A'}"></span> defineerib irratsionaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ehk irratsionaalarv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> tekitab lõike <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A|A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d677f90895eb598b46e0a31b67c3776b9b231c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:3.009ex;" alt="{\displaystyle A|A'}"></span>. </p><p>Ühendades kõik võimalikud juhtumid, võime öelda, et iga lõige <i>ratsionaalarvude vallas</i> defineerib mingi ratsionaal- või irratsionaalarvu, mis selle lõike tekitab. </p><p><i><b>Definitsioon.</b></i> <i><a href="/w/index.php?title=Mall:Razr&action=edit&redlink=1" class="new" title="Mall:Razr (pole veel kirjutatud)">Mall:Razr</a> nimetatakse iga lõiget ratsionaalarvude hulgas, mille alumises klassis ei ole suurimat elementi ja mille ülemises klassis ei ole vähimat elementi.</i> </p><p><i><b>Definitsioon.</b></i> <i><a href="/w/index.php?title=Mall:Razr&action=edit&redlink=1" class="new" title="Mall:Razr (pole veel kirjutatud)">Mall:Razr</a> nimetatakse ratsionaal- ja irratsionaalarvude hulga ühendit. Reaalarvude hulga iga elementi nimetatakse <a href="/w/index.php?title=Mall:Razr&action=edit&redlink=1" class="new" title="Mall:Razr (pole veel kirjutatud)">Mall:Razr</a>.</i> </p><p>Reaalarvude hulk on defineeritud järjestusseose suhtes lineaarselt järjestatud. </p><p><big><a href="/w/index.php?title=Mall:Razr&action=edit&redlink=1" class="new" title="Mall:Razr (pole veel kirjutatud)">Mall:Razr</a></big> <i>Reaalarvude hulk on pidev Dedekindi järgi.</i> </p><p>See lause vajab tõestust. </p><p>Liitmis- ja korrutamistehe defineeritakse reaalarvude hulgal <i>pidevuse järgi</i> (nagu ka lõpmatute kümnendmurdude teoorias). Nimelt, kahe reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> summaks nimetatakse reaalarvu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, mis rahuldab järgmist tingimust: </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,a',b,b'\;(a\leqslant \alpha \leqslant a')\land (b\leqslant \beta \leqslant b')\Rightarrow (a+a'\leqslant \gamma \leqslant b+b')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>b</mi> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⩽</mo> <mi>α</mi> <mo>⩽</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⩽</mo> <mi>β</mi> <mo>⩽</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>⩽</mo> <mi>γ</mi> <mo>⩽</mo> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,a',b,b'\;(a\leqslant \alpha \leqslant a')\land (b\leqslant \beta \leqslant b')\Rightarrow (a+a'\leqslant \gamma \leqslant b+b')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4a961909cb50903efc026c3c1684a74c5201a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.49ex; height:3.009ex;" alt="{\displaystyle \forall a,a',b,b'\;(a\leqslant \alpha \leqslant a')\land (b\leqslant \beta \leqslant b')\Rightarrow (a+a'\leqslant \gamma \leqslant b+b')}"></span> </p> </div> <p>Reaalarvude hulga pidevusest järeldub, et niisugune reaalarv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> on olemas ja on ainus. Peale selle, kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> on ratsionaalarvud, siis see definitsioon langeb kokku tavalise kahe ratsionaalarvu summa definitsiooniga. Analoogselt defineeritakse korrutamine ja tõestatakse tehete ning järjestusseose omadused. </p> <div class="mw-heading mw-heading2"><h2 id="Reaalarvude_aksiomaatikad">Reaalarvude aksiomaatikad</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=17" title="Muuda alaosa "Reaalarvude aksiomaatikad"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=17" title="Muuda alaosa "Reaalarvude aksiomaatikad" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Reaalarvude_hulk_kui_pidev_järjestatud_korpus"><span id="Reaalarvude_hulk_kui_pidev_j.C3.A4rjestatud_korpus"></span>Reaalarvude hulk kui pidev järjestatud korpus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=18" title="Muuda alaosa "Reaalarvude hulk kui pidev järjestatud korpus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=18" title="Muuda alaosa "Reaalarvude hulk kui pidev järjestatud korpus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Hulk" title="Hulk">Hulka</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> nimetatakse reaalarvude hulgaks ning selle elemente reaalarvudeks, kui on täidetud järgmine tingimuste kompleks, mida nimetatakse reaalarvude <a href="/w/index.php?title=Aksiomaatika&action=edit&redlink=1" class="new" title="Aksiomaatika (pole veel kirjutatud)">aksiomaatikaks</a>:</i> </p> <div class="mw-heading mw-heading4"><h4 id="Korpuse_aksioomid">Korpuse aksioomid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=19" title="Muuda alaosa "Korpuse aksioomid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=19" title="Muuda alaosa "Korpuse aksioomid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hulgal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on defineeritud <a href="/wiki/Kujutus" class="mw-redirect" title="Kujutus">kujutus</a> (<i><a href="/wiki/Liitmine" title="Liitmine">liitmise</a> <a href="/wiki/Binaarne_tehe" title="Binaarne tehe">binaarne tehe</a></i>) </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +:\mathbb {R} \times \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +:\mathbb {R} \times \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7f521b0e627fa1c4bf3c04a00a40a2467b0a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.234ex; height:2.343ex;" alt="{\displaystyle +:\mathbb {R} \times \mathbb {R} \to \mathbb {R} }"></span>, </p> </div> <p>mis seab igale elementide <a href="/wiki/J%C3%A4rjestatud_paar" title="Järjestatud paar">järjestatud paarile</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> hulgast <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> vastavusse mingi elemendi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> sellest samast hulgast <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>, mida nimetatakse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> <i><a href="/wiki/Summa" class="mw-redirect" title="Summa">summaks</a></i> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> on hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> elemendi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> ekvivalentne üleskirjutus). </p><p>Hulgal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on defineeritud ka kujutus (<i><a href="/wiki/Korrutamine" title="Korrutamine">korrutamise</a> tehe</i>) </p> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot :\mathbb {R} \times \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot :\mathbb {R} \times \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643ae47b3f2820f2e6c6f18ebd7dd19288392208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.073ex; height:2.176ex;" alt="{\displaystyle \cdot :\mathbb {R} \times \mathbb {R} \to \mathbb {R} }"></span>, </p> </div> <p>mis seab igale elementide <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> järjestatud paarile hulgast <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> vastavusse mingi elemendi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}"></span>, mida nimetatakse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> <i><a href="/wiki/Korrutis" class="mw-redirect" title="Korrutis">korrutiseks</a></i>. </p><p>Sealjuures kehtivad järgmised omadused: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b3dfd1544ba1048b865dc498127ad3d02803d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{1}.}"></span> <i>Liitmise <a href="/wiki/Kommutatiivsus" title="Kommutatiivsus">kommutatiivsus</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c350682e8254d03e7ee156c0aa13a9e152445bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{2}.}"></span> <i>Liitmise <a href="/wiki/Assotsiatiivsus" title="Assotsiatiivsus">assotsiatiivsus</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)=(a+b)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)=(a+b)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44038eb287a7d11c82ecf1642362bff63a012b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle a+(b+c)=(a+b)+c}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299522b2e44e7aa23abde01451f87d27201a6422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{3}.}"></span> <i>Nullelemendi olemasolu</i> Leidub <i><a href="/wiki/Nullelement" title="Nullelement">nullelemendiks</a></i> nimetatav element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b744aa80fd21e49f206ec213a0889eb81b80ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:2.176ex;" alt="{\displaystyle 0\in \mathbb {R} }"></span>, millel on omadus, et mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4564e28f0f8274644ca4e58664c0593ed48de541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle a+0=a}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db94850acb6976827c87f7747c493abd4cd45249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{4}.}"></span> <i>Vastandelemendi olemasolu.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span> korral leidub <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <i><a href="/wiki/Vastandelement" title="Vastandelement">vastandelemendiks</a></i> nimetatav element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92135fa35d83ab2e4287ba792bbf0b9a65c86276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.557ex; height:2.343ex;" alt="{\displaystyle -a\in \mathbb {R} }"></span>, millel on omadus, et</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(-a)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(-a)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bee04ca35f37fa95ee387e42959af9038fb8251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.178ex; height:2.843ex;" alt="{\displaystyle a+(-a)=0}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{5}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{5}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ae95c305570f03fabd2495275656162ce1c1f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{5}.}"></span> <i>Korrutamise kommutatiivsus.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7dede7493e0231b3ad6ff9b54f4eae954108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.911ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{6}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{6}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d02010489d5d4073a4a15d056c34ae716e585eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{6}.}"></span> <i>Korrutamise assotsiatiivsus.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3612773cca3027ce075da63c1bb466f3ee43ea5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.21ex; height:2.509ex;" alt="{\displaystyle a,b,c,\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be34a5bcecdbbd7f3d5a983e34f00bf0b80c5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.902ex; height:2.843ex;" alt="{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{7}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{7}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16962b265cba3ec9a94ddd26dfa33181d0c35ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{7}.}"></span> <i>Ühikelemendi olemasolu.</i> Leidub <i><a href="/wiki/%C3%9Chikelement" title="Ühikelement">ühikelemendiks</a></i> nimetatav element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced69ca76045409978b6c9299d4d07f86d68b05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.767ex; height:2.176ex;" alt="{\displaystyle 1\in R}"></span>, millel on omadus, et mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b8332c70318b1c764cb4bcea0a6eb274600975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.834ex; height:2.176ex;" alt="{\displaystyle a\in R}"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 1=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 1=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e5c8aeb598f9dadf4767a03328e05849f37035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.176ex;" alt="{\displaystyle a\cdot 1=a}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{8}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{8}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d50be4c99c549ab2ca11058da78004162306cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{8}.}"></span> <i>Pöördelemendi olemasolu.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} ,a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} ,a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b7cfa5365f02bae8ca3028d9337e8909a04df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.273ex; height:2.676ex;" alt="{\displaystyle a\in \mathbb {R} ,a\neq 0}"></span> korral leidub <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> <i><a href="/w/index.php?title=P%C3%B6%C3%B6rdelement&action=edit&redlink=1" class="new" title="Pöördelement (pole veel kirjutatud)">pöördelemendiks</a> nimetatav element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8280a506ddc5535b8160679c3d84b934cacb87c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.081ex; height:2.676ex;" alt="{\displaystyle a^{-1}\in \mathbb {R} }"></span>, ehk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77da3744bf9edd650bb5dc02004c95129e2bc826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle 1/a}"></span>, millel on omadus, et</i></dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot a^{-1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot a^{-1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d0141fcc95d71d4666a3ef3d084f69c72954df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.732ex; height:2.676ex;" alt="{\displaystyle a\cdot a^{-1}=1}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{9}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{9}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a30c0d7934de18a546ee8d58db5b05c75ba0e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{9}.}"></span> <i>Korrutamise <a href="/wiki/Distributiivsus" title="Distributiivsus">distributiivsus</a> liitmise suhtes.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db5a5687f2ba931e229a9a3c402f70bb49aad61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.821ex; height:2.509ex;" alt="{\displaystyle a,b,c\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8827e12f09f1ab8a5f3d7783b7357bd4cc398db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.324ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}"></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}_{10}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}_{10}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5a6ba8099dbcf543d8c385f0c57a6cbbcb7036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.363ex; height:2.509ex;" alt="{\displaystyle {\text{I}}_{10}.}"></span> <i>Korpuse mittetriviaalsus.</i> <i>Ühikelement</i> ja <i>nullelement</i> on hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> erinevad elemendid :</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bee82995e4ac724a7b5fdfb4b0d76560321e1d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 1\neq 0}"></span> </p> </div> <div class="mw-heading mw-heading4"><h4 id="Järjestuse_aksioomid"><span id="J.C3.A4rjestuse_aksioomid"></span>Järjestuse aksioomid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=20" title="Muuda alaosa "Järjestuse aksioomid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=20" title="Muuda alaosa "Järjestuse aksioomid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> elementide vahel on defineeritud <a href="/wiki/Binaarne_seos" title="Binaarne seos">binaarne seos</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leqslant }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⩽</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leqslant }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641be8b31b3bebbb9122010d73df358f4cf7203a" 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 \leqslant }"></span>, st mis tahes elementide <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> järjestatud paari korral hulgast <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on kindlaks määratud, kas seos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leqslant b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⩽</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leqslant b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f994f4b416430409bb184cfa927b85392ee9d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leqslant b}"></span> kehtib või mitte. Sealjuures kehtivad järgmised omadused. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f5c9d18c6965e04be3308013d7423e473ce6f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{1}.}"></span> <i><a href="/wiki/Refleksiivsus" title="Refleksiivsus">Refleksiivsus</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leqslant a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⩽</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leqslant a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89258c5d94db50210b896067219d2c7bf9106b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.558ex; height:2.176ex;" alt="{\displaystyle a\leqslant a}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f65f54b72927fc7b0c5f1d1e8d2cf1e2dc7113d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{2}.}"></span> <i><a href="/wiki/Antis%C3%BCmmeetria" class="mw-redirect" title="Antisümmeetria">Antisümmeetria</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\leqslant b)\land (b\leqslant a)\Rightarrow (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> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⩽</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\leqslant b)\land (b\leqslant a)\Rightarrow (a=b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92a6b7d90b19c80ac08ae0a60d750f22c55924a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.602ex; height:2.843ex;" alt="{\displaystyle (a\leqslant b)\land (b\leqslant a)\Rightarrow (a=b)}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c82a7809c73bfed9841e52fd104efe994a95f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{3}.}"></span> <i><a href="/wiki/Transitiivsus" title="Transitiivsus">Transitiivsus</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db5a5687f2ba931e229a9a3c402f70bb49aad61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.821ex; height:2.509ex;" alt="{\displaystyle a,b,c\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\leqslant b)\land (b\leqslant c)\Rightarrow (a\leqslant c)}"> <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> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⩽</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⩽</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\leqslant b)\land (b\leqslant c)\Rightarrow (a\leqslant c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d5d84f36208383ca7a4af4ae6c4a7a21a6a702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.388ex; height:2.843ex;" alt="{\displaystyle (a\leqslant b)\land (b\leqslant c)\Rightarrow (a\leqslant c)}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030e0ccc87b544c83cac3a228baa070fe78134b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{4}.}"></span> <i><a href="/wiki/Lineaarne_j%C3%A4rjestus" title="Lineaarne järjestus">Lineaarne järjestatus</a>.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\leqslant b)\lor (b\leqslant a)}"> <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> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⩽</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\leqslant b)\lor (b\leqslant a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb1ba525f3752c66a9026896f26fa7646ae897b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.853ex; height:2.843ex;" alt="{\displaystyle (a\leqslant b)\lor (b\leqslant a)}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{5}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{5}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a358d50c1bbe5917c52c8d3246eae599d0e74ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{5}.}"></span> <i>Liitmise ja järjestuse seos.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db5a5687f2ba931e229a9a3c402f70bb49aad61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.821ex; height:2.509ex;" alt="{\displaystyle a,b,c\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\leqslant b)\Rightarrow (a+c\leqslant b+c)}"> <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> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>⩽</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\leqslant b)\Rightarrow (a+c\leqslant b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670d970e945b0d9908ca98458a04df639e2371de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.579ex; height:2.843ex;" alt="{\displaystyle (a\leqslant b)\Rightarrow (a+c\leqslant b+c)}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{6}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{6}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64be6e055f1b22cb48519424e1a621a1220adea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.38ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{6}.}"></span><i>Korrutamise ja järjestuse seos.</i> Mis tahes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> korral</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\leqslant a)\land (0\leqslant b)\Rightarrow (0\leqslant a\cdot b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⩽</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⩽</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⩽</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0\leqslant a)\land (0\leqslant b)\Rightarrow (0\leqslant a\cdot b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65e705fd7ce8b8fed6f543a4005c48a6b469c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.541ex; height:2.843ex;" alt="{\displaystyle (0\leqslant a)\land (0\leqslant b)\Rightarrow (0\leqslant a\cdot b)}"></span> </p> </div> <div class="mw-heading mw-heading4"><h4 id="Pidevuse_aksioomid">Pidevuse aksioomid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=21" title="Muuda alaosa "Pidevuse aksioomid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=21" title="Muuda alaosa "Pidevuse aksioomid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/12px-Next.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/18px-Next.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Next.svg/24px-Next.svg.png 2x" data-file-width="160" data-file-height="160" /></span></span> <i>Pikemalt artiklis <a href="/w/index.php?title=Reaalarvude_hulga_pidevus&action=edit&redlink=1" class="new" title="Reaalarvude hulga pidevus (pole veel kirjutatud)">Reaalarvude hulga pidevus</a></i></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc2c92b02cd3ab9873a29aa373cfa4fc2ee9549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.22ex; height:2.509ex;" alt="{\displaystyle {\text{III}}_{1}.}"></span> Olgu antud mis tahes <a href="/wiki/Mittet%C3%BChi_hulk" class="mw-redirect" title="Mittetühi hulk">mittetühjad hulgad</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef692ec1b19585252716a62d8951d503037bf4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.176ex;" alt="{\displaystyle A\subset \mathbb {R} }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a11afcfbded2c6393d0185cf3e31814d8fb67b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.176ex;" alt="{\displaystyle B\subset \mathbb {R} }"></span>, nõnda et mis tahes kahe elemendi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}"></span> korral kehtib <a href="/wiki/V%C3%B5rratus" title="Võrratus">võrratus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leqslant b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⩽</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leqslant b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f994f4b416430409bb184cfa927b85392ee9d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leqslant b}"></span>, siis leidub niisugune arv <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d76d71407fc7c7e1978c8f4a7d393f243ea1c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.549ex; height:2.509ex;" alt="{\displaystyle \xi \in \mathbb {R} }"></span>, et kõigi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}"></span> korral kehtib seos</dd></dl> <div class="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leqslant \xi \leqslant b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⩽</mo> <mi>ξ</mi> <mo>⩽</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leqslant \xi \leqslant b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bad712c1f558cb7dd2a05dbc3fa2b14af460f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.454ex; height:2.509ex;" alt="{\displaystyle a\leqslant \xi \leqslant b}"></span></div> <div class="mw-heading mw-heading4"><h4 id="Kokkuvõte_ja_definitsioon"><span id="Kokkuv.C3.B5te_ja_definitsioon"></span>Kokkuvõte ja definitsioon</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=22" title="Muuda alaosa "Kokkuvõte ja definitsioon"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=22" title="Muuda alaosa "Kokkuvõte ja definitsioon" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neist aksioomidest piisab, et rangelt järeldada reaalarvude kõik teadaolevad omadused<sup id="cite_ref-INK6u_8-0" class="reference"><a href="#cite_note-INK6u-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>. </p><p>Tänapäeva algebra keeles tähendavad esimese rühma aksioomid, et hulk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on <a href="/wiki/Korpus_(matemaatika)" title="Korpus (matemaatika)">korpus</a>. Teise rühma aksioomid tähendavad, et hulk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> on <a href="/wiki/Lineaarselt_j%C3%A4rjestatud_hulk" class="mw-redirect" title="Lineaarselt järjestatud hulk">lineaarselt järjestatud hulk</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7674dbb8e30742657daf62c8fa50220eb024821e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.733ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{1}}"></span> – <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1657391deedbba9d950e55d9e33aa74514c62630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.733ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{4}}"></span>), kusjuures järjestusseos on kooskõlas korpuse struktuuriga (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2662806d21c9f829fb671a2256370549f84b321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.733ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{5}}"></span> – <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715d0854e26c790e086dbe8b710898e625dc292b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.733ex; height:2.509ex;" alt="{\displaystyle {\text{II}}_{6}}"></span>. Hulki, mis rahuldavad esimese ja teise rühma aksioome, nimetatakse <i><a href="/w/index.php?title=J%C3%A4rjestatud_korpus&action=edit&redlink=1" class="new" title="Järjestatud korpus (pole veel kirjutatud)">järjestatud korpusteks</a></i>. Lõpuks, viimane rühm, mis koosneb ühest aksioomist, väidab, et reaalarvude hulgal on <i><a href="/w/index.php?title=Pidevuse_omadus&action=edit&redlink=1" class="new" title="Pidevuse omadus (pole veel kirjutatud)">pidevuse omadus</a></i>, mida nimetatakse ka <i>täielikkuseks</i>. </p><p>Kokkuvõttes võib anda reaalarvude hulga definitsiooni: </p><p><i><b>Definitsioon.</b></i> <i>Reaalarvude hulgaks nimetatakse pidevat järjestatud korpust.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Reaalarvude_hulk_kui_maksimaalne_arhimeediline_järjestatud_korpus"><span id="Reaalarvude_hulk_kui_maksimaalne_arhimeediline_j.C3.A4rjestatud_korpus"></span>Reaalarvude hulk kui maksimaalne arhimeediline järjestatud korpus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=23" title="Muuda alaosa "Reaalarvude hulk kui maksimaalne arhimeediline järjestatud korpus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=23" title="Muuda alaosa "Reaalarvude hulk kui maksimaalne arhimeediline järjestatud korpus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On ka teisi reaalarvude aksiomaatikaid. Näiteks võib pidevuse aksioomi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc2c92b02cd3ab9873a29aa373cfa4fc2ee9549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.22ex; height:2.509ex;" alt="{\displaystyle {\text{III}}_{1}.}"></span> asemel kasutada mis tahes muud sellega samaväärset tingimust või tingimuste rühma. Näiteks <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilberti</a> pakutud aksiomaatikas on rühmade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20354ea5f11686221cb7f2442708c22593abd5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.176ex;" alt="{\displaystyle {\text{I}}}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ccaff226a580decc272cf1922e6e9f4f92b542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.679ex; height:2.176ex;" alt="{\displaystyle {\text{II}}}"></span> aksioomid sisuliselt samad mis ülaltoodud aksiomaatikas, kuid aksioomi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e011cd78f9bc28ae1ac7cde549cd0553b83e5f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.573ex; height:2.509ex;" alt="{\displaystyle {\text{III}}_{1}}"></span> asemel kasutatakse järgmist kaht tingimust: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{1}'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{1}'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b346d5a1a48dcdc9e0f0114a8b727c0f72e426ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.22ex; height:3.009ex;" alt="{\displaystyle {\text{III}}_{1}'.}"></span> <i><a href="/wiki/Archimedese_aksioom" title="Archimedese aksioom">Archimedese aksioom</a>.</i> Olgu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span><sup id="cite_ref-zuYaA_9-0" class="reference"><a href="#cite_note-zuYaA-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}"></span>. Siis elementi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> saab liidetavana korrata nii palju kordi, et tulemiks saadav summa on suurem kui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>:</dd></dl> <div class="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+a+\ldots +a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+a+\ldots +a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434fb94494fcc97e5dc788acde6f3fb38c31e739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.03ex; height:2.343ex;" alt="{\displaystyle a+a+\ldots +a>b}"></span> </p> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{2}'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{2}'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd6ab1a4b621352a38ba83ee4660e11ff2f2e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.22ex; height:3.009ex;" alt="{\displaystyle {\text{III}}_{2}'.}"></span> <i>Täielikkuse aksioom (Hilberti mõttes).</i> Süsteemi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> pole võimalik laiendada mitte ühegi süsteemini <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0533765ec036adbcb5cc15bb277b299906c49ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{*}}"></span>, nõnda et hulga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> elementide vaheliste endiste seoste säilides oleks <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0533765ec036adbcb5cc15bb277b299906c49ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{*}}"></span> korral täidetud kõik aksioomid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>I</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20354ea5f11686221cb7f2442708c22593abd5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.176ex;" alt="{\displaystyle {\text{I}}}"></span>–<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{II}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>II</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{II}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ccaff226a580decc272cf1922e6e9f4f92b542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.679ex; height:2.176ex;" alt="{\displaystyle {\text{II}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{III}}_{1}'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>III</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{III}}_{1}'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b346d5a1a48dcdc9e0f0114a8b727c0f72e426ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.22ex; height:3.009ex;" alt="{\displaystyle {\text{III}}_{1}'.}"></span>.</dd></dl> <p>Seega võib anda järgmise samaväärse definitsiooni: </p><p><i><b>Definitsioon.</b></i> <i>Reaalarvude hulk on maksimaalne arhimeediline järjestatud korpus.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Tarski_reaalarvude_aksiomaatika">Tarski reaalarvude aksiomaatika</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=24" title="Muuda alaosa "Tarski reaalarvude aksiomaatika"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=24" title="Muuda alaosa "Tarski reaalarvude aksiomaatika" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aastal <a href="/wiki/1936" title="1936">1936</a> esitas <a href="/w/index.php?title=Alfred_Tarski&action=edit&redlink=1" class="new" title="Alfred Tarski (pole veel kirjutatud)">Alfred Tarski</a> reaalarvude aksiomaatika, mis koosneb ainult 8 <a href="/wiki/Aksioom" title="Aksioom">aksioomist</a> ja 4 <a href="/wiki/Algm%C3%B5iste" title="Algmõiste">algmõistest</a>: reaalarvude <a href="/wiki/Hulk" title="Hulk">hulk</a> <b>R</b>, a <a href="/wiki/Binaarne_seos" title="Binaarne seos">binaarne seos</a> <a href="/wiki/T%C3%A4ielik_j%C3%A4rjestus" title="Täielik järjestus">täielik järjestus</a> hulgal <b>R</b>, mida tähistab <, <a href="/wiki/Binaarne_tehe" title="Binaarne tehe">binaarne tehe</a> <a href="/wiki/Liitmine" title="Liitmine">liitmine</a> hulgal <b>R</b>, mida tähistab +, ja <a href="/wiki/Konstant" title="Konstant">konstant</a> 1. </p><p>Seda aksiomaatikat kirjanduses mõnikord mainitakse, kuid üksikasju kunagi ei esitata, kuigi ta on ökonoomne ning tal on elegantsed <a href="/w/index.php?title=Metamatemaatika&action=edit&redlink=1" class="new" title="Metamatemaatika (pole veel kirjutatud)">metamatemaatilisi</a> omadusi. See aksiomaatika on vähe tuntud, võib-olla sellepärast, et ta on <a href="/w/index.php?title=Teist_j%C3%A4rku_loogika&action=edit&redlink=1" class="new" title="Teist järku loogika (pole veel kirjutatud)">teist järku</a>. Tarski aksiomaatikat võib vaadelda versioonina tavalisest reaalarvude definitsioonist ainsa <a href="/w/index.php?title=Dedekindi_t%C3%A4ielikkus&action=edit&redlink=1" class="new" title="Dedekindi täielikkus (pole veel kirjutatud)">Dedekindi mõttes täieliku</a> <a href="/w/index.php?title=J%C3%A4rjestatud_korpus&action=edit&redlink=1" class="new" title="Järjestatud korpus (pole veel kirjutatud)">järjestatud korpusena</a>; ent ta on muudetud tunduvalt lühemaks (näiteks aksioomid 4 ja 5 võtavad kokku tavalised neli <a href="/wiki/Abeli_r%C3%BChm" title="Abeli rühm">Abeli rühma</a> aksioomi). </p> <div class="mw-heading mw-heading4"><h4 id="Aksioomid">Aksioomid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=25" title="Muuda alaosa "Aksioomid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=25" title="Muuda alaosa "Aksioomid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Järjestuse aksioomid</i> (algmõisted: <b>R</b>, <): </p> <dl><dt>Aksioom 1</dt> <dd>Kui <i>x</i> < <i>y</i>, siis ei pea paika, et <i>y</i> < <i>x</i>. See tähendab, "<" on <a href="/w/index.php?title=As%C3%BCmmeetriline_seos&action=edit&redlink=1" class="new" title="Asümmeetriline seos (pole veel kirjutatud)">asümmeetriline seos</a>.</dd></dl> <dl><dt>Aksioom 2</dt> <dd>Kui <i>x</i> < <i>z</i>, siis leidub <i>y</i>, nii et <i>x</i> < <i>y</i> ja <i>y</i> < <i>z</i>. Teiste sõnadega, "<" on <a href="/w/index.php?title=Tihe_j%C3%A4rjestus&action=edit&redlink=1" class="new" title="Tihe järjestus (pole veel kirjutatud)">tihe</a> hulgal <b>R</b>.</dd></dl> <dl><dt>Aksioom 3</dt> <dd>"<" on <a href="/w/index.php?title=Dedekindi_t%C3%A4ielikkus&action=edit&redlink=1" class="new" title="Dedekindi täielikkus (pole veel kirjutatud)">Dedekindi mõttes täielik</a>. Formaalsemalt, kõikide <i>X</i>, <i>Y</i> ⊆ <b>R</b> korral, kui kõikide <i>x</i> ∈ <i>X</i> ja <i>y</i> ∈ <i>Y</i> korral <i>x</i> < <i>y</i>, siis leidub <i>z</i>, nii et kõikide <i>x</i> ∈ <i>X</i> ja <i>y</i> ∈ <i>Y</i> korral <i>x</i> ≤ <i>z</i> ja <i>z</i> ≤ <i>y</i>. Siin on <i>u</i> ≤ <i>v</i> lühend, mis tähendab "<i>u</i> < <i>v</i> või <i>u</i> = <i>v</i>".</dd></dl> <p><i>Liitmise aksioomid</i> (algmõisted: <b>R</b>, <, +): </p> <dl><dt>Aksioom 4</dt> <dd><i>x</i> + (<i>y</i> + <i>z</i>) = (<i>x</i> + <i>z</i>) + <i>y</i>.</dd></dl> <dl><dt>Aksioom 5</dt> <dd>Kõikide <i>x</i>, <i>y</i> korral leidub <i>z</i>, nõnda ett <i>x</i> + <i>z</i> = <i>y</i>.</dd></dl> <dl><dt>Aksioom 6</dt> <dd>Kui <i>x</i> + <i>y</i> < <i>z</i> + <i>w</i>, siis <i>x</i> < <i>z</i> või <i>y</i> < <i>w</i>.</dd></dl> <p><i>Ühikelemendi aksioomid</i> (algmõisted: <b>R</b>, <, +, 1): </p> <dl><dt>Aksioom 7</dt> <dd>1 ∈ <b>R</b>.</dd></dl> <dl><dt>Aksioom 8</dt> <dd>1 < 1 + 1.</dd></dl> <p>Nendest aksioomidest järeldub, et <b>R</b> on <a href="/w/index.php?title=Lineaarselt_j%C3%A4rjestatud_r%C3%BChm&action=edit&redlink=1" class="new" title="Lineaarselt järjestatud rühm (pole veel kirjutatud)">lineaarselt järjestatud</a> <a href="/wiki/Abeli_r%C3%BChm" title="Abeli rühm">Abeli rühm</a> liitmise suhtes koos märgitud elemendiga 1. <b>R</b> on ka <a href="/w/index.php?title=Dedekindi_t%C3%A4ielikkus&action=edit&redlink=1" class="new" title="Dedekindi täielikkus (pole veel kirjutatud)">Dedekindi mõttes täielik</a> ja <a href="/w/index.php?title=Jagatav_r%C3%BChm&action=edit&redlink=1" class="new" title="Jagatav rühm (pole veel kirjutatud)">jagatav</a>. </p><p>See aksiomaatika ei anna <a href="/wiki/Esimest_j%C3%A4rku_loogika" class="mw-redirect" title="Esimest järku loogika">esimest järku teooriat</a>, sest aksioom 3 on formuleeritud kahe <a href="/wiki/%C3%9Cldisuskvantor" title="Üldisuskvantor">üldisuskvantoriga</a> üle hulga <b>R</b> kõigi võimalike alamhulkade. </p> <div class="mw-heading mw-heading4"><h4 id="Kuidas_need_aksioomid_toovad_kaasa_korpuse">Kuidas need aksioomid toovad kaasa korpuse</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=26" title="Muuda alaosa "Kuidas need aksioomid toovad kaasa korpuse"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=26" title="Muuda alaosa "Kuidas need aksioomid toovad kaasa korpuse" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tarski visandas (mittetriviaalse) tõestuse, kuidas nendest aksioomidest ja algmõistetest tuleneb <a href="/wiki/Binaarne_tehe" title="Binaarne tehe">binaarne tehe</a> <a href="/wiki/Korrutamine" title="Korrutamine">korrutamine</a>, millel on oodatavad omadused, nõnda et <b>R</b> on täielik <a href="/w/index.php?title=J%C3%A4rjestatud_korpus&action=edit&redlink=1" class="new" title="Järjestatud korpus (pole veel kirjutatud)">järjestatud korpus</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Kirjandus">Kirjandus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=27" title="Muuda alaosa "Kirjandus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=27" title="Muuda alaosa "Kirjandus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Alfred_Tarski&action=edit&redlink=1" class="new" title="Alfred Tarski (pole veel kirjutatud)">Alfred Tarski</a>, <a href="/w/index.php?title=Jan_Tarski&action=edit&redlink=1" class="new" title="Jan Tarski (pole veel kirjutatud)">Jan Tarski</a> (1994). <i>Introduction to Logic and to the Methodology of Deductive Sciences</i>, 4. trükk, USA: OUP. <a href="/wiki/Eri:Raamatuotsimine/9780195044720" class="internal mw-magiclink-isbn">ISBN 978-0-19-504472-0</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Lõpmatu_kümnendarendus"><span id="L.C3.B5pmatu_k.C3.BCmnendarendus"></span>Lõpmatu kümnendarendus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=28" title="Muuda alaosa "Lõpmatu kümnendarendus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=28" title="Muuda alaosa "Lõpmatu kümnendarendus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Iga reaalarvu saab esitada <a href="/wiki/K%C3%BCmnendmurd" title="Kümnendmurd">kümnendmurdude</a> abil <a href="/w/index.php?title=L%C3%B5pmatu_k%C3%BCmnendarendus&action=edit&redlink=1" class="new" title="Lõpmatu kümnendarendus (pole veel kirjutatud)">lõpmatu kümnendarenduse</a> kujul; näiteks </p> <ul><li>1 = 1,0000000... või 0,99999999... <ul><li>½ = 0,5000000... või 0,49999999...</li> <li>-1/3 = -0,3333333...</li> <li>8/7 = 1,142857142857142857...</li> <li><i>e</i> = 2,718281828459045235...</li> <li>'L<i> = 0,110001000000000000000001000...</i></li></ul></li></ul> <p>Viimased kaks (<a href="/wiki/Napieri_arv" class="mw-redirect" title="Napieri arv">Napieri arv</a> ja <a href="/w/index.php?title=Liouville%27i_arv&action=edit&redlink=1" class="new" title="Liouville'i arv (pole veel kirjutatud)">Liouville'i arv</a>) on <a href="/w/index.php?title=Mitteperioodiline_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Mitteperioodiline kümnendmurd (pole veel kirjutatud)">mitteperioodilised kümnendmurrud</a> ning seetõttu <a href="/wiki/Irratsionaalarv" class="mw-redirect" title="Irratsionaalarv">irratsionaalarvud</a>, teised aga on <a href="/w/index.php?title=Perioodiline_k%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Perioodiline kümnendmurd (pole veel kirjutatud)">perioodilised kümnendmurrud</a> ning seega <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvud</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Reaalarvude_korpus">Reaalarvude korpus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=29" title="Muuda alaosa "Reaalarvude korpus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=29" title="Muuda alaosa "Reaalarvude korpus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Reaalarvude_hulk" title="Reaalarvude hulk">Reaalarvude hulk</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> moodustab oma <a href="/wiki/Aritmeetiline_tehe" class="mw-redirect" title="Aritmeetiline tehe">aritmeetiliste tehetega</a> "+" ja "·" <a href="/wiki/Korpus_(matemaatika)" title="Korpus (matemaatika)">korpuse</a> (<a href="/wiki/Reaalarvude_korpus" class="mw-redirect" title="Reaalarvude korpus">reaalarvude korpuse</a>), mis on <a href="/wiki/Kompleksarvude_korpus" class="mw-redirect" title="Kompleksarvude korpus">kompleksarvude korpuse</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span> <a href="/w/index.php?title=Alamkorpus&action=edit&redlink=1" class="new" title="Alamkorpus (pole veel kirjutatud)">alamkorpus</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ajalugu">Ajalugu</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=30" title="Muuda alaosa "Ajalugu"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=30" title="Muuda alaosa "Ajalugu" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Reaalarvu mõiste tekkis <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvu</a> mõiste laiendamisel. Vajaduse selleks tingis vajadus väljendada mis tahes <a href="/wiki/Suurus" title="Suurus">suuruse</a> <a href="/wiki/V%C3%A4%C3%A4rtus" title="Väärtus">väärtust</a> <a href="/wiki/Arv" title="Arv">arvuna</a> ning püüd laiendada tehete rakendatavust (<a href="/wiki/Juurimine" title="Juurimine">juurimine</a>, <a href="/w/index.php?title=Logaritmimine&action=edit&redlink=1" class="new" title="Logaritmimine (pole veel kirjutatud)">logaritmimine</a>, <a href="/w/index.php?title=Algebraline_v%C3%B5rrand&action=edit&redlink=1" class="new" title="Algebraline võrrand (pole veel kirjutatud)">algebraliste võrrandite</a> lahendamine). </p> <div class="mw-heading mw-heading3"><h3 id="Naiivne_reaalarvude_teooria">Naiivne reaalarvude teooria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=31" title="Muuda alaosa "Naiivne reaalarvude teooria"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=31" title="Muuda alaosa "Naiivne reaalarvude teooria" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Esimeses arenenud arvusüsteemis, mis konstrueeriti <a href="/w/index.php?title=Vanakreeka_matemaatika&action=edit&redlink=1" class="new" title="Vanakreeka matemaatika (pole veel kirjutatud)">vanakreeka matemaatikas</a>, olid ainult <a href="/wiki/Naturaalarv" title="Naturaalarv">naturaalarvud</a> ja nendes suhted (<a href="/w/index.php?title=Proportsioon&action=edit&redlink=1" class="new" title="Proportsioon (pole veel kirjutatud)">proportsioonid</a>), tänapäeva mõistes <a href="/wiki/Ratsionaalarv" title="Ratsionaalarv">ratsionaalarvud</a>. Ent peagi selgus, et <a href="/wiki/Geomeetria" title="Geomeetria">geomeetrias</a> ja <a href="/wiki/Astronoomia" title="Astronoomia">astronoomias</a> sellest ei piisa: näiteks <a href="/wiki/Ruut" title="Ruut">ruudu</a> <a href="/wiki/Diagonaal" title="Diagonaal">diagonaali</a> ja külje suhe ei ole esitatav ei naturaal- ega ratsionaalarvuna. </p><p>Olukorra lahendamiseks võttis <a href="/wiki/Eudoxos" title="Eudoxos">Eudoxos</a> kasutusele lisaks <a href="/wiki/Arv" title="Arv">arvu</a> mõistele geomeetrilise <a href="/wiki/Suurus" title="Suurus">suuruse</a> (lõigu pikkuse, pindala või ruumala) mõiste. Eudoxose teooria on meieni jõudnud <a href="/wiki/Eukleides" title="Eukleides">Eukleidese</a> esituses ("<a href="/wiki/Elemendid_(Eukleides)" title="Elemendid (Eukleides)">Elemendid</a>", 5. raamat). Eudoxose teooria on oma olemuselt reaalarvude geomeetriline mudel. Tänapäeva seisukohast on arv niisuguse lähenemise puhul kahe homogeense suuruse (näiteks uuritava suuruse ja ühiksuuruse; ühiksuurused võisid olla omavahel <a href="/w/index.php?title=%C3%9Chism%C3%B5%C3%B5duta_suurused&action=edit&redlink=1" class="new" title="Ühismõõduta suurused (pole veel kirjutatud)">ühismõõduta</a>) suhe. Eudoxos jäi siiski ustavaks varasemale traditsioonile: ta ei vaadelnud sellist suhet arvuna; sellepärast tõestatakse "Elementides" paljud teoreemid arvude omaduste kohta uuesti suuruste jaoks. Dedekindi klassikaline reaalarvude konstruktsioon on oma põhimõtetelt väga lähedane Eudoxose esitusele. Ent Eudoxose mudel on mitmes suhtes ebatäielik: näiteks ei sisalda ta <a href="/w/index.php?title=Pidevuse_aksioom&action=edit&redlink=1" class="new" title="Pidevuse aksioom (pole veel kirjutatud)">pidevuse aksioomi</a>, puudub suuruste või nende suhete aritmeetiliste tehete üldine teooria, puuduvad negatiivsed arvud. </p><p>Olukord hakkas muutuma esimestel sajanditel pKr. Juba <a href="/wiki/Diophantos" title="Diophantos">Diophantos</a> Aleksandriast vaatles erinevalt varasemast traditsioonist murde samamoodi nagu naturaalarve, ning oma "Aritmeetika" 4. raamatus ta isegi kirjutas ühe tulemuse kohta: "Arv osutub mitteratsionaalseks." Pärast antiikteaduse lõppu nihkusid esiplaanile India ja islami matemaatikud, kes pidasid iga mõõtmistulemust arvuks. Tasapisi said sellised vaated valdavaks ka keskaegses Euroopas, kus esialgu eristati ratsionaalarve ja irratsionaalarve (sõna-sõnalt "mittemõistuspäraseid" arve), mida nimetati ka kujuteldavateks, absurdseteks, kurtideks jne. Irratsionaalarvude saamine täieõiguslikeks on seotud <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevini</a> töödega. Stevin kuulutas: "Jõuame järeldusele, et pole olemas mingeid absurdseid, irratsionaalseid, ebaõigeid, seletamatuid ega kurte arve, vaid arvude seas valitseb niisugune täius ja kooskõla, et meil tuleb ööd ja päevad mõtiskleda nende hämmastava lõpetatuse üle." Tema legaliseeris teatud reservatsioonidega ka <a href="/w/index.php?title=Negatiivsed_arvud&action=edit&redlink=1" class="new" title="Negatiivsed arvud (pole veel kirjutatud)">negatiivsed arvud</a> ning töötas välja ka <a href="/wiki/K%C3%BCmnendmurd" title="Kümnendmurd">kümnendmurdude</a> teooria ja sümboolika; sellest ajast hakkasid need välja tõrjuma ebamugavaid <a href="/w/index.php?title=Kuuek%C3%BCmnendmurd&action=edit&redlink=1" class="new" title="Kuuekümnendmurd (pole veel kirjutatud)">kuuekümnendmurde</a>. </p><p>Saja aasta pärast andis <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> oma "<a href="/w/index.php?title=Universaalne_aritmeetika&action=edit&redlink=1" class="new" title="Universaalne aritmeetika (pole veel kirjutatud)">Universaalses aritmeetikas</a>" (<a href="/wiki/1707" title="1707">1707</a>) (reaal)arvu klassikalise definitsiooni mõõtmistulemusena ühiketaloni suhtes: "Arv ei ole mitte niivõrd mitme ühiku kogum kui mingi suuruse abstraktne suhe teisesse temaga homogeensesse, mis on ühikuks võetud." Kaua aega peeti seda rakenduslikku definitsiooni piisavaks, nii et reaalarvude ja reaalarvuliste funktsioonide praktika seisukohast tähtsaid omadusi ei tõestatud, vaid neid peeti intuitiivselt ilmseteks (geomeetrilistel või kinemaatilistel kaalutlustel). Näiteks peeti ilmseks asjaolu, et pidev kõver, mille punktid asetsevad eri pooltel teatud sirgest, lõikab seda sirget. Puudus ka pidevuse range definitsioon. Sellepärast oli paljudes definitsioonides vigu või ebamääraseid või liiga laiu formuleeringuid. </p><p>Isegi veel siis, kui <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a> oli välja töötanud <a href="/wiki/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs">matemaatilise analüüsi</a> range aluse, olukord ei muutunud, sest puudus reaalarvude teooria, millele matemaatiline analüüs oleks saanud tugineda. Sellepärast tegi Cauchy vigu, toetudes intuitsioonile seal, kus see viis vääradele järeldustele: ta eeldas näiteks et <a href="/w/index.php?title=Pidev_funktsioon&action=edit&redlink=1" class="new" title="Pidev funktsioon (pole veel kirjutatud)">pidevate funktsioonide</a> <a href="/wiki/Rida_(matemaatika)" title="Rida (matemaatika)">rea</a> summa on alati pidev. </p> <div class="mw-heading mw-heading3"><h3 id="Range_teooria_loomine">Range teooria loomine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=32" title="Muuda alaosa "Range teooria loomine"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=32" title="Muuda alaosa "Range teooria loomine" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Esimese katse täita lünk matemaatika alustes tegi <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> oma artiklis "Puhtanalüütiline tõestus teoreemile, et mis tahes kahe väärtuse vahel, mis annavad vastupidise märgiga tulemid, on vähemalt üks võrrandi reaalarvuline juur" (<a href="/wiki/1817" title="1817">1817</a>). Selles teedrajavas töös ei ole veel reaalarvude terviklikku süsteemi, kuid juba esitatakse pidevuse tänapäevane definitsioon ja näidatakse, et sel alusel saab pealkirjas mainitud teoreemi rangelt tõestada. Hilisemas töös "<a href="/w/index.php?title=L%C3%B5pmatuse_paradoksid&action=edit&redlink=1" class="new" title="Lõpmatuse paradoksid (pole veel kirjutatud)">Lõpmatuse paradoksid</a>" esitas Bolzano reaalarvude üldise teooria visandi, mis on ideede poolest lähedane <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantori</a> <a href="/wiki/Hulgateooria" title="Hulgateooria">hulgateooriale</a>, kuid see tema töö jäi autori eluajal avaldamata ja ilmus alles <a href="/wiki/1851" title="1851">1851</a>. Bolzano vaated olid ajast tunduvalt ees ega äratanud matemaatilises üldsuses tähelepanu. </p><p>Reaalarvu ranged teooriad rajasid 19. sajandi lõpus <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, <a href="/w/index.php?title=Richard_Dedekind&action=edit&redlink=1" class="new" title="Richard Dedekind (pole veel kirjutatud)">Richard Dedekind</a> ja <a href="/w/index.php?title=Karl_Weierstrass&action=edit&redlink=1" class="new" title="Karl Weierstrass (pole veel kirjutatud)">Karl Weierstrass</a>, samuti <a href="/w/index.php?title=Eduard_Heine&action=edit&redlink=1" class="new" title="Eduard Heine (pole veel kirjutatud)">Eduard Heine</a> ja <a href="/w/index.php?title=Charles_M%C3%A9ray&action=edit&redlink=1" class="new" title="Charles Méray (pole veel kirjutatud)">Charles Méray</a>. Weierstrass, Dedekind ja Cantor esitasid erinevad, kuid samaväärsed reaalarvude konstruktsioonid. </p> <div class="mw-heading mw-heading2"><h2 id="Vaata_ka">Vaata ka</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=33" title="Muuda alaosa "Vaata ka"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=33" title="Muuda alaosa "Vaata ka" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kompleksarv" title="Kompleksarv">Kompleksarv</a></li> <li><a href="/w/index.php?title=Aleksandrovi_sirge&action=edit&redlink=1" class="new" title="Aleksandrovi sirge (pole veel kirjutatud)">Aleksandrovi sirge</a></li> <li><a href="/w/index.php?title=Suslini_sirge&action=edit&redlink=1" class="new" title="Suslini sirge (pole veel kirjutatud)">Suslini sirge</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Viited">Viited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reaalarv&veaction=edit&section=34" title="Muuda alaosa "Viited"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Reaalarv&action=edit&section=34" title="Muuda alaosa "Viited" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Кантор-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Кантор_1-0">1,0</a></sup> <sup><a href="#cite_ref-Кантор_1-1">1,1</a></sup></span> <span class="reference-text">Кантор Г.<i>Труды по теории множеств</i>, toim А. Н. Колмогоров, Ф. А. Медведев, А. П. Юшкевич, Moskva, НАУКА, 1985, <i>Классики науки</i>, lk 9–10}}</span> </li> <li id="cite_note-Арнольд-2"><span class="mw-cite-backlink"><a href="#cite_ref-Арнольд_2-0">↑</a></span> <span class="reference-text">Арнольд И. В. <i>Теоретическая арифметика</i>, lk 277</span> </li> <li id="cite_note-0TL20-3"><span class="mw-cite-backlink"><a href="#cite_ref-0TL20_3-0">↑</a></span> <span class="reference-text">Tegelikult tegi Cauchy kindlaks <a href="/wiki/Rida_(matemaatika)" title="Rida (matemaatika)">rea</a> <a href="/w/index.php?title=Koonduv_rida&action=edit&redlink=1" class="new" title="Koonduv rida (pole veel kirjutatud)">koonduvuse</a> tingimuse, mis samuti kannab tema nime, kuid kumbki nendest kriteeriumidest järeldub hõlpsasti teisest.</span> </li> <li id="cite_note-Пути-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Пути_4-0">4,0</a></sup> <sup><a href="#cite_ref-Пути_4-1">4,1</a></sup></span> <span class="reference-text">Даан-Дальмедико А., Пейффер Ж. <i>Пути и лабиринты. Очерки по истории математики</i>, lk 287–289.</span> </li> <li id="cite_note-Nsa5G-5"><span class="mw-cite-backlink"><a href="#cite_ref-Nsa5G_5-0">↑</a></span> <span class="reference-text">Mõnikord vaadeldakse selleks, et vastavus reaalarvude hulga ja lõpmatute kümnendmurdude hulga vahel oleks üksühene, mitte kõiki, vaid ainult <i>lubatavaid</i> kümnendmurde, mõeldes nende all kõiki neid, millel ei ole ainult üheksast koosnevat perioodi ja mis ei ole murd<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0,00\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>00</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0,00\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd18b8b39cb5d2d0419c07425044883ddf6c0bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.44ex; height:2.509ex;" alt="{\displaystyle -0,00\ldots }"></span>.</span> </li> <li id="cite_note-f5k4l-6"><span class="mw-cite-backlink"><a href="#cite_ref-f5k4l_6-0">↑</a></span> <span class="reference-text">Рыбников К. А. <i>История математики</i>, kd 2, lk 197.</span> </li> <li id="cite_note-QMGcp-7"><span class="mw-cite-backlink"><a href="#cite_ref-QMGcp_7-0">↑</a></span> <span class="reference-text">Venekeelne tõlge "<a rel="nofollow" class="external text" href="http://www.mathesis.ru/book/dedekind4">Непрерывность и иррациональные числа</a>"</span> </li> <li id="cite_note-INK6u-8"><span class="mw-cite-backlink"><a href="#cite_ref-INK6u_8-0">↑</a></span> <span class="reference-text">Кудрявцев Л. Д. <i>Курс математического анализа</i>, kd 1.</span> </li> <li id="cite_note-zuYaA-9"><span class="mw-cite-backlink"><a href="#cite_ref-zuYaA_9-0">↑</a></span> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a>0)\;{\overset {\text{def}}{\Leftrightarrow }}\;(a\geqslant 0)\land (a\neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⇔</mo> <mtext>def</mtext> </mover> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⩾</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a>0)\;{\overset {\text{def}}{\Leftrightarrow }}\;(a\geqslant 0)\land (a\neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3f0fc24fcb36b7a1c9f3ef4e739a5c7df9111d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.097ex; height:4.176ex;" alt="{\displaystyle (a>0)\;{\overset {\text{def}}{\Leftrightarrow }}\;(a\geqslant 0)\land (a\neq 0)}"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Pärit leheküljelt "<a dir="ltr" href="https://et.wikipedia.org/w/index.php?title=Reaalarv&oldid=6601291">https://et.wikipedia.org/w/index.php?title=Reaalarv&oldid=6601291</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Eri:Kategooriad" title="Eri:Kategooriad">Kategooria</a>: <ul><li><a href="/wiki/Kategooria:Arvud" title="Kategooria:Arvud">Arvud</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Peidetud kategooria: <ul><li><a href="/wiki/Kategooria:ISBN-v%C3%B5lulingiga_lehek%C3%BCljed" title="Kategooria:ISBN-võlulingiga leheküljed">ISBN-võlulingiga leheküljed</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Selle lehekülje viimane muutmine: 09:58, 10. märts 2024.</li> <li id="footer-info-copyright">Tekst on kasutatav vastavalt Creative Commonsi litsentsile "<a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.et">Autorile viitamine + jagamine samadel tingimustel</a>"; sellele võivad lisanduda täiendavad tingimused. 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