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Matematikaren arloak azpiatal </button> <ul id="toc-Matematikaren_arloak-sublist" class="vector-toc-list"> <li id="toc-Zenbakien_teoria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zenbakien_teoria"> <div class="vector-toc-text"> 2.1 Zenbakien teoria </div> </a> <ul id="toc-Zenbakien_teoria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometria"> <div class="vector-toc-text"> 2.2 Geometria </div> </a> <ul id="toc-Geometria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aljebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aljebra"> <div class="vector-toc-text"> 2.3 Aljebra </div> </a> <ul id="toc-Aljebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kalkulua_eta_analisia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kalkulua_eta_analisia"> <div class="vector-toc-text"> 2.4 Kalkulua eta analisia </div> </a> <ul id="toc-Kalkulua_eta_analisia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matematika_diskretua" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matematika_diskretua"> <div class="vector-toc-text"> 2.5 Matematika diskretua </div> </a> <ul id="toc-Matematika_diskretua-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logika_matematikoa_eta_multzo-teoria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logika_matematikoa_eta_multzo-teoria"> <div class="vector-toc-text"> 2.6 Logika matematikoa eta multzo-teoria </div> </a> <ul id="toc-Logika_matematikoa_eta_multzo-teoria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estatistika_eta_erabakiak_hartzearen_zientziak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estatistika_eta_erabakiak_hartzearen_zientziak"> <div class="vector-toc-text"> 2.7 Estatistika eta erabakiak hartzearen zientziak </div> </a> <ul id="toc-Estatistika_eta_erabakiak_hartzearen_zientziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matematika_konputazionala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matematika_konputazionala"> <div class="vector-toc-text"> 2.8 Matematika konputazionala </div> </a> <ul id="toc-Matematika_konputazionala-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arlo_formal,_metodologiko_eta_estetikoak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arlo_formal,_metodologiko_eta_estetikoak"> <div class="vector-toc-text"> 3 Arlo formal, metodologiko eta estetikoak </div> </a> <button aria-controls="toc-Arlo_formal,_metodologiko_eta_estetikoak-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Erakutsi/ezkutatu Arlo formal, metodologiko eta estetikoak azpiatal </button> <ul id="toc-Arlo_formal,_metodologiko_eta_estetikoak-sublist" class="vector-toc-list"> <li id="toc-Notazioa,_hizkuntza_eta_arauak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notazioa,_hizkuntza_eta_arauak"> <div class="vector-toc-text"> 3.1 Notazioa, hizkuntza eta arauak </div> </a> <ul id="toc-Notazioa,_hizkuntza_eta_arauak-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Matematikari_ospetsuak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matematikari_ospetsuak"> <div class="vector-toc-text"> 4 Matematikari ospetsuak </div> </a> <button aria-controls="toc-Matematikari_ospetsuak-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Erakutsi/ezkutatu Matematikari ospetsuak azpiatal </button> <ul id="toc-Matematikari_ospetsuak-sublist" class="vector-toc-list"> <li id="toc-Grezia_Klasikoa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grezia_Klasikoa"> <div class="vector-toc-text"> 4.1 Grezia Klasikoa </div> </a> <ul id="toc-Grezia_Klasikoa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Erdi_Aroa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Erdi_Aroa"> <div class="vector-toc-text"> 4.2 Erdi Aroa </div> </a> <ul id="toc-Erdi_Aroa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aro_Berria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aro_Berria"> <div class="vector-toc-text"> 4.3 Aro Berria </div> </a> <ul id="toc-Aro_Berria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aro_Garaikidea" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aro_Garaikidea"> <div class="vector-toc-text"> 4.4 Aro Garaikidea </div> </a> <ul id="toc-Aro_Garaikidea-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Erreferentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erreferentziak"> <div class="vector-toc-text"> 5 Erreferentziak </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> 6 Bibliografia </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> 7 Kanpo estekak </div> </a> <ul id="toc-Kanpo_estekak-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="Edukiak" 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="Eduki taularen ikusgarritasuna aldatu" > <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" > Eduki taularen ikusgarritasuna aldatu </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">Matematika</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="Joan beste hizkuntza batean idatzitako artikulu batera. 254 hizkuntzatan eskuragarri." > <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-254" aria-hidden="true" > 254 hizkuntza </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/Wiskunde" title="Wiskunde – afrikaansa" lang="af" hreflang="af" data-title="Wiskunde" data-language-autonym="Afrikaans" data-language-local-name="afrikaansa" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mathematik" title="Mathematik – Suitzako alemana" lang="gsw" hreflang="gsw" data-title="Mathematik" data-language-autonym="Alemannisch" data-language-local-name="Suitzako alemana" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B5%E1%88%9D%E1%88%85%E1%88%AD%E1%89%B0_%E1%88%82%E1%88%B3%E1%89%A5" title="ትምህርተ ሂሳብ – amharera" lang="am" hreflang="am" data-title="ትምህርተ ሂሳብ" data-language-autonym="አማርኛ" data-language-local-name="amharera" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Matematicas" title="Matematicas – aragoiera" lang="an" hreflang="an" data-title="Matematicas" data-language-autonym="Aragonés" data-language-local-name="aragoiera" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/R%C4%ABmcr%C3%A6ft" title="Rīmcræft – Old English" lang="ang" hreflang="ang" data-title="Rīmcræft" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – angikera" lang="anp" hreflang="anp" data-title="गणित" data-language-autonym="अंगिका" data-language-local-name="angikera" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="رياضيات – arabiera" lang="ar" hreflang="ar" data-title="رياضيات" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D9%85%D8%A7%D8%B7" title="لماط – Moroccan Arabic" lang="ary" hreflang="ary" data-title="لماط" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="رياضيات – Egyptian Arabic" lang="arz" hreflang="arz" data-title="رياضيات" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – assamera" lang="as" hreflang="as" data-title="গণিত" data-language-autonym="অসমীয়া" data-language-local-name="assamera" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Matem%C3%A1tiques" title="Matemátiques – asturiera" lang="ast" hreflang="ast" data-title="Matemátiques" data-language-autonym="Asturianu" data-language-local-name="asturiera" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-avk mw-list-item"><a href="https://avk.wikipedia.org/wiki/Solokseropa" title="Solokseropa – Kotava" lang="avk" hreflang="avk" data-title="Solokseropa" data-language-autonym="Kotava" data-language-local-name="Kotava" class="interlanguage-link-target">Kotava</a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – awadhiera" lang="awa" hreflang="awa" data-title="गणित" data-language-autonym="अवधी" data-language-local-name="awadhiera" class="interlanguage-link-target">अवधी</a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jakhu" title="Jakhu – aimara" lang="ay" hreflang="ay" data-title="Jakhu" data-language-autonym="Aymar aru" data-language-local-name="aimara" class="interlanguage-link-target">Aymar aru</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyaziyyat" title="Riyaziyyat – azerbaijanera" lang="az" hreflang="az" data-title="Riyaziyyat" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="ریاضیات – South Azerbaijani" lang="azb" hreflang="azb" data-title="ریاضیات" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – baxkirera" lang="ba" hreflang="ba" data-title="Математика" data-language-autonym="Башҡортса" data-language-local-name="baxkirera" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Mathematik" title="Mathematik – Bavarian" lang="bar" hreflang="bar" data-title="Mathematik" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Matemat%C4%97ka" title="Matematėka – Samogitian" lang="sgs" hreflang="sgs" data-title="Matematėka" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Matematika" title="Matematika – Central Bikol" lang="bcl" hreflang="bcl" data-title="Matematika" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – bielorrusiera" lang="be" hreflang="be" data-title="Матэматыка" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматыка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – bulgariera" lang="bg" hreflang="bg" data-title="Математика" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Bhojpuri" lang="bh" hreflang="bh" data-title="गणित" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bi mw-list-item"><a href="https://bi.wikipedia.org/wiki/Matematikis" title="Matematikis – bislama" lang="bi" hreflang="bi" data-title="Matematikis" data-language-autonym="Bislama" data-language-local-name="bislama" class="interlanguage-link-target">Bislama</a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Matamatika" title="Matamatika – Banjar" lang="bjn" hreflang="bjn" data-title="Matamatika" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target">Banjar</a></li><li class="interlanguage-link interwiki-blk mw-list-item"><a href="https://blk.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သင်္ချာ – Pa'O" lang="blk" hreflang="blk" data-title="သင်္ချာ" data-language-autonym="ပအိုဝ်ႏဘာႏသာႏ" data-language-local-name="Pa'O" class="interlanguage-link-target">ပအိုဝ်ႏဘာႏသာႏ</a></li><li class="interlanguage-link interwiki-bm mw-list-item"><a href="https://bm.wikipedia.org/wiki/Matematiki" title="Matematiki – bambarera" lang="bm" hreflang="bm" data-title="Matematiki" data-language-autonym="Bamanankan" data-language-local-name="bambarera" class="interlanguage-link-target">Bamanankan</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – bengalera" lang="bn" hreflang="bn" data-title="গণিত" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%82" title="རྩིས་རིག – tibetera" lang="bo" hreflang="bo" data-title="རྩིས་རིག" data-language-autonym="བོད་ཡིག" data-language-local-name="tibetera" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bpy mw-list-item"><a href="https://bpy.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – Bishnupriya" lang="bpy" hreflang="bpy" data-title="গণিত" data-language-autonym="বিষ্ণুপ্রিয়া মণিপুরী" data-language-local-name="Bishnupriya" class="interlanguage-link-target">বিষ্ণুপ্রিয়া মণিপুরী</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Matematik" title="Matematik – bretoiera" lang="br" hreflang="br" data-title="Matematik" data-language-autonym="Brezhoneg" data-language-local-name="bretoiera" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matematika" title="Matematika – bosniera" lang="bs" hreflang="bs" data-title="Matematika" data-language-autonym="Bosanski" data-language-local-name="bosniera" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-bug mw-list-item"><a href="https://bug.wikipedia.org/wiki/%E1%A8%86%E1%A8%88%E1%A8%9B%E1%A8%86%E1%A8%88%E1%A8%97%E1%A8%80" title="ᨆᨈᨛᨆᨈᨗᨀ – buginera" lang="bug" hreflang="bug" data-title="ᨆᨈᨛᨆᨈᨗᨀ" data-language-autonym="Basa Ugi" data-language-local-name="buginera" class="interlanguage-link-target">Basa Ugi</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D0%B3%D0%BE%D0%B9_%D1%83%D1%85%D0%B0%D0%B0%D0%BD" title="Тоогой ухаан – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Тоогой ухаан" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matem%C3%A0tiques" title="Matemàtiques – katalana" lang="ca" hreflang="ca" data-title="Matemàtiques" data-language-autonym="Català" data-language-local-name="katalana" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Matematica" title="Matematica – Chavacano" lang="cbk" hreflang="cbk" data-title="Matematica" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target">Chavacano de Zamboanga</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/S%C3%B3-h%C5%8Fk" title="Só-hŏk – Mindong" lang="cdo" hreflang="cdo" data-title="Só-hŏk" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – txetxenera" lang="ce" hreflang="ce" data-title="Математика" data-language-autonym="Нохчийн" data-language-local-name="txetxenera" class="interlanguage-link-target">Нохчийн</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Isip" title="Isip – cebuanoera" lang="ceb" hreflang="ceb" data-title="Isip" data-language-autonym="Cebuano" data-language-local-name="cebuanoera" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-ch mw-list-item"><a href="https://ch.wikipedia.org/wiki/Matematika" title="Matematika – txamorroera" lang="ch" hreflang="ch" data-title="Matematika" data-language-autonym="Chamoru" data-language-local-name="txamorroera" class="interlanguage-link-target">Chamoru</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9" title="ماتماتیک – erdialdeko kurduera" lang="ckb" hreflang="ckb" data-title="ماتماتیک" data-language-autonym="کوردی" data-language-local-name="erdialdeko kurduera" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Matematica" title="Matematica – korsikera" lang="co" hreflang="co" data-title="Matematica" data-language-autonym="Corsu" data-language-local-name="korsikera" class="interlanguage-link-target">Corsu</a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Riyaziyat" title="Riyaziyat – Crimean Tatar" lang="crh" hreflang="crh" data-title="Riyaziyat" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target">Qırımtatarca</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematika" title="Matematika – txekiera" lang="cs" hreflang="cs" data-title="Matematika" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Matematika" title="Matematika – Kashubian" lang="csb" hreflang="csb" data-title="Matematika" data-language-autonym="Kaszëbsczi" data-language-local-name="Kashubian" class="interlanguage-link-target">Kaszëbsczi</a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%9C%D0%B0%D1%B3%D0%B8%D0%BC%D0%B0%D1%82%D1%97%D0%BA%D0%B0" title="Маѳиматїка – elizako eslaviera" lang="cu" hreflang="cu" data-title="Маѳиматїка" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="elizako eslaviera" class="interlanguage-link-target">Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – txuvaxera" lang="cv" hreflang="cv" data-title="Математика" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mathemateg" title="Mathemateg – galesa" lang="cy" hreflang="cy" data-title="Mathemateg" data-language-autonym="Cymraeg" data-language-local-name="galesa" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://da.wikipedia.org/wiki/Matematik" title="Matematik – daniera" lang="da" hreflang="da" data-title="Matematik" data-language-autonym="Dansk" data-language-local-name="daniera" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-dag mw-list-item"><a href="https://dag.wikipedia.org/wiki/Laasabu_malibu" title="Laasabu malibu – Dagbani" lang="dag" hreflang="dag" data-title="Laasabu malibu" data-language-autonym="Dagbanli" data-language-local-name="Dagbani" class="interlanguage-link-target">Dagbanli</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://de.wikipedia.org/wiki/Mathematik" title="Mathematik – alemana" lang="de" hreflang="de" data-title="Mathematik" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-din mw-list-item"><a href="https://din.wikipedia.org/wiki/Aku%C3%ABnk%C3%A4%C5%8B" title="Akuënkäŋ – Dinka" lang="din" hreflang="din" data-title="Akuënkäŋ" data-language-autonym="Thuɔŋjäŋ" data-language-local-name="Dinka" class="interlanguage-link-target">Thuɔŋjäŋ</a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Matematik" title="Matematik – Zazaki" lang="diq" hreflang="diq" data-title="Matematik" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target">Zazaki</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Matematika" title="Matematika – behe-sorabiera" lang="dsb" hreflang="dsb" data-title="Matematika" data-language-autonym="Dolnoserbski" data-language-local-name="behe-sorabiera" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Matematik" title="Matematik – Central Dusun" lang="dtp" hreflang="dtp" data-title="Matematik" data-language-autonym="Kadazandusun" data-language-local-name="Central Dusun" class="interlanguage-link-target">Kadazandusun</a></li><li class="interlanguage-link interwiki-dv mw-list-item"><a href="https://dv.wikipedia.org/wiki/%DE%83%DE%A8%DE%94%DE%A7%DE%9F%DE%A8%DE%87%DE%B0%DE%94%DE%A7%DE%8C%DE%AA" title="ރިޔާޟިއްޔާތު – dhivehia" lang="dv" hreflang="dv" data-title="ރިޔާޟިއްޔާތު" data-language-autonym="ދިވެހިބަސް" data-language-local-name="dhivehia" class="interlanguage-link-target">ދިވެހިބަސް</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά – greziera" lang="el" hreflang="el" data-title="Μαθηματικά" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Matem%C3%A2tica" title="Matemâtica – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Matemâtica" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mathematics" title="Mathematics – ingelesa" lang="en" hreflang="en" data-title="Mathematics" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Matematiko" title="Matematiko – esperantoa" lang="eo" hreflang="eo" data-title="Matematiko" data-language-autonym="Esperanto" data-language-local-name="esperantoa" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – gaztelania" lang="es" hreflang="es" data-title="Matemáticas" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatika" title="Matemaatika – estoniera" lang="et" hreflang="et" data-title="Matemaatika" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Extremaduran" lang="ext" hreflang="ext" data-title="Matemáticas" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="ریاضیات – persiera" lang="fa" hreflang="fa" data-title="ریاضیات" data-language-autonym="فارسی" data-language-local-name="persiera" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matematiikka" title="Matematiikka – finlandiera" lang="fi" hreflang="fi" data-title="Matematiikka" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Mat%C3%B5maatiga" title="Matõmaatiga – Võro" lang="vro" hreflang="vro" data-title="Matõmaatiga" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Fika" title="Fika – fijiera" lang="fj" hreflang="fj" data-title="Fika" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijiera" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/St%C3%B8ddfr%C3%B8%C3%B0i" title="Støddfrøði – faroera" lang="fo" hreflang="fo" data-title="Støddfrøði" data-language-autonym="Føroyskt" data-language-local-name="faroera" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques" title="Mathématiques – frantsesa" lang="fr" hreflang="fr" data-title="Mathématiques" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-frp mw-list-item"><a href="https://frp.wikipedia.org/wiki/Mat%C3%A8matiques" title="Matèmatiques – Arpitan" lang="frp" hreflang="frp" data-title="Matèmatiques" data-language-autonym="Arpetan" data-language-local-name="Arpitan" class="interlanguage-link-target">Arpetan</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Matematiik" title="Matematiik – iparraldeko frisiera" lang="frr" hreflang="frr" data-title="Matematiik" data-language-autonym="Nordfriisk" data-language-local-name="iparraldeko frisiera" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Matematiche" title="Matematiche – friulera" lang="fur" hreflang="fur" data-title="Matematiche" data-language-autonym="Furlan" data-language-local-name="friulera" class="interlanguage-link-target">Furlan</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Wiskunde" title="Wiskunde – mendebaldeko frisiera" lang="fy" hreflang="fy" data-title="Wiskunde" data-language-autonym="Frysk" data-language-local-name="mendebaldeko frisiera" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Matamaitic" title="Matamaitic – irlandera" lang="ga" hreflang="ga" data-title="Matamaitic" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Gan" lang="gan" hreflang="gan" data-title="數學" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Mat%C3%A9matik" title="Matématik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Matématik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Matamataig" title="Matamataig – Eskoziako gaelikoa" lang="gd" hreflang="gd" data-title="Matamataig" data-language-autonym="Gàidhlig" data-language-local-name="Eskoziako gaelikoa" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – galiziera" lang="gl" hreflang="gl" data-title="Matemáticas" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapykuaa" title="Papapykuaa – guaraniera" lang="gn" hreflang="gn" data-title="Papapykuaa" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraniera" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-gom mw-list-item"><a href="https://gom.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4%E0%B3%8D" title="ಗಣಿತ್ – Goan Konkani" lang="gom" hreflang="gom" data-title="ಗಣಿತ್" data-language-autonym="गोंयची कोंकणी / Gõychi Konknni" data-language-local-name="Goan Konkani" class="interlanguage-link-target">गोंयची कोंकणी / Gõychi Konknni</a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Matematika" title="Matematika – gorontaloera" lang="gor" hreflang="gor" data-title="Matematika" data-language-autonym="Bahasa Hulontalo" data-language-local-name="gorontaloera" class="interlanguage-link-target">Bahasa Hulontalo</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%97%E0%AA%A3%E0%AA%BF%E0%AA%A4" title="ગણિત – gujaratera" lang="gu" hreflang="gu" data-title="ગણિત" data-language-autonym="ગુજરાતી" data-language-local-name="gujaratera" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Maddaght" title="Maddaght – manxera" lang="gv" hreflang="gv" data-title="Maddaght" data-language-autonym="Gaelg" data-language-local-name="manxera" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Lissafi" title="Lissafi – hausa" lang="ha" hreflang="ha" data-title="Lissafi" data-language-autonym="Hausa" data-language-local-name="hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Su-ho%CC%8Dk" title="Su-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Su-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Makemakika" title="Makemakika – hawaiiera" lang="haw" hreflang="haw" data-title="Makemakika" data-language-autonym="Hawaiʻi" data-language-local-name="hawaiiera" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה – hebreera" lang="he" hreflang="he" data-title="מתמטיקה" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – hindia" lang="hi" hreflang="hi" data-title="गणित" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematics" title="Mathematics – Fiji Hindi" lang="hif" hreflang="hif" data-title="Mathematics" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matematika" title="Matematika – kroaziera" lang="hr" hreflang="hr" data-title="Matematika" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Matematik" title="Matematik – Haitiko kreolera" lang="ht" hreflang="ht" data-title="Matematik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitiko kreolera" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematika" title="Matematika – hungariera" lang="hu" hreflang="hu" data-title="Matematika" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1" title="Մաթեմատիկա – armeniera" lang="hy" hreflang="hy" data-title="Մաթեմատիկա" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://ia.wikipedia.org/wiki/Mathematica" title="Mathematica – interlingua" lang="ia" hreflang="ia" data-title="Mathematica" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Matematik" title="Matematik – ibanera" lang="iba" hreflang="iba" data-title="Matematik" data-language-autonym="Jaku Iban" data-language-local-name="ibanera" class="interlanguage-link-target">Jaku Iban</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matematika" title="Matematika – indonesiera" lang="id" hreflang="id" data-title="Matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Matematica" title="Matematica – interlingue" lang="ie" hreflang="ie" data-title="Matematica" data-language-autonym="Interlingue" data-language-local-name="interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/%E1%BB%8Cm%C3%BAm%C3%BA-%C3%B3n%C3%BA%E1%BB%8Dg%E1%BB%A5g%E1%BB%A5" title="Ọmúmú-ónúọgụgụ – igboera" lang="ig" hreflang="ig" data-title="Ọmúmú-ónúọgụgụ" data-language-autonym="Igbo" data-language-local-name="igboera" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Matematika" title="Matematika – ilocanoera" lang="ilo" hreflang="ilo" data-title="Matematika" data-language-autonym="Ilokano" data-language-local-name="ilocanoera" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – ingushera" lang="inh" hreflang="inh" data-title="Математика" data-language-autonym="ГӀалгӀай" data-language-local-name="ingushera" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Matematiko" title="Matematiko – idoa" lang="io" hreflang="io" data-title="Matematiko" data-language-autonym="Ido" data-language-local-name="idoa" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0i" title="Stærðfræði – islandiera" lang="is" hreflang="is" data-title="Stærðfræði" data-language-autonym="Íslenska" data-language-local-name="islandiera" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matematica" title="Matematica – italiera" lang="it" hreflang="it" data-title="Matematica" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – japoniera" lang="ja" hreflang="ja" data-title="数学" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatix" title="Matimatix – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Matimatix" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/cmaci" title="cmaci – lojbana" lang="jbo" hreflang="jbo" data-title="cmaci" data-language-autonym="La .lojban." data-language-local-name="lojbana" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-jv badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://jv.wikipedia.org/wiki/Mat%C3%A9matika" title="Matématika – javera" lang="jv" hreflang="jv" data-title="Matématika" data-language-autonym="Jawa" data-language-local-name="javera" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკა – georgiera" lang="ka" hreflang="ka" data-title="მათემატიკა" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Matematika" title="Matematika – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Matematika" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target">Qaraqalpaqsha</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tusnakt" title="Tusnakt – kabiliera" lang="kab" hreflang="kab" data-title="Tusnakt" data-language-autonym="Taqbaylit" data-language-local-name="kabiliera" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-kge mw-list-item"><a href="https://kge.wikipedia.org/wiki/Matematika" title="Matematika – Komering" lang="kge" hreflang="kge" data-title="Matematika" data-language-autonym="Kumoring" data-language-local-name="Komering" class="interlanguage-link-target">Kumoring</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – kazakhera" lang="kk" hreflang="kk" data-title="Математика" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kl mw-list-item"><a href="https://kl.wikipedia.org/wiki/Matematikki" title="Matematikki – groenlandiera" lang="kl" hreflang="kl" data-title="Matematikki" data-language-autonym="Kalaallisut" data-language-local-name="groenlandiera" class="interlanguage-link-target">Kalaallisut</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%82%E1%9E%8E%E1%9E%B7%E1%9E%8F%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6" title="គណិតវិទ្យា – khemerera" lang="km" hreflang="km" data-title="គណិតវិទ្យា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khemerera" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ – kannada" lang="kn" hreflang="kn" data-title="ಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99" title="수학 – koreera" lang="ko" hreflang="ko" data-title="수학" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – karachayera-balkarera" lang="krc" hreflang="krc" data-title="Математика" data-language-autonym="Къарачай-малкъар" data-language-local-name="karachayera-balkarera" class="interlanguage-link-target">Къарачай-малкъар</a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/%DA%AF%D8%B1%D9%9B%D8%A0%D9%86%D8%AF%D9%8F%D8%AA" title="گرٛؠندُت – kaxmirera" lang="ks" hreflang="ks" data-title="گرٛؠندُت" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="kaxmirera" class="interlanguage-link-target">कॉशुर / کٲشُر</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Matemat%C3%AEk" title="Matematîk – kurduera" lang="ku" hreflang="ku" data-title="Matematîk" data-language-autonym="Kurdî" data-language-local-name="kurduera" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Awgrym" title="Awgrym – kornubiera" lang="kw" hreflang="kw" data-title="Awgrym" data-language-autonym="Kernowek" data-language-local-name="kornubiera" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BC%D0%BE/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Википедия:Сомо/Математика – kirgizera" lang="ky" hreflang="ky" data-title="Википедия:Сомо/Математика" data-language-autonym="Кыргызча" data-language-local-name="kirgizera" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://la.wikipedia.org/wiki/Mathematica" title="Mathematica – latina" lang="la" hreflang="la" data-title="Mathematica" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lad mw-list-item"><a href="https://lad.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – ladinoa" lang="lad" hreflang="lad" data-title="Matemátika" data-language-autonym="Ladino" data-language-local-name="ladinoa" class="interlanguage-link-target">Ladino</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mathematik" title="Mathematik – luxenburgera" lang="lb" hreflang="lb" data-title="Mathematik" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxenburgera" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – lezginera" lang="lez" hreflang="lez" data-title="Математика" data-language-autonym="Лезги" data-language-local-name="lezginera" class="interlanguage-link-target">Лезги</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Matematica" title="Matematica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Matematica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Ekibalangulo" title="Ekibalangulo – luganda" lang="lg" hreflang="lg" data-title="Ekibalangulo" data-language-autonym="Luganda" data-language-local-name="luganda" class="interlanguage-link-target">Luganda</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Mathematiek" title="Mathematiek – limburgera" lang="li" hreflang="li" data-title="Mathematiek" data-language-autonym="Limburgs" data-language-local-name="limburgera" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Matematica" title="Matematica – liguriera" lang="lij" hreflang="lij" data-title="Matematica" data-language-autonym="Ligure" data-language-local-name="liguriera" class="interlanguage-link-target">Ligure</a></li><li class="interlanguage-link interwiki-lld mw-list-item"><a href="https://lld.wikipedia.org/wiki/Matematica" title="Matematica – Ladin" lang="lld" hreflang="lld" data-title="Matematica" data-language-autonym="Ladin" data-language-local-name="Ladin" class="interlanguage-link-target">Ladin</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matematega" title="Matematega – Lombard" lang="lmo" hreflang="lmo" data-title="Matematega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ຄະນິດສາດ – laosera" lang="lo" hreflang="lo" data-title="ຄະນິດສາດ" data-language-autonym="ລາວ" data-language-local-name="laosera" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematika" title="Matematika – lituaniera" lang="lt" hreflang="lt" data-title="Matematika" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tika" title="Matemātika – letoniera" lang="lv" hreflang="lv" data-title="Matemātika" data-language-autonym="Latviešu" data-language-local-name="letoniera" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mad mw-list-item"><a href="https://mad.wikipedia.org/wiki/Matematika" title="Matematika – madurera" lang="mad" hreflang="mad" data-title="Matematika" data-language-autonym="Madhurâ" data-language-local-name="madurera" class="interlanguage-link-target">Madhurâ</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – maithilia" lang="mai" hreflang="mai" data-title="गणित" data-language-autonym="मैथिली" data-language-local-name="maithilia" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-map-bms mw-list-item"><a href="https://map-bms.wikipedia.org/wiki/Matematika" title="Matematika – Banyumasan" lang="jv-x-bms" hreflang="jv-x-bms" data-title="Matematika" data-language-autonym="Basa Banyumasan" data-language-local-name="Banyumasan" class="interlanguage-link-target">Basa Banyumasan</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Matematika" title="Matematika – malgaxea" lang="mg" hreflang="mg" data-title="Matematika" data-language-autonym="Malagasy" data-language-local-name="malgaxea" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5" title="Математике – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Математике" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Matematika" title="Matematika – minangkabauera" lang="min" hreflang="min" data-title="Matematika" data-language-autonym="Minangkabau" data-language-local-name="minangkabauera" class="interlanguage-link-target">Minangkabau</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – mazedoniera" lang="mk" hreflang="mk" data-title="Математика" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ഗണിതം – malabarera" lang="ml" hreflang="ml" data-title="ഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="malabarera" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA" title="Математик – mongoliera" lang="mn" hreflang="mn" data-title="Математик" data-language-autonym="Монгол" data-language-local-name="mongoliera" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%86%EA%AF%A9%EA%AF%81%EA%AF%A4%EA%AF%A1%EA%AF%82%EA%AF%A3%EA%AF%9F" title="ꯆꯩꯁꯤꯡꯂꯣꯟ – manipurera" lang="mni" hreflang="mni" data-title="ꯆꯩꯁꯤꯡꯂꯣꯟ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="manipurera" class="interlanguage-link-target">ꯃꯤꯇꯩ ꯂꯣꯟ</a></li><li class="interlanguage-link interwiki-mnw mw-list-item"><a href="https://mnw.wikipedia.org/wiki/%E1%80%9E%E1%81%9A%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သၚ်္ချာ – Mon" lang="mnw" hreflang="mnw" data-title="သၚ်္ချာ" data-language-autonym="ဘာသာမန်" data-language-local-name="Mon" class="interlanguage-link-target">ဘာသာမန်</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – marathera" lang="mr" hreflang="mr" data-title="गणित" data-language-autonym="मराठी" data-language-local-name="marathera" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matematik" title="Matematik – malaysiera" lang="ms" hreflang="ms" data-title="Matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Matematika" title="Matematika – maltera" lang="mt" hreflang="mt" data-title="Matematika" data-language-autonym="Malti" data-language-local-name="maltera" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – mirandesa" lang="mwl" hreflang="mwl" data-title="Matemática" data-language-autonym="Mirandés" data-language-local-name="mirandesa" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သင်္ချာ – birmaniera" lang="my" hreflang="my" data-title="သင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmaniera" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – erziera" lang="myv" hreflang="myv" data-title="Математика" data-language-autonym="Эрзянь" data-language-local-name="erziera" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%B1%D9%82%D9%85" title="رقم – mazandarandera" lang="mzn" hreflang="mzn" data-title="رقم" data-language-autonym="مازِرونی" data-language-local-name="mazandarandera" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-nah mw-list-item"><a href="https://nah.wikipedia.org/wiki/Tlap%C5%8Dhualmatiliztli" title="Tlapōhualmatiliztli – Nahuatl" lang="nah" hreflang="nah" data-title="Tlapōhualmatiliztli" data-language-autonym="Nāhuatl" data-language-local-name="Nahuatl" class="interlanguage-link-target">Nāhuatl</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Mathematik" title="Mathematik – behe-alemana" lang="nds" hreflang="nds" data-title="Mathematik" data-language-autonym="Plattdüütsch" data-language-local-name="behe-alemana" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – behe-saxoiera" lang="nds-NL" hreflang="nds-NL" data-title="Wiskunde" data-language-autonym="Nedersaksies" data-language-local-name="behe-saxoiera" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – nepalera" lang="ne" hreflang="ne" data-title="गणित" data-language-autonym="नेपाली" data-language-local-name="nepalera" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="ल्याःज्या – newarera" lang="new" hreflang="new" data-title="ल्याःज्या" data-language-autonym="नेपाल भाषा" data-language-local-name="newarera" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-nia mw-list-item"><a href="https://nia.wikipedia.org/wiki/Matematika" title="Matematika – niasera" lang="nia" hreflang="nia" data-title="Matematika" data-language-autonym="Li Niha" data-language-local-name="niasera" class="interlanguage-link-target">Li Niha</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – nederlandera" lang="nl" hreflang="nl" data-title="Wiskunde" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematikk" title="Matematikk – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematikk" title="Matematikk – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Matematikk" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Matematike" title="Matematike – Novial" lang="nov" hreflang="nov" data-title="Matematike" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Caltchul" title="Caltchul – Norman" lang="nrf" hreflang="nrf" data-title="Caltchul" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target">Nouormand</a></li><li class="interlanguage-link interwiki-nv mw-list-item"><a href="https://nv.wikipedia.org/wiki/A%C5%82hii%CA%BCn%C3%ADn%C3%A1%CA%BCiidz%C3%B3%C3%B3h" title="Ałhiiʼnínáʼiidzóóh – navajoera" lang="nv" hreflang="nv" data-title="Ałhiiʼnínáʼiidzóóh" data-language-autonym="Diné bizaad" data-language-local-name="navajoera" class="interlanguage-link-target">Diné bizaad</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Matematicas" title="Matematicas – okzitaniera" lang="oc" hreflang="oc" data-title="Matematicas" data-language-autonym="Occitan" data-language-local-name="okzitaniera" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-olo mw-list-item"><a href="https://olo.wikipedia.org/wiki/Matematiekku" title="Matematiekku – Livvi-Karelian" lang="olo" hreflang="olo" data-title="Matematiekku" data-language-autonym="Livvinkarjala" data-language-local-name="Livvi-Karelian" class="interlanguage-link-target">Livvinkarjala</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Herrega" title="Herrega – oromoera" lang="om" hreflang="om" data-title="Herrega" data-language-autonym="Oromoo" data-language-local-name="oromoera" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%97%E0%AC%A3%E0%AC%BF%E0%AC%A4" title="ଗଣିତ – oriya" lang="or" hreflang="or" data-title="ଗଣିତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="oriya" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C3%A6" title="Математикæ – osetiera" lang="os" hreflang="os" data-title="Математикæ" data-language-autonym="Ирон" data-language-local-name="osetiera" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਗਣਿਤ – punjabera" lang="pa" hreflang="pa" data-title="ਗਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabera" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pag mw-list-item"><a href="https://pag.wikipedia.org/wiki/Matematiks" title="Matematiks – pangasinanera" lang="pag" hreflang="pag" data-title="Matematiks" data-language-autonym="Pangasinan" data-language-local-name="pangasinanera" class="interlanguage-link-target">Pangasinan</a></li><li class="interlanguage-link interwiki-pam mw-list-item"><a href="https://pam.wikipedia.org/wiki/Matematica" title="Matematica – pampangera" lang="pam" hreflang="pam" data-title="Matematica" data-language-autonym="Kapampangan" data-language-local-name="pampangera" class="interlanguage-link-target">Kapampangan</a></li><li class="interlanguage-link interwiki-pap mw-list-item"><a href="https://pap.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – papiamentoa" lang="pap" hreflang="pap" data-title="Matemátika" data-language-autonym="Papiamentu" data-language-local-name="papiamentoa" class="interlanguage-link-target">Papiamentu</a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Mat%C3%A9matikes" title="Matématikes – Picard" lang="pcd" hreflang="pcd" data-title="Matématikes" data-language-autonym="Picard" data-language-local-name="Picard" class="interlanguage-link-target">Picard</a></li><li class="interlanguage-link interwiki-pi mw-list-item"><a href="https://pi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%82" title="गणितं – Pali" lang="pi" hreflang="pi" data-title="गणितं" data-language-autonym="पालि" data-language-local-name="Pali" class="interlanguage-link-target">पालि</a></li><li class="interlanguage-link interwiki-pih mw-list-item"><a href="https://pih.wikipedia.org/wiki/Maethamatiks" title="Maethamatiks – Norfuk / Pitkern" lang="pih" hreflang="pih" data-title="Maethamatiks" data-language-autonym="Norfuk / Pitkern" data-language-local-name="Norfuk / Pitkern" class="interlanguage-link-target">Norfuk / Pitkern</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Matematyka" title="Matematyka – poloniera" lang="pl" hreflang="pl" data-title="Matematyka" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – Piedmontese" lang="pms" hreflang="pms" data-title="Matemàtica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ریاضی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B4%D9%85%DB%90%D8%B1%D9%BE%D9%88%D9%87%D9%86%D9%87" title="شمېرپوهنه – paxtunera" lang="ps" hreflang="ps" data-title="شمېرپوهنه" data-language-autonym="پښتو" data-language-local-name="paxtunera" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – portugesa" lang="pt" hreflang="pt" data-title="Matemática" data-language-autonym="Português" data-language-local-name="portugesa" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupay_yachay" title="Yupay yachay – kitxua" lang="qu" hreflang="qu" data-title="Yupay yachay" data-language-autonym="Runa Simi" data-language-local-name="kitxua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matematic%C4%83" title="Matematică – errumaniera" lang="ro" hreflang="ro" data-title="Matematică" data-language-autonym="Română" data-language-local-name="errumaniera" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/Mathematic%C3%A2" title="Mathematicâ – aromaniera" lang="rup" hreflang="rup" data-title="Mathematicâ" data-language-autonym="Armãneashti" data-language-local-name="aromaniera" class="interlanguage-link-target">Armãneashti</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – errusiera" lang="ru" hreflang="ru" data-title="Математика" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D0%BA%D0%B0" title="Математіка – Rusyn" lang="rue" hreflang="rue" data-title="Математіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D" title="गणितम् – sanskritoa" lang="sa" hreflang="sa" data-title="गणितम्" data-language-autonym="संस्कृतम्" data-language-local-name="sanskritoa" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – sakhera" lang="sah" hreflang="sah" data-title="Математика" data-language-autonym="Саха тыла" data-language-local-name="sakhera" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AE%E1%B1%9E%E1%B1%A0%E1%B1%B7%E1%B1%9F" title="ᱮᱞᱠᱷᱟ – santalera" lang="sat" hreflang="sat" data-title="ᱮᱞᱠᱷᱟ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santalera" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sc badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://sc.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – sardiniera" lang="sc" hreflang="sc" data-title="Matemàtica" data-language-autonym="Sardu" data-language-local-name="sardiniera" class="interlanguage-link-target">Sardu</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Matim%C3%A0tica" title="Matimàtica – siziliera" lang="scn" hreflang="scn" data-title="Matimàtica" data-language-autonym="Sicilianu" data-language-local-name="siziliera" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Mathematics" title="Mathematics – eskoziera" lang="sco" hreflang="sco" data-title="Mathematics" data-language-autonym="Scots" data-language-local-name="eskoziera" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="رياضي – sindhia" lang="sd" hreflang="sd" data-title="رياضي" data-language-autonym="سنڌي" data-language-local-name="sindhia" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matematika" title="Matematika – serbokroaziera" lang="sh" hreflang="sh" data-title="Matematika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tusnakt" title="Tusnakt – tachelhita" lang="shi" hreflang="shi" data-title="Tusnakt" data-language-autonym="Taclḥit" data-language-local-name="tachelhita" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ගණිතය – sinhala" lang="si" hreflang="si" data-title="ගණිතය" data-language-autonym="සිංහල" data-language-local-name="sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematics" title="Mathematics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematika" title="Matematika – eslovakiera" lang="sk" hreflang="sk" data-title="Matematika" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Saraiki" lang="skr" hreflang="skr" data-title="ریاضی" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target">سرائیکی</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matematika" title="Matematika – esloveniera" lang="sl" hreflang="sl" data-title="Matematika" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sm mw-list-item"><a href="https://sm.wikipedia.org/wiki/Matematika" title="Matematika – samoera" lang="sm" hreflang="sm" data-title="Matematika" data-language-autonym="Gagana Samoa" data-language-local-name="samoera" class="interlanguage-link-target">Gagana Samoa</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Masvomhu" title="Masvomhu – shonera" lang="sn" hreflang="sn" data-title="Masvomhu" data-language-autonym="ChiShona" data-language-local-name="shonera" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tiromaal" title="Tiromaal – somaliera" lang="so" hreflang="so" data-title="Tiromaal" data-language-autonym="Soomaaliga" data-language-local-name="somaliera" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matematika" title="Matematika – albaniera" lang="sq" hreflang="sq" data-title="Matematika" data-language-autonym="Shqip" data-language-local-name="albaniera" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – serbiera" lang="sr" hreflang="sr" data-title="Математика" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-srn mw-list-item"><a href="https://srn.wikipedia.org/wiki/Sabi_fu_Teri" title="Sabi fu Teri – sranan tongoa" lang="srn" hreflang="srn" data-title="Sabi fu Teri" data-language-autonym="Sranantongo" data-language-local-name="sranan tongoa" class="interlanguage-link-target">Sranantongo</a></li><li class="interlanguage-link interwiki-ss mw-list-item"><a href="https://ss.wikipedia.org/wiki/Tekubala" title="Tekubala – swatiera" lang="ss" hreflang="ss" data-title="Tekubala" data-language-autonym="SiSwati" data-language-local-name="swatiera" class="interlanguage-link-target">SiSwati</a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Mathematik" title="Mathematik – Saterland Frisian" lang="stq" hreflang="stq" data-title="Mathematik" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target">Seeltersk</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Matematika" title="Matematika – sundanera" lang="su" hreflang="su" data-title="Matematika" data-language-autonym="Sunda" data-language-local-name="sundanera" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematik" title="Matematik – suediera" lang="sv" hreflang="sv" data-title="Matematik" data-language-autonym="Svenska" data-language-local-name="suediera" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Hisabati" title="Hisabati – swahilia" lang="sw" hreflang="sw" data-title="Hisabati" data-language-autonym="Kiswahili" data-language-local-name="swahilia" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Matymatyka" title="Matymatyka – Silesian" lang="szl" hreflang="szl" data-title="Matymatyka" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="கணிதம் – tamilera" lang="ta" hreflang="ta" data-title="கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="tamilera" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tcy mw-list-item"><a href="https://tcy.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ – Tulu" lang="tcy" hreflang="tcy" data-title="ಗಣಿತ" data-language-autonym="ತುಳು" data-language-local-name="Tulu" class="interlanguage-link-target">ತುಳು</a></li><li class="interlanguage-link interwiki-tdd mw-list-item"><a href="https://tdd.wikipedia.org/wiki/%E1%A5%99%E1%A5%A3%E1%A5%AD%E1%A5%B0_%E1%A5%98%E1%A5%99%E1%A5%B3" title="ᥙᥣᥭᥰ ᥘᥙᥳ – Tai Nuea" lang="tdd" hreflang="tdd" data-title="ᥙᥣᥭᥰ ᥘᥙᥳ" data-language-autonym="ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ" data-language-local-name="Tai Nuea" class="interlanguage-link-target">ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B0%A3%E0%B0%BF%E0%B0%A4%E0%B0%82" title="గణితం – telugua" lang="te" hreflang="te" data-title="గణితం" data-language-autonym="తెలుగు" data-language-local-name="telugua" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-tet mw-list-item"><a href="https://tet.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – tetuma" lang="tet" hreflang="tet" data-title="Matemátika" data-language-autonym="Tetun" data-language-local-name="tetuma" class="interlanguage-link-target">Tetun</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82" title="Риёзиёт – tajikera" lang="tg" hreflang="tg" data-title="Риёзиёт" data-language-autonym="Тоҷикӣ" data-language-local-name="tajikera" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="คณิตศาสตร์ – thailandiera" lang="th" hreflang="th" data-title="คณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Matematika" title="Matematika – turkmenera" lang="tk" hreflang="tk" data-title="Matematika" data-language-autonym="Türkmençe" data-language-local-name="turkmenera" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Matematika" title="Matematika – tagaloa" lang="tl" hreflang="tl" data-title="Matematika" data-language-autonym="Tagalog" data-language-local-name="tagaloa" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tly mw-list-item"><a href="https://tly.wikipedia.org/wiki/Rijozijot" title="Rijozijot – Talysh" lang="tly" hreflang="tly" data-title="Rijozijot" data-language-autonym="Tolışi" data-language-local-name="Talysh" class="interlanguage-link-target">Tolışi</a></li><li class="interlanguage-link interwiki-tn mw-list-item"><a href="https://tn.wikipedia.org/wiki/Dipalo" title="Dipalo – tswanera" lang="tn" hreflang="tn" data-title="Dipalo" data-language-autonym="Setswana" data-language-local-name="tswanera" class="interlanguage-link-target">Setswana</a></li><li class="interlanguage-link interwiki-tpi mw-list-item"><a href="https://tpi.wikipedia.org/wiki/Ol_matematik" title="Ol matematik – tok pisin" lang="tpi" hreflang="tpi" data-title="Ol matematik" data-language-autonym="Tok Pisin" data-language-local-name="tok pisin" class="interlanguage-link-target">Tok Pisin</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematik" title="Matematik – turkiera" lang="tr" hreflang="tr" data-title="Matematik" data-language-autonym="Türkçe" data-language-local-name="turkiera" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Dyondzo-Tinhlayo" title="Dyondzo-Tinhlayo – tsongera" lang="ts" hreflang="ts" data-title="Dyondzo-Tinhlayo" data-language-autonym="Xitsonga" data-language-local-name="tsongera" class="interlanguage-link-target">Xitsonga</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – tatarera" lang="tt" hreflang="tt" data-title="Математика" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarera" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – udmurtera" lang="udm" hreflang="udm" data-title="Математика" data-language-autonym="Удмурт" data-language-local-name="udmurtera" class="interlanguage-link-target">Удмурт</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – ukrainera" lang="uk" hreflang="uk" data-title="Математика" data-language-autonym="Українська" data-language-local-name="ukrainera" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – urdua" lang="ur" hreflang="ur" data-title="ریاضی" data-language-autonym="اردو" data-language-local-name="urdua" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematika" title="Matematika – uzbekera" lang="uz" hreflang="uz" data-title="Matematika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekera" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Matem%C3%A0tega" title="Matemàtega – veneziera" lang="vec" hreflang="vec" data-title="Matemàtega" data-language-autonym="Vèneto" data-language-local-name="veneziera" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Matematik" title="Matematik – Veps" lang="vep" hreflang="vep" data-title="Matematik" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_h%E1%BB%8Dc" title="Toán học – vietnamera" lang="vi" hreflang="vi" data-title="Toán học" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Wiskunde" title="Wiskunde – West Flemish" lang="vls" hreflang="vls" data-title="Wiskunde" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-vo badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://vo.wikipedia.org/wiki/Matemat" title="Matemat – volapük" lang="vo" hreflang="vo" data-title="Matemat" data-language-autonym="Volapük" data-language-local-name="volapük" class="interlanguage-link-target">Volapük</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Matematike" title="Matematike – valoniera" lang="wa" hreflang="wa" data-title="Matematike" data-language-autonym="Walon" data-language-local-name="valoniera" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Matematika" title="Matematika – warayera" lang="war" hreflang="war" data-title="Matematika" data-language-autonym="Winaray" data-language-local-name="warayera" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Xayma" title="Xayma – wolofera" lang="wo" hreflang="wo" data-title="Xayma" data-language-autonym="Wolof" data-language-local-name="wolofera" class="interlanguage-link-target">Wolof</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – wu txinera" lang="wuu" hreflang="wuu" data-title="数学" data-language-autonym="吴语" data-language-local-name="wu txinera" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – kalmykera" lang="xal" hreflang="xal" data-title="Математика" data-language-autonym="Хальмг" data-language-local-name="kalmykera" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Mathematics" title="I-Mathematics – xhosera" lang="xh" hreflang="xh" data-title="I-Mathematics" data-language-autonym="IsiXhosa" data-language-local-name="xhosera" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკა – Mingrelian" lang="xmf" hreflang="xmf" data-title="მათემატიკა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7" title="מאטעמאטיק – yiddisha" lang="yi" hreflang="yi" data-title="מאטעמאטיק" data-language-autonym="ייִדיש" data-language-local-name="yiddisha" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Mathim%C3%A1t%C3%ADk%C3%AC" title="Mathimátíkì – jorubera" lang="yo" hreflang="yo" data-title="Mathimátíkì" data-language-autonym="Yorùbá" data-language-local-name="jorubera" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Soqyoz" title="Soqyoz – Zhuang" lang="za" hreflang="za" data-title="Soqyoz" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target">Vahcuengh</a></li><li class="interlanguage-link interwiki-zea mw-list-item"><a href="https://zea.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Zeelandic" lang="zea" hreflang="zea" data-title="Wiskunde" data-language-autonym="Zeêuws" data-language-local-name="Zeelandic" class="interlanguage-link-target">Zeêuws</a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B5%93%E2%B5%99%E2%B5%8F%E2%B4%B0%E2%B4%BD%E2%B5%9C" title="ⵜⵓⵙⵏⴰⴽⵜ – amazigera estandarra" lang="zgh" hreflang="zgh" data-title="ⵜⵓⵙⵏⴰⴽⵜ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="amazigera estandarra" class="interlanguage-link-target">ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – txinera" lang="zh" hreflang="zh" data-title="数学" data-language-autonym="中文" data-language-local-name="txinera" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="數學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-ha%CC%8Dk" title="Sò͘-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – kantonera" lang="yue" hreflang="yue" data-title="數學" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Umchazazibalo" title="Umchazazibalo – zuluera" lang="zu" hreflang="zu" data-title="Umchazazibalo" data-language-autonym="IsiZulu" data-language-local-name="zuluera" class="interlanguage-link-target">IsiZulu</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q395#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</a></div> </div> 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href="/wiki/Euklides" title="Euklides">Euklides</a> eskolak ematen, Atenasko eskola margolanaren xehetasunean (<a href="/wiki/Rafael" title="Rafael">Rafael</a>).</figcaption></figure> Matematika ezagutza-arlo bat da. Bertan sartzen diren gaien artean daude zenbakiak, formulak eta erlazionatutako egiturak, formak eta hauek biltzen dituzten espazioak, eta kantitateak eta euren aldaketak. Gai horiek matematika modernoan irudikatzen dira, <a href="/wiki/Zenbakien_teoria" title="Zenbakien teoria">zenbakien teoriaren</a><a href="#cite_note-1">[1]</a>, <a href="/wiki/Aljebra" title="Aljebra">aljebraren</a><a href="#cite_note-2">[2]</a>, <a href="/wiki/Geometria" title="Geometria">geometriaren</a> eta <a href="/wiki/Analisi_matematiko" title="Analisi matematiko">analisiaren</a><a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> azpi-diziplina handiekin, hurrenez hurren. Matematikariek ez dute adostasun orokorrik beren <a href="/wiki/Diziplina_akademiko" title="Diziplina akademiko">diziplina akademikoaren</a> definizio komun bati buruz. Matematika-jardueraren zatirik handienean, <a href="/w/index.php?title=Objektu_matematiko&action=edit&redlink=1" class="new" title="Objektu matematiko (sortu gabe)">objektu abstraktuen</a> propietateak aurkitu behar dira, eta horiek <a href="/wiki/Froga_matematiko" title="Froga matematiko">frogatzeko</a> <a href="/wiki/Arrazoimen" class="mw-redirect" title="Arrazoimen">arrazoimen</a> hutsa erabili. Objektu horiek naturaren abstrakzioak dira, edo, matematika modernoetan, zenbait propietate dituzten entitateetan, <a href="/wiki/Axioma" title="Axioma">axioma</a> deritzenetan. Froga bat da <a href="/wiki/Arrazoibide_deduktibo" title="Arrazoibide deduktibo">arau deduktiboen</a> aplikazioen segida bat jadanik ezarrita dauden emaitzetan. Emaitza horiek aldez aurretik frogatutako <a href="/wiki/Teorema" title="Teorema">teoremak</a>, axiomak eta, naturaren abstrakzioaren kasuan, kontuan hartutako teoriaren abiapuntutzat jotzen diren oinarrizko propietate batzuk dituzte<a href="#cite_note-5">[5]</a>. Matematika funtsezkoa da <a href="/wiki/Natura-zientziak" title="Natura-zientziak">natur zientzietan</a>, <a href="/wiki/Ingeniaritza" title="Ingeniaritza">ingeniaritzan</a>, <a href="/wiki/Medikuntza" title="Medikuntza">medikuntzan</a>, <a href="/wiki/Finantza" class="mw-redirect" title="Finantza">finantzetan</a>, <a href="/wiki/Informatika" title="Informatika">informatikan</a> eta <a href="/wiki/Gizarte-zientzia" class="mw-redirect" title="Gizarte-zientzia">gizarte-zientzietan</a>. Nahiz eta matematika asko erabiltzen den fenomenoak modelizatzeko, matematikaren funtsezko egiak ez dira edozein esperimentazio zientifikorekiko independenteak. Matematikaren arlo batzuk, hala nola <a href="/wiki/Estatistika" title="Estatistika">estatistika</a> eta <a href="/wiki/Jokoen_teoria" class="mw-redirect" title="Jokoen teoria">jokoen teoria</a>, korrelazio estuan garatzen dira beren aplikazioekin, eta <a href="/wiki/Matematika_aplikatu" class="mw-redirect" title="Matematika aplikatu">matematika aplikatuen</a> epigrafean biltzen dira. Beste arlo batzuk edozein aplikaziotatik kanpo garatzen dira (eta horregatik esaten zaie <a href="/wiki/Matematika_puru" class="mw-redirect" title="Matematika puru">matematika puru</a>), baina askotan geroago aplikazio praktikoak aurkitzen dituzte<a href="#cite_note-FOOTNOTEPeterson200112-6">[6]</a><a href="#cite_note-7">[7]</a>. <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">Zenbaki osoen</a> <a href="/wiki/Faktorizazio" title="Faktorizazio">faktorizazioaren</a> arazoak, adibidez, K.a. 300. urtean <a href="/wiki/Euklides" title="Euklides">Euklidesen</a> hasi zenak, ez zuen aplikazio praktikorik <a href="/wiki/RSA_kriptosistema" class="mw-redirect" title="RSA kriptosistema">RSA kriptosisteman</a> erabili aurretik, orain asko erabiltzen baita <a href="/wiki/Sare_informatiko" class="mw-redirect" title="Sare informatiko">sare informatikoen</a> segurtasunerako. Historikoki, frogapenaren kontzeptua eta hari lotutako zorroztasun matematikoa lehen aldiz agertu ziren <a href="/wiki/Greziar_matematika" title="Greziar matematika">greziar matematikan</a>, batez ere Euklidesen <a href="/wiki/Euklidesen_Elementuak" title="Euklidesen Elementuak">Elementuetan</a><a href="#cite_note-8">[8]</a>. Hasieratik, matematika funtsean geometrian eta <a href="/wiki/Aritmetika" title="Aritmetika">aritmetikan</a> (<a href="/wiki/Zenbaki_natural" class="mw-redirect" title="Zenbaki natural">zenbaki naturalen</a> eta <a href="/wiki/Zatiki" class="mw-redirect" title="Zatiki">zatikien</a> manipulazioan) banatu zen, XVI. eta XVII. mendeetara arte, aljebra eta <a href="/wiki/Kalkulu_infinitesimal" title="Kalkulu infinitesimal">kalkulu infinitesimala</a> eremu berri gisa sartu zirenean. Ordutik aurrera, berrikuntza matematikoen eta aurkikuntza zientifikoen arteko elkarreraginak bi horien garapena azkar handitzea ekarri du<a href="#cite_note-9">[9]</a>. XIX. mendearen amaieran, matematikaren sorrerako krisiak <a href="/w/index.php?title=Metodo_axiomatiko&action=edit&redlink=1" class="new" title="Metodo axiomatiko (sortu gabe)">metodo axiomatikoaren</a> sistematizaziora eraman zuen<a href="#cite_note-10">[10]</a>, zeinak iragarri baitzuen izugarri handitu zirela matematika-arloen kopurua eta haien aplikazio-eremuak. Matematika-irakasgaien egungo sailkapenak lehen mailako 60 matematika-arlo baino gehiago aipatzen ditu. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=1" title="Aldatu atal hau: «Historia»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=1" title="Edit section's source code: Historia">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Matematikaren_historia" title="Matematikaren historia">Matematikaren historia</a>»</div> Nahiz eta matematika izena Grezia klasikotik datorkigun, <a href="/wiki/Babiloniar_Inperio" class="mw-redirect" title="Babiloniar Inperio">Babiloniar Inperioko</a> garaiko aztarnek, jadanik zenbatzeko sistema konplexuak izateaz gainera 3.mailako zenbait ekuazio ebazteko metodoak bazituztela erakutsi dute. Ezaguna da baita ere antzinako egiptoarrek <a href="/wiki/Nilo" title="Nilo">Nilo</a> ibai hertzeko lurraldeak mugatzeko trigonometria erabiltzen zutela. Geometrian ere aurreratuak ziren, piramideak eraikitzeko ezinbestekoa. Antzinaroan, gaur egungo matematikaren arlo batzuen oinarriak ezarri baziren ere, grekoak izan ziren ordura arte ezagutzen zena, <a href="/wiki/Zientzia" title="Zientzia">zientzia</a> arrazional eta estrukturatu batean bihurtzen lehenak. Haiek ideia abstraktuak aztertu zituzten, nahiz eta hasiera batean aplikazio praktikorik ez eduki. Garai honetakoak dira lehenengo teorema geometrikoak, <a href="/wiki/Pitagorasen_teorema" title="Pitagorasen teorema">Pitagorasen teorema</a> kasu. <a href="/wiki/Geometria_euklidear" title="Geometria euklidear">Geometria euklidearraren</a> oinarri diren <a href="/wiki/Euklidesen_axiomak" class="mw-redirect" title="Euklidesen axiomak">Euklidesen bost axiomak</a> ere garai hartakoak dira. <a href="/wiki/Europa" title="Europa">Europan</a> <a href="/wiki/XIII._mendea" title="XIII. mendea">XIII. mendea</a> arte ez zen aurrerapen garrantzitsurik gertatu. Mende horietan zehar batez ere indiar eta txinatar matematikariek garatu zuten matematika. Indiarrak izan ziren <a href="/wiki/Zero" title="Zero">zero</a> <a href="/wiki/Zenbaki" title="Zenbaki">zenbakia</a> matematikan sartu zutenak (K.o. 650) eta txinatarrak aldiz, zenbaki negatiboak. Aipatzekoak dira ere <a href="/wiki/VIII._mendea" title="VIII. mendea">VIII</a>. mendetik <a href="/wiki/XVII._mendea" title="XVII. mendea">XVII</a>.era [islamiar]]rek egindako ekarpenak. <a href="/wiki/1202" title="1202">1202</a>. urtean <a href="/wiki/Leonardo_Pisano" title="Leonardo Pisano">Leonardo Fibonacci</a> matematikari italiarrak, islamiar herrialdeetako maisuengandik ikasitakoaren argitalpenak, Europa mailan, greziarren garaitik ematen zen lehen aurrera pausua izan zen. Hala eta guztiz ere <a href="/wiki/XVI._mendea" title="XVI. mendea">XVI. mendean</a> hasi zen benetako pizkundea, ekuazio kubikoen soluzio orokorraren aurkikuntzarekin. <a href="/wiki/XVII._mendea" title="XVII. mendea">XVII. mendean</a> <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartesek</a> geometria analitikoa garatu zuen, haren ekarpen nabarmenena, haren ohorez ezagutzen diren <a href="/wiki/Kartesiar_koordenatu" class="mw-redirect" title="Kartesiar koordenatu">ardatz kartesiarrak</a> izanik. Mende honetan ere, <a href="/wiki/Newton" class="mw-disambig" title="Newton">Newton</a> eta <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibnizek</a>, bakoitza bere aldetik, <a href="/wiki/Kalkulu_diferentzial" title="Kalkulu diferentzial">kalkulu diferentziala</a> garatuko dute. <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> eta <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascalek</a> probabilitate eta konbinatoriaren lehen lan formalak zabaldu zituzten. <a href="/wiki/XVIII._mendea" title="XVIII. mendea">XVIII. mendeko</a> protagonista <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> izan zen. Aurkikuntza eta garapen izugarriak egiteaz aparte, Eulerrek eragin handia izan zuen notazio matematikoaren estandarizazioan. Esaterako, <a href="/wiki/John_Napier" title="John Napier">John Napierrek</a> ikertutako konstanteari lehen aldiz e bezala agertu zuen eta zirkunferentzia batek diametroarekin duen erlazioari pi deitu zion. <a href="/wiki/XIX._mendea" title="XIX. mendea">XIX. mendeak</a> berriz <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> matematikari alemaniarraren lanak ezagutuko ditu. Askorentzat historiako matematikaririk argiena den honek, geometria, aldagai konplexuzko funtzioetan eta serieen konbergentzian egin zituen aurrerapen izugarriak. Honetaz aparte teorema garrantzitsuak ere frogatu zituen, hala nola, <a href="/wiki/Aljebraren_oinarrizko_teorema" title="Aljebraren oinarrizko teorema">Aljebraren oinarrizko teorema</a>. Garai honetan ere, Euklidesen 5. axioma baztertuz, lehen aldiz geometria ez-euklidearrari buruz hitz egin zen. Gero, <a href="/wiki/George_Boole" title="George Boole">George Boolek</a> garrantzi handiagoa hartuko zuen 0 eta 1ean oinarritutako <a href="/w/index.php?title=Aljebra_boolear&action=edit&redlink=1" class="new" title="Aljebra boolear (sortu gabe)">Aljebra boolearra</a> garatu zuen. <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> matematikari frantziarrak, 16 urte zituela, matematikaren arlo berria garatu zuen, <a href="/wiki/Galoisen_teoria" title="Galoisen teoria">Galoisen teoria</a> deiturikoa. Orokorrean esan daiteke matematika abstraktuago bilakatu zela mende honetan. Garai honetan ere sortu ziren lehen matematika elkarteak. Esaterako <a href="/wiki/1865" title="1865">1865ean</a>, Londresko matematika elkartea, <a href="/wiki/1872" title="1872">1872an</a> Frantziako matematika elkartea edo 1889an Amerikako matematika elkartea. <div class="mw-heading mw-heading2"><h2 id="Matematikaren_arloak">Matematikaren arloak</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=2" title="Aldatu atal hau: «Matematikaren arloak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=2" title="Edit section's source code: Matematikaren arloak">aldatu iturburu kodea</a>]</div> <a href="/wiki/Errenazimentu" class="mw-redirect" title="Errenazimentu">Errenazimentuaren</a> aurretik, matematika bi arlo handitan banatzen zen: <a href="/wiki/Aritmetika" title="Aritmetika">aritmetika</a>, zenbakien manipulazioari buruzkoa, eta <a href="/wiki/Geometria" title="Geometria">geometria</a>, formen azterketari buruzkoa<a href="#cite_note-11">[11]</a>. <a href="/wiki/Sasizientzia" title="Sasizientzia">Sasizientzia</a> mota batzuk, hala nola <a href="/wiki/Numerologia" title="Numerologia">numerologia</a> eta <a href="/wiki/Astrologia" title="Astrologia">astrologia</a>, ez ziren argi bereizten matematikatik<a href="#cite_note-12">[12]</a>. Errenazimentuan beste bi eremu agertu ziren. <a href="/w/index.php?title=Notazio_matematiko&action=edit&redlink=1" class="new" title="Notazio matematiko (sortu gabe)">Notazio matematikoak</a> <a href="/wiki/Aljebra" title="Aljebra">aljebra</a> ekarri zuen, zeina, oro har, formulak aztertu eta manipulatzean datza. <a href="/wiki/Kalkulu" class="mw-redirect mw-disambig" title="Kalkulu">Kalkulua</a>, <a href="/wiki/Kalkulu_diferentzial" title="Kalkulu diferentzial">kalkulu diferentzialaren</a> eta <a href="/wiki/Kalkulu_integral" class="mw-redirect" title="Kalkulu integral">kalkulu integralaren</a> bi azpieremuek osatua, <a href="/wiki/Funtzio_jarraitu" title="Funtzio jarraitu">funtzio jarraituen</a> azterketa da. Funtzio horiek kopuru aldakorren (aldagaien bidez adieraziak) arteko <a href="/wiki/Sistema_ez_lineal" title="Sistema ez lineal">erlazio ez-linealak</a> modelizatzen dituzte. Lau eremu nagusitan banatze hori –aritmetika, geometria, aljebra eta kalkulua<a href="#cite_note-13">[13]</a>– XIX. mendearen bukaera arte mantendu zen. Orduan, matematikariek aztertu zituzten <a href="/wiki/Zeruko_mekanika" title="Zeruko mekanika">zeruko mekanika</a> eta <a href="/wiki/Solidoen_mekanika" class="mw-redirect" title="Solidoen mekanika">solidoen mekanika</a> bezalako arloak, baina orain <a href="/wiki/Fisika" title="Fisika">fisikatzat</a> hartzen dira<a href="#cite_note-14">[14]</a>. <a href="/wiki/Konbinatoria" title="Konbinatoria">Konbinatoria</a> historiaren zati handi batean aztertu da, baina ez zen matematikatik bereizitako adar bihurtu XVII. mendera arte<a href="#cite_note-15">[15]</a>. XIX. mendearen amaieran, <a href="/w/index.php?title=Matematikaren_fundamentuak&action=edit&redlink=1" class="new" title="Matematikaren fundamentuak (sortu gabe)">matematikaren fundazio-krisiak</a> eta, ondorioz, <a href="/w/index.php?title=Metodo_axiomatiko&action=edit&redlink=1" class="new" title="Metodo axiomatiko (sortu gabe)">metodo axiomatikoaren</a> sistematizazioak matematikaren arlo berrien leherketa eragin zuten. 2020ko <a href="/w/index.php?title=Matematika_Irakasgaien_Sailkapena&action=edit&redlink=1" class="new" title="Matematika Irakasgaien Sailkapena (sortu gabe)">Matematika Irakasgaien Sailkapenak</a> lehen mailako hirurogeita hiru arlo ditu gutxienez. Arlo horietako batzuk zatiketa zaharrenari dagozkio, hala nola zenbakien teoriari (goiko aritmetikaren izen modernoa) eta geometriari. Lehen mailako beste eremu batzuek "geometria" dute beren izenean, edo normalean geometriaren zatitzat hartzen dira. Aljebra eta kalkulua ez dira lehen mailako area gisa agertzen, lehen mailako zenbait eremutan banatzen dira, hurrenez hurren. Lehen mailako beste arlo batzuk XX. mendean sortu ziren edo lehenago matematikatzat hartu ez ziren, hala nola <a href="/wiki/Logika_matematiko" title="Logika matematiko">logika matematikoa</a> eta fundamentuak<a href="#cite_note-16">[16]</a>. <div class="mw-heading mw-heading3"><h3 id="Zenbakien_teoria">Zenbakien teoria</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=3" title="Aldatu atal hau: «Zenbakien teoria»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=3" title="Edit section's source code: Zenbakien teoria">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Zenbakien_teoria" title="Zenbakien teoria">Zenbakien teoria</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Spirale_Ulam_150.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/220px-Spirale_Ulam_150.jpg" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/330px-Spirale_Ulam_150.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/440px-Spirale_Ulam_150.jpg 2x" data-file-width="750" data-file-height="752" /></a><figcaption><a href="/w/index.php?title=Ulamen_espirala&action=edit&redlink=1" class="new" title="Ulamen espirala (sortu gabe)">Ulamen espiralak</a> <a href="/wiki/Zenbaki_lehen" title="Zenbaki lehen">zenbaki lehenen</a> distribuzioa erakusten du. <a href="/w/index.php?title=Hardy_eta_Littlewooden_F_aierua&action=edit&redlink=1" class="new" title="Hardy eta Littlewooden F aierua (sortu gabe)">Hardy eta Littlewooden F aieruak</a> diagonalen egiturari buruzko proposamen bat egiten du.</figcaption></figure> <a href="/wiki/Zenbakien_teoria" title="Zenbakien teoria">Zenbakien teoria</a> <a href="/wiki/Zenbaki" title="Zenbaki">zenbakien</a> manipulazioarekin hasi zen, hau da, <a href="/wiki/Zenbaki_arrunt" title="Zenbaki arrunt">zenbaki arruntekin</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df81a020f988536b900afeb19accc316516b1b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} )}">, eta geroago, <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">zenbaki osoetara</a> zabaldu zen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7c394740649bbc95dc603df09ceebaac387c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} )}"> eta <a href="/wiki/Zenbaki_arrazional" title="Zenbaki arrazional">zenbaki arrazionaletara</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a721495ab8dcae6b57e2b3e33402f94d57f7b9b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.617ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} )}">. Antzina, zenbakien teoriari <a href="/wiki/Aritmetika" title="Aritmetika">aritmetika</a> deitzen zitzaion, baina gaur egun termino hori batez ere <a href="/wiki/Zenbakizko_analisi" title="Zenbakizko analisi">zenbakizko kalkuluei</a> erreferentzia egiteko erabiltzen da. Zenbakien teoria <a href="/wiki/Babilonia" class="mw-redirect" title="Babilonia">Babilonia</a> zaharrean hasi zen, eta, seguru asko, <a href="/wiki/Txina" class="mw-redirect" title="Txina">Txinan</a> ere. Zenbakien bi teorialari garrantzitsu Greziako <a href="/wiki/Euklides" title="Euklides">Euklides</a> eta Alexandriako <a href="/wiki/Diofanto" class="mw-redirect" title="Diofanto">Diofanto</a> izan ziren. Zenbakien teoriaren forma abstraktuko azterketa modernoa, neurri handi batean, <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermati</a> eta <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Eulerri</a> egozten zaie. <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> eta <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gaussen</a> ekarpenekin lortu zuen garapen osoa. Erraz enuntziatzen diren zenbakizko problema askok metodo sofistikatuak behar dituzten soluzioak dituzte, askotan matematika guztietakoak. Adibide aipagarri bat <a href="/wiki/Fermaten_azken_teorema" title="Fermaten azken teorema">Fermaten azken teorema</a> da. Aieru hori 1637an enuntziatu zuen Pierre de Fermatek, baina 1994 arte ez zuen frogatu <a href="/wiki/Andrew_Wiles" class="mw-redirect" title="Andrew Wiles">Andrew Wilesek</a>, <a href="/wiki/Geometria_aljebraiko" title="Geometria aljebraiko">geometria aljebraikoaren</a> eskemen teoria, <a href="/wiki/Kategorien_teoria" title="Kategorien teoria">kategorien teoria</a> eta <a href="/wiki/Aljebra_homologiko" title="Aljebra homologiko">aljebra homologikoa</a> bezalako tresnak erabilita. Beste adibide bat <a href="/wiki/Goldbachen_aieru" title="Goldbachen aieru">Goldbachen aierua</a> da: 2 baino handiagoa den zenbaki oso bat bi zenbaki lehenen batura dela dio. <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbachek</a> 1742an enuntziatua, oraindik ez da frogatu ahalegin handiak egin arren. Zenbakien teoriak zenbait azpiarlo ditu, hala nola <a href="/wiki/Zenbakien_teoria_analitiko" class="mw-redirect" title="Zenbakien teoria analitiko">zenbakien teoria analitikoa</a>, <a href="/wiki/Zenbakien_teoria_aljebraiko" class="mw-redirect" title="Zenbakien teoria aljebraiko">zenbakien teoria aljebraikoa</a>, <a href="/w/index.php?title=Zenbakien_geometria&action=edit&redlink=1" class="new" title="Zenbakien geometria (sortu gabe)">zenbakien geometria</a> (metodoetan oinarritzen dena), <a href="/wiki/Ekuazio_diofantiko" class="mw-redirect" title="Ekuazio diofantiko">ekuazio diofantikoak</a> eta <a href="/w/index.php?title=Transzendentziaren_teoria&action=edit&redlink=1" class="new" title="Transzendentziaren teoria (sortu gabe)">transzendentziaren teoria</a> (problemei begira). <div class="mw-heading mw-heading3"><h3 id="Geometria">Geometria</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=4" title="Aldatu atal hau: «Geometria»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=4" title="Edit section's source code: Geometria">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Geometria" title="Geometria">Geometria</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Triangles_(spherical_geometry).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/220px-Triangles_%28spherical_geometry%29.jpg" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/330px-Triangles_%28spherical_geometry%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/440px-Triangles_%28spherical_geometry%29.jpg 2x" data-file-width="2489" data-file-height="2048" /></a><figcaption><a href="/wiki/Esfera" title="Esfera">Esfera</a> baten gainazalean, geometria euklidearra hurbilketa lokal gisa baino ez da aplikatzen. Eskala handiagoan, triangelu baten angeluen batura ez da 180º</figcaption></figure> <a href="/wiki/Geometria" title="Geometria">Geometria</a> matematikaren adarrik zaharrenetako bat da. Hasieran, <a href="/w/index.php?title=Forma_(geometria)&action=edit&redlink=1" class="new" title="Forma (geometria) (sortu gabe)">forma</a>, <a href="/wiki/Lerro_(geometria)" title="Lerro (geometria)">lerro</a>, <a href="/wiki/Angelu_(geometria)" title="Angelu (geometria)">angelu</a> eta <a href="/wiki/Zirkulu" title="Zirkulu">zirkuluei</a> buruzko errezeta enpirikoak egin ziren, batez ere <a href="/wiki/Topografia" title="Topografia">topografiaren</a> eta <a href="/wiki/Arkitektura" title="Arkitektura">arkitekturaren</a> beharretarako, baina ordutik beste azpieremu askotan loratu da<a href="#cite_note-:0-17">[17]</a>. Funtsezko berrikuntza bat <a href="/wiki/Antzinako_Grezia" title="Antzinako Grezia">Antzinako Grezian</a> <a href="/wiki/Froga_matematiko" title="Froga matematiko">froga</a> kontzeptua sartzea izan zen, baieztapen guztiak frogatzea eskatzen duena. Adibidez, ez da nahikoa neurketa bidez egiaztatzea bi <a href="/wiki/Luzera" class="mw-redirect" title="Luzera">luzera</a> berdinak direla; haien berdintasuna frogatu behar da aldez aurretik onartutako emaitzetatik (<a href="/wiki/Teorema" title="Teorema">teoremak</a>) eta oinarrizko baieztapen batzuetatik abiatuta. Oinarrizko enuntziatuak ez dira frogatu behar bistakoak direlako (<a href="/w/index.php?title=Postulatu&action=edit&redlink=1" class="new" title="Postulatu (sortu gabe)">postulatuak</a>) edo aztergaiaren definizioaren parte direlako (<a href="/wiki/Axioma" title="Axioma">axiomak</a>). Printzipio hori funtsezkoa zen matematika guztientzat, eta lehen aldiz egin zen geometriarako, eta Euklidesek sistematizatu zuen K.a. 300. urte inguruan <a href="/wiki/Euklidesen_Elementuak" title="Euklidesen Elementuak">Elementuak</a> liburuan<a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a>. <a href="/wiki/Geometria_euklidear" title="Geometria euklidear">Geometria euklidearra</a> <a href="/w/index.php?title=Plano_euklidear&action=edit&redlink=1" class="new" title="Plano euklidear (sortu gabe)">plano euklidearreko</a> (geometria laua) eta hiru dimentsioko <a href="/wiki/Espazio_euklidear" class="mw-redirect" title="Espazio euklidear">espazio euklidearreko</a> lerro, plano eta zirkuluetatik abiatuta eraikitako formak eta haien antolaerak aztertzea da<a href="#cite_note-:0-17">[17]</a>. Geometria euklidearra XVII. mendera arte garatu zen metodo- eta irismen-aldaketarik gabe, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartesek</a> gaur egun <a href="/wiki/Koordenatu_kartesiar" title="Koordenatu kartesiar">koordenatu kartesiar</a> deritzenak sartu zituenean. Horrek <a href="/w/index.php?title=Paradigma-aldaketa&action=edit&redlink=1" class="new" title="Paradigma-aldaketa (sortu gabe)">paradigma-aldaketa</a> garrantzitsu bat ekarri zuen: <a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">zenbaki errealak</a> <a href="/wiki/Zuzen-segmentu" class="mw-redirect" title="Zuzen-segmentu">zuzen-segmentuen</a> luzera gisa definitu ordez (ikus <a href="/wiki/Zuzen_erreal" title="Zuzen erreal">zuzen erreal</a>), <a href="/wiki/Koordenatu" class="mw-redirect" title="Koordenatu">koordenatuen</a> bidez puntuak adierazteko aukera eman zuen, zenbakiak baitira. <a href="/wiki/Aljebra" title="Aljebra">Aljebra</a> (eta geroago <a href="/wiki/Kalkulu_(matematikaren_adarra)" title="Kalkulu (matematikaren adarra)">kalkulua</a>) problema geometrikoak ebazteko erabil daiteke. Geometria bi azpieremutan banatu zen: <a href="/w/index.php?title=Geometria_sintetiko&action=edit&redlink=1" class="new" title="Geometria sintetiko (sortu gabe)">geometria sintetikoa</a>, metodo geometriko hutsak erabiltzen dituena, eta <a href="/wiki/Geometria_analitiko" title="Geometria analitiko">geometria analitikoa</a>, koordenatuak sistematikoki erabiltzen dituena<a href="#cite_note-20">[20]</a>. Geometria analitikoari esker, zirkuluekin eta zuzenekin zerikusirik ez duten <a href="/wiki/Kurba_(matematika)" title="Kurba (matematika)">kurbak</a> azter daitezke. Kurba horiek <a href="/wiki/Funtzio_(matematika)" title="Funtzio (matematika)">funtzioen</a> grafiko gisa defini daitezke, eta haien azterketak <a href="/wiki/Geometria_diferentzial" title="Geometria diferentzial">geometria diferentziala</a> ekarri zuen. <a href="/w/index.php?title=Ekuazio_inplizitu&action=edit&redlink=1" class="new" title="Ekuazio inplizitu (sortu gabe)">Ekuazio inplizitu</a> gisa ere defini daitezke, askotan <a href="/w/index.php?title=Ekuazio_polinomiko&action=edit&redlink=1" class="new" title="Ekuazio polinomiko (sortu gabe)">ekuazio polinomikoak</a> (<a href="/wiki/Geometria_aljebraiko" title="Geometria aljebraiko">geometria aljebraikoa</a> eragin zutenak). Geometria analitikoari esker, hiru dimentsiotik gorako espazio euklidearrak ere kontuan har daitezke<a href="#cite_note-:0-17">[17]</a>. XIX. mendean, matematikariek <a href="/wiki/Geometria_ez-euklidear" title="Geometria ez-euklidear">geometria ez-euklidearrak</a> aurkitu zituzten, <a href="/w/index.php?title=Paraleloen_postulatu&action=edit&redlink=1" class="new" title="Paraleloen postulatu (sortu gabe)">paraleloen postulatuari</a> jarraitzen ez diotenak. Postulatu horren egiazkotasuna zalantzan jartzean, uste izan da aurkikuntza hori bat zetorrela <a href="/wiki/Russellen_paradoxa" title="Russellen paradoxa">Russellen paradoxarekin</a>, matematikaren fundazio-krisia azaltzean. Krisiaren alderdi hori metodo axiomatikoa sistematizatuz ebatzi zen, eta aukeratutako axiomen egia ez dela problema matematiko bat hartu zen<a href="#cite_note-21">[21]</a>. Era berean, metodo axiomatikoari esker, lortutako geometriak azter daitezke, bai axiomak aldatuz, bai espazioaren eraldaketa jakin batzuen ondorioz <a href="/w/index.php?title=Inbariante&action=edit&redlink=1" class="new" title="Inbariante (sortu gabe)">aldatzen ez diren propietateak</a> kontuan hartuz<a href="#cite_note-22">[22]</a>. Gaur egun, honako azpieremuak daude geometriaren baitan: <ul><li><a href="/w/index.php?title=Geometria_proiektibo&action=edit&redlink=1" class="new" title="Geometria proiektibo (sortu gabe)">Geometria proiektiboak</a>, XVI. mendean <a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desarguesek</a> sortua, geometria euklidearra handitzen du, eta puntu batzuk gehitzen ditu <a href="/wiki/Infinitu" title="Infinitu">infinituan</a>, non zuzen paraleloak gurutzatzen baitiren. Horrek <a href="/w/index.php?title=Geometria_klasiko&action=edit&redlink=1" class="new" title="Geometria klasiko (sortu gabe)">geometria klasikoaren</a> alderdi asko sinplifikatzen ditu <a href="/wiki/Lerro_paralelo" class="mw-redirect" title="Lerro paralelo">lerro paraleloen</a> eta ebakiduren tratamendua bateratzean.</li> <li><a href="/w/index.php?title=Geometria_afin&action=edit&redlink=1" class="new" title="Geometria afin (sortu gabe)">Geometria afina</a>, paraleloei dagozkion propietateen azterketa eta luzeraren kontzeptuarekiko independenteak.</li> <li><a href="/wiki/Geometria_diferentzial" title="Geometria diferentzial">Geometria diferentziala</a>, kurben, azaleren eta haien orokortzeen azterketa, funtzio bereizgarrien bidez definitzen direnak.</li> <li><a href="/wiki/Barietate_(matematika)" title="Barietate (matematika)">Barietateen teoria</a>, nahitaez espazio handiago batean sartuta ez dauden formen azterketa.</li> <li><a href="/wiki/Riemannen_geometria" title="Riemannen geometria">Riemannen geometria</a>, espazio kurboetako distantzia-propietateen azterketa.</li> <li><a href="/wiki/Geometria_aljebraiko" title="Geometria aljebraiko">Geometria aljebraikoa</a>, kurben, azaleren eta haien orokortzeen azterketa, polinomioen bidez definitzen direnak.</li> <li><a href="/wiki/Topologia" title="Topologia">Topologia</a>, deformazio jarraituetan mantentzen diren propietateen azterketa. <ul><li><a href="/wiki/Topologia_aljebraiko" title="Topologia aljebraiko">Topologia aljebraikoa</a>, topologian metodo aljebraikoak erabiltzea, batez ere <a href="/wiki/Aljebra_homologiko" title="Aljebra homologiko">aljebra homologikoa</a>.</li></ul></li> <li><a href="/w/index.php?title=Geometria_diskretu&action=edit&redlink=1" class="new" title="Geometria diskretu (sortu gabe)">Geometria diskretua</a>, geometriako konfigurazio finituen azterketa.</li> <li><a href="/w/index.php?title=Geometria_konbexu&action=edit&redlink=1" class="new" title="Geometria konbexu (sortu gabe)">Geometria konbexua</a>, <a href="/wiki/Multzo_konbexu" title="Multzo konbexu">multzo konbexuen</a> azterketa, bere garrantzia hartzen duena optimizazioan dituzten aplikazioetatik.</li> <li><a href="/w/index.php?title=Geometria_konplexu&action=edit&redlink=1" class="new" title="Geometria konplexu (sortu gabe)">Geometria konplexua</a>, zenbaki errealak <a href="/wiki/Zenbaki_konplexu" title="Zenbaki konplexu">zenbaki konplexuekin</a> ordezkatuz lortzen den geometria.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Aljebra">Aljebra</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=5" title="Aldatu atal hau: «Aljebra»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=5" title="Edit section's source code: Aljebra">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Aljebra" title="Aljebra">Aljebra</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Quadratic_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/220px-Quadratic_formula.svg.png" decoding="async" width="220" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/330px-Quadratic_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/440px-Quadratic_formula.svg.png 2x" data-file-width="402" data-file-height="124" /></a><figcaption><a href="/w/index.php?title=Formula_koadratiko&action=edit&redlink=1" class="new" title="Formula koadratiko (sortu gabe)">Formula koadratikoa</a>, <a href="/wiki/Bigarren_mailako_ekuazio" title="Bigarren mailako ekuazio">bigarren mailako ekuazioak</a> ebazteko erabiltzen dena.</figcaption></figure> <a href="/wiki/Aljebra" title="Aljebra">Aljebra</a> <a href="/wiki/Ekuazio" title="Ekuazio">ekuazioak</a> eta formulak manipulatzeko artea da. <a href="/wiki/Diofanto" class="mw-redirect" title="Diofanto">Diofanto</a> (III. mendea) eta <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">al-Khwarizmi</a> (IX. mendea) izan ziren aljebraren bi aitzindari nagusiak<a href="#cite_note-23">[23]</a><a href="#cite_note-FOOTNOTEKleiner20073–5-24">[24]</a>. Diofantok zenbaki natural ezezagunek parte hartzen zuten ekuazio batzuk ebatzi zituen, erlazio berriak ondorioztatuz ebazpena lortu arte. Al-Khwarizmik ekuazioak eraldatzeko metodo sistematikoak sartu zituen, hala nola termino bat ekuazio batetik bestera eramatea. Aljebra terminoa al-jabr arabiar hitzetik dator, "zati hautsien bilera" esan nahi baitu<a href="#cite_note-25">[25]</a>, eta metodo horietako bat izendatzeko erabili zuen bere tratatu nagusiaren izenburuan: <a href="/wiki/Al-Kitab_al-Jabr_wa-l-Muqabala" title="Al-Kitab al-Jabr wa-l-Muqabala">Al-Kitab al-Jabr wa-l-Muqabala</a>. Aljebra <a href="/w/index.php?title=Fran%C3%A7ois_Vi%C3%A8tere&action=edit&redlink=1" class="new" title="François Viètere (sortu gabe)">François Vièterekin</a> (1540-1603) baino ez zen eremu berezi bihurtu, hark aldagaiak erabili zituen zenbaki ezezagunak edo zehaztu gabeak adierazteko<a href="#cite_note-26">[26]</a>. Aldagaien bidez, <a href="/wiki/Formula_(matematika)" title="Formula (matematika)">matematika-formulen</a> bidez irudikatutako zenbakiekin egin behar diren eragiketak deskriba ditzakete matematikariek. XIX. mendera arte, aljebra, nagusiki, <a href="/wiki/Ekuazio_lineal" title="Ekuazio lineal">ekuazio linealak</a> (gaur egun aljebra lineala) eta <a href="/w/index.php?title=Ekuazio_polinomiko&action=edit&redlink=1" class="new" title="Ekuazio polinomiko (sortu gabe)">ekuazio polinomikoak</a> ezezagun bakar batean aztertzean oinarritzen zen, <a href="/wiki/Ekuazio_aljebraiko" title="Ekuazio aljebraiko">ekuazio aljebraiko</a> deritzenak (oraindik erabiltzen den terminoa, anbiguoa izan arren). XIX. mendean, matematikariak aldagaiak erabiltzen hasi ziren zenbakiez bestelako gauzak irudikatzeko (hala nola <a href="/wiki/Matrize" title="Matrize">matrizeak</a>, <a href="/wiki/Aritmetika_modular" title="Aritmetika modular">aritmetika modularra</a> eta <a href="/w/index.php?title=Transformazio_geometriko&action=edit&redlink=1" class="new" title="Transformazio geometriko (sortu gabe)">transformazio geometrikoak</a>), eta, askotan, baliagarriak izaten dira eragiketa aritmetikoen orokortzeak. <a href="/wiki/Egitura_aljebraiko" title="Egitura aljebraiko">Egitura aljebraikoaren</a> kontzeptuak gai horri heltzen dio<a href="#cite_note-FOOTNOTEKleiner200779-101-27">[27]</a>. <a href="/wiki/Multzo" title="Multzo">Multzo</a> horren elementuak ez dira zehazten, multzoko elementuetan eragiten duten eragiketetan eta eragiketa horiek jarraitu behar dituzten arauetan. Hala, aljebraren eremua handitu egin zen, egitura aljebraikoak aztertu arte. Aljebraren objektu horri <a href="/w/index.php?title=Aljebra_moderno&action=edit&redlink=1" class="new" title="Aljebra moderno (sortu gabe)">aljebra modernoa</a> edo <a href="/wiki/Aljebra_abstraktu" title="Aljebra abstraktu">aljebra abstraktua</a> deitu zitzaion, <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noetherren</a> eraginak eta lanek ezarri bezala<a href="#cite_note-28">[28]</a> (azken termino hori batez ere hezkuntza-testuinguru batean agertzen da, aljebra elementalaren aurka, zeina formulak manipulatzeko modurik zaharrenaz arduratzen baita). Egitura aljebraiko mota batzuek propietate erabilgarriak dituzte, eta askotan funtsezkoak, matematikaren arlo askotan. Haren azterketa aljebraren zati autonomo bihurtu zen, eta hauek biltzen ditu: <ul><li><a href="/wiki/Talde-teoria" title="Talde-teoria">talde-teoria</a>;</li> <li><a href="/wiki/Eremuen_teoria" class="mw-redirect" title="Eremuen teoria">eremuen teoria</a>;</li> <li><a href="/wiki/Bektore-espazio" class="mw-redirect" title="Bektore-espazio">bektore-espazioak</a>, batez ere aljebra linealaren azterketa bera dutenak;</li> <li><a href="/wiki/Eraztunen_teoria" title="Eraztunen teoria">eraztunen teoria</a>;</li> <li><a href="/wiki/Aljebra_kommutatibo" class="mw-redirect" title="Aljebra kommutatibo">aljebra kommutatiboa</a> <a href="/w/index.php?title=Eraztun_kommutatibo&action=edit&redlink=1" class="new" title="Eraztun kommutatibo (sortu gabe)">eraztun kommutatiboen</a> azterketa da, polinomioen azterketa barne hartzen du eta geometria aljebraikoaren zati fundazionala da;</li> <li><a href="/wiki/Aljebra_homologiko" title="Aljebra homologiko">aljebra homologikoa</a>;</li> <li><a href="/w/index.php?title=Lieren_aljebra&action=edit&redlink=1" class="new" title="Lieren aljebra (sortu gabe)">Lieren aljebra</a> eta <a href="/w/index.php?title=Lieren_taldeen_teoria&action=edit&redlink=1" class="new" title="Lieren taldeen teoria (sortu gabe)">Lieren taldeen teoria</a>;</li> <li><a href="/wiki/Booleren_aljebra" title="Booleren aljebra">Booleren aljebra</a>, ordenagailuen egitura logikoa aztertzeko asko erabiltzen dena.</li></ul> <a href="/w/index.php?title=Aljebra_unibertsal&action=edit&redlink=1" class="new" title="Aljebra unibertsal (sortu gabe)">Aljebra unibertsalaren</a> eta <a href="/wiki/Kategorien_teoria" title="Kategorien teoria">kategorien teoriaren</a> helburua da egitura aljebraiko motak objektu matematiko gisa aztertzea<a href="#cite_note-29">[29]</a>. Azken hori egitura matematiko guztiei aplikatzen zaie (ez aljebraikoei bakarrik). Jatorrian, <a href="/wiki/Aljebra_homologiko" title="Aljebra homologiko">aljebra homologikoarekin</a> batera, <a href="/w/index.php?title=Objektu_ez-aljebraiko&action=edit&redlink=1" class="new" title="Objektu ez-aljebraiko (sortu gabe)">objektu ez-aljebraikoak</a>, hala nola <a href="/wiki/Espazio_topologiko" title="Espazio topologiko">espazio topologikoak</a>, aztertzeko aukera emateko sartu zen; aplikazio-eremu zehatz horri <a href="/wiki/Topologia_aljebraiko" title="Topologia aljebraiko">topologia aljebraikoa</a> deitzen zaio<a href="#cite_note-30">[30]</a>. <div class="mw-heading mw-heading3"><h3 id="Kalkulua_eta_analisia">Kalkulua eta analisia</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=6" title="Aldatu atal hau: «Kalkulua eta analisia»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=6" title="Edit section's source code: Kalkulua eta analisia">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Kalkulu_(matematikaren_adarra)" title="Kalkulu (matematikaren adarra)">Kalkulu (matematikaren adarra)</a>» eta «<a href="/wiki/Analisi_matematiko" title="Analisi matematiko">Analisi matematiko</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Cauchy_sequence_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/220px-Cauchy_sequence_illustration.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/330px-Cauchy_sequence_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/440px-Cauchy_sequence_illustration.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption><a href="/wiki/Cauchyren_segida" title="Cauchyren segida">Cauchyren segida</a>, arbitrarioki gertuago dauden elementuz osatua.</figcaption></figure> XVII. mendeko <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> eta <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> matematikariek modu independentean eta aldi berean sortu zuten <a href="/wiki/Kalkulu_infinitesimal" title="Kalkulu infinitesimal">kalkulu infinitesimala</a>. Funtsean, elkarren mende dauden aldagaien zerrenda aztertzea da<a href="#cite_note-31">[31]</a>. XVIII. mendean, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Eulerrek</a> handitu egin zuen kalkulua, funtzio kontzeptua eta beste emaitza asko sartuz. Gaur egun, "kalkulua" batez ere teoria horren oinarrizko zatiari dagokio, eta "analisia" eskuarki zati aurreratuetan erabiltzen da<a href="#cite_note-32">[32]</a>. Analisia, era berean, honela banatzen da: <a href="/wiki/Analisi_erreal" title="Analisi erreal">analisi erreala</a>, non aldagaiek zenbaki errealak adierazten baitituzte, eta <a href="/wiki/Analisi_konplexu" title="Analisi konplexu">analisi konplexua</a>, non aldagaiek zenbaki konplexuak adierazten baitituzte. Analisiak matematikaren beste arlo batzuek partekatutako azpiarlo asko biltzen ditu, besteak beste: <ul><li><a href="/wiki/Aldagai_anitzeko_kalkulu" class="mw-redirect" title="Aldagai anitzeko kalkulu">Aldagai anitzeko kalkulua</a></li> <li><a href="/wiki/Analisi_funtzional" title="Analisi funtzional">Analisi funtzionala</a>, non aldagaiek funtzio aldakorrak adierazten baitituzte;</li> <li><a href="/wiki/Integral" title="Integral">Integrazioa</a>, <a href="/w/index.php?title=Neurriaren_teoria&action=edit&redlink=1" class="new" title="Neurriaren teoria (sortu gabe)">neurriaren teoria</a> eta <a href="/w/index.php?title=Potentzialaren_teoria&action=edit&redlink=1" class="new" title="Potentzialaren teoria (sortu gabe)">potentzialaren teoria</a>; horiek guztiak probabilitatearen teoriarekin lotura estua dute;</li> <li><a href="/wiki/Ekuazio_diferentzial_arrunt" title="Ekuazio diferentzial arrunt">Ekuazio diferentzial arruntak</a>;</li> <li><a href="/wiki/Ekuazio_diferentzial_partzial" title="Ekuazio diferentzial partzial">Ekuazio diferentzial partzialak</a>;</li> <li><a href="/wiki/Zenbakizko_analisi" title="Zenbakizko analisi">Zenbakizko analisia</a>, aplikazio askotan sortzen diren ekuazio diferentzial arrunt eta partzialen soluzioak ordenagailuetan kalkulatzeari eskainia.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Matematika_diskretua">Matematika diskretua</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=7" title="Aldatu atal hau: «Matematika diskretua»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=7" title="Edit section's source code: Matematika diskretua">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Matematika_diskretu" title="Matematika diskretu">Matematika diskretu</a>»</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Markovkate_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/220px-Markovkate_01.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/330px-Markovkate_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/440px-Markovkate_01.svg.png 2x" data-file-width="563" data-file-height="563" /></a><figcaption>Bi-egoerako <a href="/wiki/Markoven_kate" class="mw-redirect" title="Markoven kate">Markoven kate</a> baten diagrama. Egoera bakoitzak 'A' edo 'E' letrak daramatza. Zenbakiak batetik bestera aldatzeko probabilitatea dira.</figcaption></figure> <a href="/wiki/Matematika_diskretu" title="Matematika diskretu">Matematika diskretuak</a>, zentzu zabalean, objektu matematiko indibidualen eta <a href="/wiki/Multzo_zenbakigarri" title="Multzo zenbakigarri">multzo zenbakigarrien</a> azterketa dira. Adibide bat zenbaki oso guztien multzoa da<a href="#cite_note-33">[33]</a>. Aztergaiak diskretuak direnez, kalkulu eta analisi matematikoko metodoak ez dira zuzenean aplikatzen. <a href="/wiki/Algoritmo" title="Algoritmo">Algoritmoek</a> –bereziki haien inplementazioa eta <a href="/w/index.php?title=Konplexutasun_konputazional&action=edit&redlink=1" class="new" title="Konplexutasun konputazional (sortu gabe)">konplexutasun konputazionala</a>– funtsezko zeregina dute matematika diskretuetan<a href="#cite_note-34">[34]</a>. <a href="/wiki/Lau_koloreen_teorema" title="Lau koloreen teorema">Lau koloreen teorema</a> eta <a href="/w/index.php?title=Esferen_paketatze_optimo&action=edit&redlink=1" class="new" title="Esferen paketatze optimo (sortu gabe)">esferen paketatze optimoa</a> XX. mendearen bigarren erdian ebatzitako matematika diskretuen problema nagusietako bi izan ziren<a href="#cite_note-35">[35]</a>. <a href="/wiki/P_vs_NP_problema" title="P vs NP problema">P vs NP problema</a>, gaur egun soluziorik gabe dagoena, matematika diskretuentzat ere garrantzitsua da, haren konponbideak arazo zail askori eragin baitiezaieke konputazioaren ikuspuntutik<a href="#cite_note-36">[36]</a>. Matematika diskretuen barruan sartzen dira: <ul><li><a href="/wiki/Konbinatoria" title="Konbinatoria">Konbinatoria</a>: emandako muga batzuk asetzen dituzten objektu matematikoak zerrendatzeko artea. Jatorrian, objektu horiek multzo jakin baten elementuak edo azpimultzoak ziren; hori hainbat objektutara zabaldu da, eta horrek lotura handia ezartzen du konbinatoriaren eta matematika diskretuaren beste zati batzuen artean. Adibidez, <a href="/w/index.php?title=Geometria_diskretu&action=edit&redlink=1" class="new" title="Geometria diskretu (sortu gabe)">geometria diskretuak</a> forma geometrikoen konfigurazioen zenbaketa barne hartzen du.</li> <li><a href="/wiki/Grafo_teoria" title="Grafo teoria">Grafoen</a> eta <a href="/w/index.php?title=Hipergrafo&action=edit&redlink=1" class="new" title="Hipergrafo (sortu gabe)">hipergrafoen</a> teoria</li> <li><a href="/w/index.php?title=Kodetzearen_teoria&action=edit&redlink=1" class="new" title="Kodetzearen teoria (sortu gabe)">Kodetzearen teoria</a>, <a href="/w/index.php?title=Erroreak_zuzentzeko_kode&action=edit&redlink=1" class="new" title="Erroreak zuzentzeko kode (sortu gabe)">erroreak zuzentzeko kodeak</a> eta <a href="/wiki/Kriptografia" title="Kriptografia">kriptografiaren</a> zati bat barne.</li> <li><a href="/w/index.php?title=Matroide&action=edit&redlink=1" class="new" title="Matroide (sortu gabe)">Matroideen</a> teoria</li> <li><a href="/w/index.php?title=Geometria_diskretu&action=edit&redlink=1" class="new" title="Geometria diskretu (sortu gabe)">Geometria diskretua</a></li> <li>Probabilitate-banaketa diskretuak</li> <li><a href="/wiki/Jokoen_teoria" class="mw-redirect" title="Jokoen teoria">Jokoen teoria</a> (joko jarraituak ere aztertzen diren arren, ohikoenak, <a href="/wiki/Xake" title="Xake">xakea</a> eta <a href="/wiki/Poker" title="Poker">pokerra</a> kasu, diskretuak dira)</li> <li><a href="/w/index.php?title=Optimizazio_diskretu&action=edit&redlink=1" class="new" title="Optimizazio diskretu (sortu gabe)">Optimizazio diskretua</a>, <a href="/w/index.php?title=Optimizazio_konbinatorio&action=edit&redlink=1" class="new" title="Optimizazio konbinatorio (sortu gabe)">optimizazio konbinatorioa</a>, <a href="/w/index.php?title=Programazio_oso&action=edit&redlink=1" class="new" title="Programazio oso (sortu gabe)">programazio osoa</a> eta <a href="/w/index.php?title=Murrizketadun_programazio&action=edit&redlink=1" class="new" title="Murrizketadun programazio (sortu gabe)">murrizketadun programazioa</a> barne.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Logika_matematikoa_eta_multzo-teoria">Logika matematikoa eta multzo-teoria</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=8" title="Aldatu atal hau: «Logika matematikoa eta multzo-teoria»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=8" title="Edit section's source code: Logika matematikoa eta multzo-teoria">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Logika_matematiko" title="Logika matematiko">Logika matematiko</a>» eta «<a href="/wiki/Multzo-teoria" title="Multzo-teoria">Multzo-teoria</a>»</div> <a href="/wiki/Logika_matematiko" title="Logika matematiko">Logika matematikoaren</a> bi gaiak eta <a href="/wiki/Multzo-teoria" title="Multzo-teoria">multzo-teoria</a> XIX. mendearen bukaeratik daude matematikan<a href="#cite_note-37">[37]</a><a href="#cite_note-38">[38]</a>. Garai horren aurretik, multzoak ez ziren objektu matematikotzat hartzen, eta <a href="/wiki/Logika" title="Logika">logika</a>, nahiz eta matematika-erakustaldietarako erabili, <a href="/wiki/Filosofia" title="Filosofia">filosofiarena</a> zen eta matematikariek ez zuten berariaz aztertzen<a href="#cite_note-39">[39]</a>. <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantorrek</a> <a href="/wiki/Multzo_infinitu" title="Multzo infinitu">multzo infinituak</a> aztertu baino lehen, matematikariek ez zituzten onartzen bilduma <a href="/w/index.php?title=Benetan_infinitu&action=edit&redlink=1" class="new" title="Benetan infinitu (sortu gabe)">benetan infinituak</a>, eta <a href="/wiki/Infinitu" title="Infinitu">infinitua</a> zenbaketa amaigabe baten emaitza zela uste zuten. Cantorren lanak mindu egin zituen matematikari asko, multzoak benetan infinitutzat jotzeagatik ez ezik<a href="#cite_note-40">[40]</a>, horrek infinituaren tamaina desberdinak dakartzala frogatzeagatik ere, <a href="/wiki/Cantorren_Argudio_Diagonala" title="Cantorren Argudio Diagonala">Cantorren Argudio Diagonalaren</a> arabera. Horrek Cantorren multzoen teoriari buruzko eztabaida ekarri zuen. Aldi berean, matematikaren zenbait arlotan ondorioztatu zen oinarrizko matematika-objektuen antzinako definizio intuitiboak ez zirela nahikoak matematika-zehaztasuna bermatzeko. Definizio intuitibo horien adibideak dira "multzo bat objektu-bilduma bat da", "zenbaki arrunta da zenbatzeko erabiltzen dena", "puntu bat zero luzera duen forma bat da norabide guztietan", "kurba bat da mugitzen ari den puntu batek utzitako arrastoa", etab. Hori matematikaren fundazio-krisi bihurtu zen<a href="#cite_note-41">[41]</a>. Denborarekin, matematikaren korronte nagusian ebatzi zen, metodo axiomatikoa sistematizatuz <a href="/w/index.php?title=Zermelo%E2%80%93Fraenkelen_axiomak&action=edit&redlink=1" class="new" title="Zermelo–Fraenkelen axiomak (sortu gabe)">multzoen teoria formalizatu</a> baten barruan. Oro har, objektu matematiko bakoitza antzeko objektu guztien multzoak eta objektu horiek izan behar dituzten propietateek definitzen dute. Adibidez, <a href="/w/index.php?title=Peanoren_aritmetika&action=edit&redlink=1" class="new" title="Peanoren aritmetika (sortu gabe)">Peanoren aritmetikan</a>, zenbaki arruntak honela definitzen dira: "zero zenbaki bat da", "zenbaki bakoitzak oinordeko bakarra du", "zenbaki bakoitzak zero izan ezik aurrekari bakarra du" eta arrazoitzeko arau batzuk<a href="#cite_note-42">[42]</a>. Errealitatearen <a href="/w/index.php?title=Abstrakzio_matematiko&action=edit&redlink=1" class="new" title="Abstrakzio matematiko (sortu gabe)">abstrakzio matematiko</a> hori <a href="/w/index.php?title=Formalismo&action=edit&redlink=1" class="new" title="Formalismo (sortu gabe)">formalismoaren</a> filosofia modernoan islatzen da, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbertek</a> 1910 inguruan sortu zuen bezala<a href="#cite_note-43">[43]</a>. Horrela definitutako objektuen "izaera" matematikariek filosofoen esku uzten duten problema filosofikoa da, nahiz eta matematikari askok izaera horri buruzko iritziak izan eta beren iritzia -batzuetan "<a href="/wiki/Intuizio" title="Intuizio">intuizioa</a>" deiturikoa- erabili haien azterketa eta frogak gidatzeko. Ikuspegiak aukera ematen du "logikoak" (hau da, kenketa-arau baimenduen multzoak), teoremak, probak eta abar kontuan hartzeko. objektu matematiko gisa, eta haiei buruzko teoremak frogatzea. Adibidez, <a href="/wiki/G%C3%B6delen_ez-osotasunaren_teoremak" title="Gödelen ez-osotasunaren teoremak">Gödelen ez-osotasunaren teoremek</a>, oro har, esaten dute ezen, zenbaki naturalak dituen sistema formal tinko orotan, badaude egiazkoak diren teoremak (hau da, sistema solidoago batean froga daitezkeenak), baina sistemaren barruan ezin direla frogatu<a href="#cite_note-44">[44]</a>. Matematikaren oinarrien ikuspegi hori zalantzan jarri zuten XX. mendearen lehen erdian <a href="/w/index.php?title=L._E._J._Brouwer&action=edit&redlink=1" class="new" title="L. E. J. Brouwer (sortu gabe)">Brouwer</a> buru zuten matematikariek, <a href="/w/index.php?title=Logika_intuizionista&action=edit&redlink=1" class="new" title="Logika intuizionista (sortu gabe)">logika intuizionista</a> sustatu baitzuten, ez baitzuen esplizituki <a href="/wiki/Hirugarrena_baztertzearen_printzipio" title="Hirugarrena baztertzearen printzipio">hirugarrena baztertzearen printzipio</a><a href="#cite_note-45">[45]</a><a href="#cite_note-46">[46]</a>. Problema eta eztabaida horiek <a href="/wiki/Logika_matematikoa" class="mw-redirect" title="Logika matematikoa">logika matematikoaaren</a> hedapen zabala ekarri zuten, azpieremu hauekin: <a href="/w/index.php?title=Ereduen_teoria&action=edit&redlink=1" class="new" title="Ereduen teoria (sortu gabe)">ereduen teoria</a> (teoria logiko batzuk beste teoria batzuen barruan modelatzea), <a href="/w/index.php?title=Probaren_teoria&action=edit&redlink=1" class="new" title="Probaren teoria (sortu gabe)">probaren teoria</a>, <a href="/w/index.php?title=Tipoen_teoria&action=edit&redlink=1" class="new" title="Tipoen teoria (sortu gabe)">tipoen teoria</a>, <a href="/wiki/Konputagarritasunaren_teoria" title="Konputagarritasunaren teoria">konputagarritasunaren teoria</a> eta <a href="/wiki/Konplexutasun_konputazionalaren_teoria" title="Konplexutasun konputazionalaren teoria">konplexutasun konputazionalaren teoria</a>. Nahiz eta logika matematikoaren alderdi horiek ordenagailuen gorakada baino lehen sartu ziren, <a href="/wiki/Konpiladore" title="Konpiladore">konpiladoreen</a> diseinuan, programen ziurtagirian, probetan parte hartu zutenek eta informatikaren beste alderdi batzuetan, teoria logiko horiek hedatzen lagundu zuten<a href="#cite_note-47">[47]</a>. <div class="mw-heading mw-heading3"><h3 id="Estatistika_eta_erabakiak_hartzearen_zientziak">Estatistika eta erabakiak hartzearen zientziak</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=9" title="Aldatu atal hau: «Estatistika eta erabakiak hartzearen zientziak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=9" title="Edit section's source code: Estatistika eta erabakiak hartzearen zientziak">aldatu iturburu kodea</a>]</div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Estatistika" title="Estatistika">Estatistika</a>» eta «<a href="/wiki/Probabilitate" title="Probabilitate">Probabilitate</a>»</div> <a href="/wiki/Estatistika" title="Estatistika">Estatistikaren</a> eremua <a href="/w/index.php?title=Datu-lagin&action=edit&redlink=1" class="new" title="Datu-lagin (sortu gabe)">datu-laginak</a> bildu eta tratatzeko erabiltzen den aplikazio matematikoa da, metodo matematikoetan oinarritutako prozedurak erabiliz, bereziki <a href="/wiki/Probabilitate" title="Probabilitate">probabilitatearen</a> teoria. Estatistikoek <a href="/wiki/Ausa" title="Ausa">ausazko</a> laginketak edo ausazko <a href="/wiki/Esperimentu" class="mw-redirect" title="Esperimentu">esperimentuak</a> dituzten datuak sortzen dituzte<a href="#cite_note-48">[48]</a>. Lagin edo esperimentu estatistiko baten diseinuak zehaztuko ditu erabiliko diren metodo analitikoak. <a href="/w/index.php?title=Behaketa-azterketeta&action=edit&redlink=1" class="new" title="Behaketa-azterketeta (sortu gabe)">Behaketa-azterketetatik</a> datozen datuen analisia <a href="/w/index.php?title=Eredu_estatistiko&action=edit&redlink=1" class="new" title="Eredu estatistiko (sortu gabe)">eredu estatistikoen</a> eta <a href="/wiki/Inferentzia" class="mw-disambig" title="Inferentzia">inferentziaren</a> teoriaren bidez egiten da, <a href="/w/index.php?title=Ereduen_hautapen&action=edit&redlink=1" class="new" title="Ereduen hautapen (sortu gabe)">ereduen hautapena</a> eta <a href="/wiki/Zenbatespen" title="Zenbatespen">zenbatespena</a> erabiliz. Ondoren, lortutako ereduak eta iragarpenak datu berriekin probatu behar dira. Teoria estatistikoak <a href="/w/index.php?title=Erabaki-problema&action=edit&redlink=1" class="new" title="Erabaki-problema (sortu gabe)">erabaki-problemak</a> aztertzen ditu, hala nola ekintza estatistiko baten <a href="/wiki/Arrisku" class="mw-redirect" title="Arrisku">arriskua</a> minimizatzea (<a href="/w/index.php?title=Espero_den_galera&action=edit&redlink=1" class="new" title="Espero den galera (sortu gabe)">espero den galera</a>), hala nola prozedura baten erabilera <a href="/w/index.php?title=Parametro-estimazio&action=edit&redlink=1" class="new" title="Parametro-estimazio (sortu gabe)">parametro-estimazioan</a>, <a href="/wiki/Hipotesi-proba" title="Hipotesi-proba">hipotesiak egiaztatzean</a> eta <a href="/w/index.php?title=Hautaketa_algoritmo&action=edit&redlink=1" class="new" title="Hautaketa algoritmo (sortu gabe)">onena hautatzean</a>. Estatistika matematikoaren eremu tradizional horietan, estatistika-erabakiaren problema bat formulatzeko, <a href="/w/index.php?title=Funtzio_objektibo&action=edit&redlink=1" class="new" title="Funtzio objektibo (sortu gabe)">funtzio objektibo</a> bat minimizatu behar da, hala nola itxarondako galera edo <a href="/wiki/Kostu" title="Kostu">kostua</a>, murrizketa espezifikoen pean. Adibidez, <a href="/wiki/Inkesta" title="Inkesta">inkesta</a> baten diseinuak <a href="/w/index.php?title=Konfiantza-maila&action=edit&redlink=1" class="new" title="Konfiantza-maila (sortu gabe)">konfiantza-maila</a> jakin bat duen <a href="/wiki/Populazio_(estatistika)" title="Populazio (estatistika)">populazioaren</a> <a href="/wiki/Batezbesteko" title="Batezbesteko">batezbestekoa</a> kalkulatzearen kostua minimizatzea eskatzen du<a href="#cite_note-49">[49]</a>. <a href="/wiki/Optimizazio_(matematika)" title="Optimizazio (matematika)">Optimizazioaren</a> erabilera dela eta, estatistikaren teoria matematikoa beste erabaki-zientzia batzuekin gainjartzen da, hala nola <a href="/wiki/Ikerketa_operatibo" class="mw-redirect" title="Ikerketa operatibo">ikerketa operatiboarekin</a>, <a href="/wiki/Kontrolaren_teoria" title="Kontrolaren teoria">kontrolaren teoriarekin</a> eta <a href="/wiki/Ekonomia_matematiko" title="Ekonomia matematiko">ekonomia matematikoarekin</a><a href="#cite_note-50">[50]</a>. <div class="mw-heading mw-heading3"><h3 id="Matematika_konputazionala">Matematika konputazionala</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=10" title="Aldatu atal hau: «Matematika konputazionala»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=10" title="Edit section's source code: Matematika konputazionala">aldatu iturburu kodea</a>]</div> <a href="/w/index.php?title=Matematika_konputazional&action=edit&redlink=1" class="new" title="Matematika konputazional (sortu gabe)">Matematika konputazionala</a> giza zenbaki-gaitasunerako handiegiak izan ohi diren problema matematikoen azterketa da<a href="#cite_note-51">[51]</a><a href="#cite_note-52">[52]</a>. Zenbakizko analisiak <a href="/wiki/Analisi_funtzional" title="Analisi funtzional">analisi funtzionalaren</a> eta <a href="/wiki/Hurbilketaren_teoria" class="mw-redirect" title="Hurbilketaren teoria">hurbilketaren teoriaren</a> bidez analisi-problemetarako metodoak aztertzen ditu; zenbakizko analisiak, oro har, hurbilketaren eta diskretizazioaren azterketa barne hartzen du, eta arreta berezia jartzen die <a href="/wiki/Biribiltze" title="Biribiltze">biribiltze</a>-akatsei<a href="#cite_note-53">[53]</a>. Zenbakizko analisiak eta, zabalago, konputazio zientifikoak matematika-zientziaren gai ez-analitikoak ere aztertzen dituzte, batez ere <a href="/w/index.php?title=Aljebra_numeriko_lineal&action=edit&redlink=1" class="new" title="Aljebra numeriko lineal (sortu gabe)">matrizeen</a> eta <a href="/wiki/Grafoen_teoria" class="mw-redirect" title="Grafoen teoria">grafoen teoria</a> algoritmikoa. Matematika konputazionalaren beste arlo batzuk <a href="/w/index.php?title=Aljebra_konputazional&action=edit&redlink=1" class="new" title="Aljebra konputazional (sortu gabe)">aljebra konputazionala</a> eta <a href="/w/index.php?title=Kalkulu_sinboliko&action=edit&redlink=1" class="new" title="Kalkulu sinboliko (sortu gabe)">kalkulu sinbolikoa</a> dira. <div class="mw-heading mw-heading2"><h2 id="Arlo_formal,_metodologiko_eta_estetikoak">Arlo formal, metodologiko eta estetikoak</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=11" title="Aldatu atal hau: «Arlo formal, metodologiko eta estetikoak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=11" title="Edit section's source code: Arlo formal, metodologiko eta estetikoak">aldatu iturburu kodea</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Notazioa,_hizkuntza_eta_arauak">Notazioa, hizkuntza eta arauak</h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=12" title="Aldatu atal hau: «Notazioa, hizkuntza eta arauak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=12" title="Edit section's source code: Notazioa, hizkuntza eta arauak">aldatu iturburu kodea</a>]</div> Gaur egun erabiltzen den notazio matematikoaren gehiengoa ez zen XVIII. mende arte asmatu. Hori baino lehen, matematikak hitzez idazten ziren, horrek matematikaren aurreratzea mugatzen zuen. XVIII. mendean, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>, gaur egun erabiltzen diren notazio askoren asmatzaile izan zen. Notazio modernoak asko errazten dizkie matematikak profesionalei, baina hasiberriei, ordea, zail egiten zaie. Notazioak matematikak murrizten ditu erabat, hainbat ikurrek informazio asko izatea egiten du. Notazio musikala bezala, notazio matematiko modernoak ere sintaxi zorrotz bat badu eta beste modu batera idaztea zaila izango litzatekeen informazioa kodifikatzen du. Hizkuntza matematikoa ere zaila izan daiteke hasiberrientzat. Edo eta soilik bezalako hitzek, egunerokotasuneko hizkuntzan duten esanahia baino zehatzagoa dute. Gainera, <a href="/wiki/Multzo_ireki" title="Multzo ireki">ireki</a> eta <a href="/wiki/Gorputz_(matematika)" title="Gorputz (matematika)">gorputz</a> esanahi matematiko oso zehatza dute. Jargoi matematikoak, edo hizkuntza matematikoak, termino teknikoak sartzen ditu homeomorfismoa eta integrazioa bezalakoak. Notazioa eta jargoia erabiltzeko arrazoia hizkuntza matematikoak egunerokotasuneko hizkuntzak baina zehaztasun gehiago behar duela da. Matematikoek hizkuntza eta logikaren zehaztasuna «zorroztasunatzat» jotzen dute. Zorroztasuna <a href="/wiki/Frogapen_matematiko" class="mw-redirect" title="Frogapen matematiko">frogapen matematiko</a> baten ezinbesteko baldintza da. Matematikoek haien teoremak axiomei bitartez arrazoiketa sistematiko bat jarraitzea nahi dute. Horrek teorema okerrak saihesteko balio du, intuizio hutseginkorretan oinarrituak izan direnak, historian zehar hainbatetan gertatu dena. Zorroztasun maila aldatzen joan da denboran zehar: grekoek argumentu zehatzak bilatzen zituzten, baina <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newtonen</a> garaian erabilitako metodoak ez ziren hain zorrotzak. Newtonek erabiltzen zituen definizioei datxikien arazoek analisi arduratsu eta XIX. mendeko frogapenak sustatu zituen. Gaur egun, matematikoak haien artean babesten jarraitzen dira ordenagailuz lagunduriko frogapenen bidez. <a href="/wiki/Axioma" title="Axioma">Axioma</a> bat tradizionalki «begi bistako egia» bezala interpretatzen da, baina ikuskera horrek arazoak ditu. Esparru formalean, axioma bat sinbolo kate bat besterik ez da, sistema axiomatiko batetik deribatu diren formula guztien testuinguruan berezko esanahia duena. <div class="mw-heading mw-heading2"><h2 id="Matematikari_ospetsuak"><a href="/wiki/Matematikari" title="Matematikari">Matematikari ospetsuak</a></h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=13" title="Aldatu atal hau: «Matematikari ospetsuak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=13" title="Edit section's source code: Matematikari ospetsuak">aldatu iturburu kodea</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Kapitolinischer_Pythagoras.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Kapitolinischer_Pythagoras.jpg/120px-Kapitolinischer_Pythagoras.jpg" decoding="async" width="120" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Kapitolinischer_Pythagoras.jpg/180px-Kapitolinischer_Pythagoras.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Kapitolinischer_Pythagoras.jpg/240px-Kapitolinischer_Pythagoras.jpg 2x" data-file-width="449" data-file-height="599" /></a><figcaption><a href="/wiki/Pitagoras" title="Pitagoras">Pitagoras</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Grezia_Klasikoa"><a href="/wiki/Grezia_klasikoa" title="Grezia klasikoa">Grezia Klasikoa</a></h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=14" title="Aldatu atal hau: «Grezia Klasikoa»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=14" title="Edit section's source code: Grezia Klasikoa">aldatu iturburu kodea</a>]</div> <ul><li><a href="/wiki/Tales" title="Tales">Tales Miletokoa</a></li> <li><a href="/wiki/Pitagoras" title="Pitagoras">Pitagoras</a></li> <li><a href="/wiki/Euklides" title="Euklides">Euklides</a></li> <li><a href="/wiki/Arkimedes" title="Arkimedes">Arkimedes</a></li> <li><a href="/wiki/Hipatia_Alexandriakoa" title="Hipatia Alexandriakoa">Hipatia</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Erdi_Aroa"><a href="/wiki/Erdi_Aroa" title="Erdi Aroa">Erdi Aroa</a></h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=15" title="Aldatu atal hau: «Erdi Aroa»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=15" title="Edit section's source code: Erdi Aroa">aldatu iturburu kodea</a>]</div> <a href="/wiki/V._mendea" title="V. mendea">V</a> <dl><dd><ul><li><a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a></li></ul></dd></dl> <a href="/wiki/XII._mendea" title="XII. mendea">XII</a> <dl><dd><ul><li><a href="/wiki/Bhaskara_II.a" title="Bhaskara II.a">Bhaskara II.a</a></li> <li><a href="/wiki/Leonardo_Pisano" title="Leonardo Pisano">Leonardo Fibonacci</a></li></ul></dd></dl> <a href="/wiki/XV._mendea" title="XV. mendea">XV</a> <dl><dd><ul><li><a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a></li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Aro_Berria"><a href="/wiki/Aro_Modernoa" title="Aro Modernoa">Aro Berria</a></h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=16" title="Aldatu atal hau: «Aro Berria»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=16" title="Edit section's source code: Aro Berria">aldatu iturburu kodea</a>]</div> <a href="/wiki/XVI._mendea" title="XVI. mendea">XVI</a> <dl><dd><ul><li><a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" title="Niccolò Fontana Tartaglia">Tartaglia</a></li> <li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a></li> <li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a></li></ul></dd></dl> <a href="/wiki/XVII._mendea" title="XVII. mendea">XVII</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Pierre_de_Fermat.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pierre_de_Fermat.jpg/120px-Pierre_de_Fermat.jpg" decoding="async" width="120" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pierre_de_Fermat.jpg/180px-Pierre_de_Fermat.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Pierre_de_Fermat.jpg/240px-Pierre_de_Fermat.jpg 2x" data-file-width="299" data-file-height="400" /></a><figcaption><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></figcaption></figure> <dl><dd><ul><li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Rene Descartes</a></li> <li><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jacob Bernoulli</a></li></ul></dd></dl> <a href="/wiki/XVIII._mendea" title="XVIII. mendea">XVIII</a> <dl><dd><ul><li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean D'Alembert</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph Louis Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sophie_Germain" title="Sophie Germain">Sophie Germain</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Johann Gauss</a></li> <li><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Aro_Garaikidea"><a href="/wiki/Aro_Garaikidea" title="Aro Garaikidea">Aro Garaikidea</a></h3>[<a href="/w/index.php?title=Matematika&veaction=edit&section=17" title="Aldatu atal hau: «Aro Garaikidea»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=17" title="Edit section's source code: Aro Garaikidea">aldatu iturburu kodea</a>]</div> <a href="/wiki/XIX._mendea" title="XIX. mendea">XIX</a> <dl><dd><ul><li><a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Evariste Galois</a></li> <li><a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Henri_L%C3%A9on_Lebesgue" title="Henri Léon Lebesgue">Henri Lebesgue</a></li></ul></dd></dl> <a href="/wiki/XX._mendea" title="XX. mendea">XX</a> <dl><dd><ul><li><a href="/wiki/John_Von_Neumann" class="mw-redirect" title="John Von Neumann">John Von Neumann</a></li> <li><a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a></li> <li><a href="/wiki/Edward_Lorenz" title="Edward Lorenz">Edward Lorenz</a></li> <li><a href="/wiki/John_Nash" class="mw-redirect" title="John Nash">John Nash</a></li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=18" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=18" title="Edit section's source code: Erreferentziak">aldatu iturburu kodea</a>]</div> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> (Ingelesez) <a rel="nofollow" class="external text" href="https://www.oed.com/start;jsessionid=F63D9062481407016B6DABAC98917982?authRejection=true&url=%2Fview%2FEntry%2F114974">«Home : Oxford English Dictionary»</a> www.oed.com (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> (Ingelesez) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tCXxf4vbXCcC&pg=PA4">Mathematical Logic. </a> Courier Dover Publications <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-486-15789-4" title="Berezi:BookSources/978-0-486-15789-4">978-0-486-15789-4</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> (Ingelesez) LaTorre, Donald R.; Kenelly, John W.; Reed, Iris B.; Carpenter, Laurel R.; Harris, Cynthia R.. (2011-01-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1Ebu2Tij4QsC&pg=PA2">Calculus Concepts: An Informal Approach to the Mathematics of Change. </a> Cengage Learning <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-1-4390-4957-0" title="Berezi:BookSources/978-1-4390-4957-0">978-1-4390-4957-0</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> (Ingelesez) Ramana. (2007-09-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XCRC6BeKhIIC&pg=SA2%E2%80%93PA10">Applied Mathematics. </a> McGraw-Hill Education (India) Pvt Limited <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-07-066753-2" title="Berezi:BookSources/978-0-07-066753-2">978-0-07-066753-2</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221105145638/https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof">«(PDF) Abstract Cognition and the Nature of Mathematical Proof»</a> web.archive.org 2022-11-05 (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-FOOTNOTEPeterson200112-6"><a href="#cite_ref-FOOTNOTEPeterson200112_6-0">↑</a> <a href="#CITEREFPeterson2001">Peterson 2001</a>, 12 orr. . </li> <li id="cite_note-7"><a href="#cite_ref-7">↑</a> (Ingelesez) Wigner, Eugene P.. (1960-02). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160130102">«The unreasonable effectiveness of mathematics in the natural sciences. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959»</a> Communications on Pure and Applied Mathematics 13 (1): 1–14.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002%2Fcpa.3160130102">10.1002/cpa.3160130102</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> <a rel="nofollow" class="external text" href="http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Wise/essay7/essay7.htm">«Euclidís Elements, by far his most famous and important work, is a comprehensive collection of the mathematical knowledge discovered by the classical Greeks, and thus represents a mathematical history of the age just prior to Euclid and the development o»</a> jwilson.coe.uga.edu (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> (Ingelesez) Alexander, Amir. (2011-09). <a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/10.1086/661620">«The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?»</a> Isis 102 (3): 475–480.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1086%2F661620">10.1086/661620</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0021-1753">0021-1753</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> Kleiner, Israel. (1991-12-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1080/0025570X.1991.11977625">«Rigor and Proof in Mathematics: A Historical Perspective»</a> Mathematics Magazine 64 (5): 291–314.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F0025570X.1991.11977625">10.1080/0025570X.1991.11977625</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0025-570X">0025-570X</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> (Ingelesez) Bell, E. T.. (2012-09-11). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4lTCAgAAQBAJ&pg=PA3">The Development of Mathematics. </a> Courier Corporation <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-486-15228-8" title="Berezi:BookSources/978-0-486-15228-8">978-0-486-15228-8</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-12"><a href="#cite_ref-12">↑</a> (Ingelesez) Tiwari, Sarju. (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-yyDJUD0DNwC&pg=PA27">Mathematics in History, Culture, Philosophy, and Science: From Ancient Times to Modern Age. </a> Mittal Publications <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-81-7099-404-6" title="Berezi:BookSources/978-81-7099-404-6">978-81-7099-404-6</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-13"><a href="#cite_ref-13">↑</a> (Ingelesez) Restivo, S.. (2013-12-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V0RuCQAAQBAJ&pg=PA14">Mathematics in Society and History: Sociological Inquiries. </a> Springer Science & Business Media <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-94-011-2944-2" title="Berezi:BookSources/978-94-011-2944-2">978-94-011-2944-2</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-14"><a href="#cite_ref-14">↑</a> (Ingelesez) Musielak, Dora. (2022-11-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZY2ZEAAAQBAJ&pg=PA2">Leonhard Euler and the Foundations of Celestial Mechanics. </a> Springer Nature <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-3-031-12322-1" title="Berezi:BookSources/978-3-031-12322-1">978-3-031-12322-1</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-15"><a href="#cite_ref-15">↑</a> (Ingelesez) Biggs, N. L. (1979-05-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/0315086079900740">«The roots of combinatorics»</a> Historia Mathematica 6 (2): 109–136.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2F0315-0860%2879%2990074-0">10.1016/0315-0860(79)90074-0</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0315-0860">0315-0860</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-16"><a href="#cite_ref-16">↑</a> <a rel="nofollow" class="external text" href="https://msc2020.org/">«MSC 2020»</a> msc2020.org (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-:0-17">↑ <a href="#cite_ref-:0_17-0">a</a> <a href="#cite_ref-:0_17-1">b</a> <a href="#cite_ref-:0_17-2">c</a> Straume, Eldar. (2014-09-03). <a rel="nofollow" class="external text" href="http://arxiv.org/abs/1409.1140">«A Survey of the Development of Geometry up to 1870»</a> arXiv:1409.1140 [math] (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-18"><a href="#cite_ref-18">↑</a> (Ingelesez) Hilbert, David. (1902). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0AA8AQAAMAAJ">The Foundations of Geometry. </a> Open Court Publishing Company (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-19"><a href="#cite_ref-19">↑</a> (Ingelesez) Hartshorne, Robin. (2013-11-11). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA9">Geometry: Euclid and Beyond. </a> Springer Science & Business Media <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-387-22676-7" title="Berezi:BookSources/978-0-387-22676-7">978-0-387-22676-7</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-20"><a href="#cite_ref-20">↑</a> (Ingelesez) Boyer, Carl B.. (2012-06-28). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2T4i5fXZbOYC&pg=PA74">History of Analytic Geometry. </a> Courier Corporation <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-486-15451-0" title="Berezi:BookSources/978-0-486-15451-0">978-0-486-15451-0</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-21"><a href="#cite_ref-21">↑</a> Stump, David J.. (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1162/posc_a_00532">«Reconstructing the Unity of Mathematics circa 1900»</a> Perspectives on Science 5 (3): 383–417.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1162%2Fposc_a_00532">10.1162/posc_a_00532</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/1063-6145">1063-6145</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-22"><a href="#cite_ref-22">↑</a> (Ingelesez) <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/">«Non-Euclidean geometry»</a> Maths History (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-23"><a href="#cite_ref-23">↑</a> (Ingelesez) Christianidis, Jean; Oaks, Jeffrey. (2013-05-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0315086012000742">«Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria»</a> Historia Mathematica 40 (2): 127–163.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.hm.2012.09.001">10.1016/j.hm.2012.09.001</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0315-0860">0315-0860</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-FOOTNOTEKleiner20073–5-24"><a href="#cite_ref-FOOTNOTEKleiner20073–5_24-0">↑</a> <a href="#CITEREFKleiner2007">Kleiner 2007</a>, 3–5 orr. . </li> <li id="cite_note-25"><a href="#cite_ref-25">↑</a> (Ingelesez) <a rel="nofollow" class="external text" href="https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas">«Where algebra got its x from, and Xmas its X»</a> South China Morning Post 2018-12-21 (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-26"><a href="#cite_ref-26">↑</a> (Ingelesez) Oaks, Jeffrey A.. (2018-05-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s00407-018-0208-0">«François Viète’s revolution in algebra»</a> Archive for History of Exact Sciences 72 (3): 245–302.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2Fs00407-018-0208-0">10.1007/s00407-018-0208-0</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/1432-0657">1432-0657</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-FOOTNOTEKleiner200779-101-27"><a href="#cite_ref-FOOTNOTEKleiner200779-101_27-0">↑</a> <a href="#CITEREFKleiner2007">Kleiner 2007</a>, 79-101 orr. . </li> <li id="cite_note-28"><a href="#cite_ref-28">↑</a> (Ingelesez) Corry, Leo. (2012-12-06). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8G0FCAAAQBAJ&pg=PA247">Modern Algebra and the Rise of Mathematical Structures. </a> Birkhäuser <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-3-0348-7917-0" title="Berezi:BookSources/978-3-0348-7917-0">978-3-0348-7917-0</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-29"><a href="#cite_ref-29">↑</a> (Ingelesez) Beziau, J. Y.; Costa-Leite, Alexandre. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZoRN9T1GUVwC&pg=PA3">Perspectives on Universal Logic. </a> Polimetrica s.a.s. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-88-7699-077-9" title="Berezi:BookSources/978-88-7699-077-9">978-88-7699-077-9</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-30"><a href="#cite_ref-30">↑</a> (Ingelesez) Krömer, Ralph. (2007-06-25). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=41bHxtHxjUAC&pg=PR20">Tool and Object: A History and Philosophy of Category Theory. </a> Springer Science & Business Media <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-3-7643-7524-9" title="Berezi:BookSources/978-3-7643-7524-9">978-3-7643-7524-9</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-31"><a href="#cite_ref-31">↑</a> Franklin, James. (2017-07). <a rel="nofollow" class="external text" href="http://scholarship.claremont.edu/jhm/vol7/iss2/18/">«Discrete and Continuous: A Fundamental Dichotomy in Mathematics»</a> Journal of Humanistic Mathematics 7 (2): 355–378.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.5642%2Fjhummath.201702.18">10.5642/jhummath.201702.18</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-32"><a href="#cite_ref-32">↑</a> (Ingelesez) Rosenstein, Joseph G.. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EvuQdO3h-DQC&pg=PA121">Discrete Mathematics in the Schools. </a> American Mathematical Soc. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-8218-8578-9" title="Berezi:BookSources/978-0-8218-8578-9">978-0-8218-8578-9</a>. (Noiz kontsultatua: 2023-05-14). </li> <li id="cite_note-33"><a href="#cite_ref-33">↑</a> Franklin, James. (2017-07-11). <a rel="nofollow" class="external text" href="https://scholarship.claremont.edu/jhm/vol7/iss2/18">«Discrete and Continuous: A Fundamental Dichotomy in Mathematics»</a> Journal of Humanistic Mathematics 7 (2): 355–378.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.5642%2Fjhummath.201702.18">10.5642/jhummath.201702.18</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/2159-8118">2159-8118</a>. (Noiz kontsultatua: 2023-05-22). </li> <li id="cite_note-34"><a href="#cite_ref-34">↑</a> (Ingelesez) Rosenstein, Joseph G.. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EvuQdO3h-DQC&pg=PA121">Discrete Mathematics in the Schools. </a> American Mathematical Soc. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-8218-8578-9" title="Berezi:BookSources/978-0-8218-8578-9">978-0-8218-8578-9</a>. (Noiz kontsultatua: 2023-05-22). </li> <li id="cite_note-35"><a href="#cite_ref-35">↑</a> (Ingelesez) Downey, Rod. (2014-05). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fYgaBQAAQBAJ&pg=PA260">Turing's Legacy: Developments from Turing's Ideas in Logic. </a> Cambridge University Press <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-1-107-04348-0" title="Berezi:BookSources/978-1-107-04348-0">978-1-107-04348-0</a>. (Noiz kontsultatua: 2023-05-22). </li> <li id="cite_note-36"><a href="#cite_ref-36">↑</a> Sipser, Michael. (1992-07-01). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/10.1145/129712.129771">«The history and status of the P versus NP question»</a> Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing (Association for Computing Machinery): 603–618.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1145%2F129712.129771">10.1145/129712.129771</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-89791-511-3" title="Berezi:BookSources/978-0-89791-511-3">978-0-89791-511-3</a>. (Noiz kontsultatua: 2023-05-22). </li> <li id="cite_note-37"><a href="#cite_ref-37">↑</a> Ferreirós, José. (2023). Zalta, Edward N. ed. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/sum2023/entries/settheory-early/">«The Early Development of Set Theory»</a> The Stanford Encyclopedia of Philosophy (Metaphysics Research Lab, Stanford University) (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-38"><a href="#cite_ref-38">↑</a> Ferreirós, José. (2023). Zalta, Edward N. ed. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/sum2023/entries/settheory-early/">«The Early Development of Set Theory»</a> The Stanford Encyclopedia of Philosophy (Metaphysics Research Lab, Stanford University) (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-39"><a href="#cite_ref-39">↑</a> (Ingelesez) Ferreirós, José. (2001-12). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/abs/road-to-modern-logican-interpretation/FA262E2F30A6DA5F9376D724C333A68A">«The Road to Modern Logic—An Interpretation»</a> Bulletin of Symbolic Logic 7 (4): 441–484.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F2687794">10.2307/2687794</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/1079-8986">1079-8986</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-40"><a href="#cite_ref-40">↑</a> (Ingelesez) Magazine, Natalie Wolchover,Quanta. <a rel="nofollow" class="external text" href="https://www.scientificamerican.com/article/infinity-logic-law/">«Dispute over Infinity Divides Mathematicians»</a> Scientific American (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-41"><a href="#cite_ref-41">↑</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221009074025/https://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf">«Wayback Machine»</a> web.archive.org 2022-10-09 (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-42"><a href="#cite_ref-42">↑</a> (Ingelesez) Hamilton, A. G.. (1982). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OXfmTHXvRXMC&pg=PA3">Numbers, Sets and Axioms: The Apparatus of Mathematics. </a> Cambridge University Press <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-521-28761-6" title="Berezi:BookSources/978-0-521-28761-6">978-0-521-28761-6</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-43"><a href="#cite_ref-43">↑</a> (Ingelesez) Snapper, Ernst. (1979-09-01). <a rel="nofollow" class="external text" href="http://www.jstor.org/stable/10.2307/2689412?origin=crossref">«The Three Crises in Mathematics: Logicism, Intuitionism and Formalism»</a> Mathematics Magazine 52 (4): 207.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F2689412">10.2307/2689412</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-44"><a href="#cite_ref-44">↑</a> Raattkainen, Panu. (2005-12-01). <a rel="nofollow" class="external text" href="https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm?ref=doi">«On the Philosophical Relevance of Godel's Incompleteness Theorems:»</a> Revue internationale de philosophie n° 234 (4): 513–534.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.3917%2Frip.234.0513">10.3917/rip.234.0513</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0048-8143">0048-8143</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-45"><a href="#cite_ref-45">↑</a> Moschovakis, Joan. (2022). Zalta, Edward N. ed. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/win2022/entries/logic-intuitionistic/">«Intuitionistic Logic»</a> The Stanford Encyclopedia of Philosophy (Metaphysics Research Lab, Stanford University) (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-46"><a href="#cite_ref-46">↑</a> (Ingelesez) McCarty, Charles. (2006-09-01). <a rel="nofollow" class="external text" href="https://journals.openedition.org/philosophiascientiae/411">«At the Heart of Analysis: Intuitionism and Philosophy»</a> Philosophia Scientiæ. Travaux d'histoire et de philosophie des sciences (CS 6): 81–94.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.4000%2Fphilosophiascientiae.411">10.4000/philosophiascientiae.411</a>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/1281-2463">1281-2463</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-47"><a href="#cite_ref-47">↑</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210303115643/https://www.cs.cmu.edu/~rwh/papers/unreasonable/basl.pdf">«Wayback Machine»</a> web.archive.org 2021-03-03 (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-48"><a href="#cite_ref-48">↑</a> Rao, Calyampudi Radhakrishna. (1999). Statistics and truth: putting chance to work. (2., enl. ed., reprinted. argitaraldia) World Scientific <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-981-02-3111-8" title="Berezi:BookSources/978-981-02-3111-8">978-981-02-3111-8</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-49"><a href="#cite_ref-49">↑</a> Arthanari, T. S.; Dodge, Yadolah. (1981). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalprog0000arth">Mathematical programming in statistics. </a> Wiley <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-471-08073-2" title="Berezi:BookSources/978-0-471-08073-2">978-0-471-08073-2</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-50"><a href="#cite_ref-50">↑</a> Kelly, F. P., ed. (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitystati0000unse_r1l1">Probability, statistics and optimization: a tribute to Peter Whittle. </a> Wiley <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-471-94829-2" title="Berezi:BookSources/978-0-471-94829-2">978-0-471-94829-2</a>. (Noiz kontsultatua: 2023-06-03). </li> <li id="cite_note-51"><a href="#cite_ref-51">↑</a> (Ingelesez) <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Extras/Computational_mathematics/">«G I Marchuk's plenary: ICM 1970»</a> Maths History (Noiz kontsultatua: 2023-05-23). </li> <li id="cite_note-52"><a href="#cite_ref-52">↑</a> (Ingelesez) Phua, Kang Hoh; Loe, Kia Fock. (1991-09-10). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jYNIDwAAQBAJ&pg=PA28">Singapore Supercomputing Conference '90: Supercomputing For Strategic Advantage. </a> World Scientific <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-981-4555-99-9" title="Berezi:BookSources/978-981-4555-99-9">978-981-4555-99-9</a>. (Noiz kontsultatua: 2023-05-23). </li> <li id="cite_note-53"><a href="#cite_ref-53">↑</a> Gowers, Timothy. (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/princetoncompani0000unse_i8p7">The Princeton companion to mathematics. </a> Princeton university press <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-691-11880-2" title="Berezi:BookSources/978-0-691-11880-2">978-0-691-11880-2</a>. (Noiz kontsultatua: 2023-06-03). </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=19" title="Aldatu atal hau: «Bibliografia»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=19" title="Edit section's source code: Bibliografia">aldatu iturburu kodea</a>]</div> <style data-mw-deduplicate="TemplateStyles:r8009455">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin" style=""> <ul><li> Bouleau, Nicolas. (1999). Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur. L'Harmattan <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-2-7384-8125-2" title="Berezi:BookSources/978-2-7384-8125-2">978-2-7384-8125-2</a>..</li> <li> Boyer, Carl Benjamin. (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/n3/mode/2up">A History of Mathematics. </a> (2nd. argitaraldia) <a href="/w/index.php?title=Wiley_(argitaletxea)&action=edit&redlink=1" class="new" title="Wiley (argitaletxea) (sortu gabe)">Wiley</a> <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-471-54397-8" title="Berezi:BookSources/978-0-471-54397-8">978-0-471-54397-8</a>..</li> <li> Eves, Howard. (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohi0000eves">An Introduction to the History of Mathematics. </a> (6th. argitaraldia) Saunders <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-03-029558-4" title="Berezi:BookSources/978-0-03-029558-4">978-0-03-029558-4</a>..</li> <li> Kleiner, Izraïl'. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RTLRBK-wj6wC&pg=PR1">A History of Abstract Algebra. </a> Springer Science & Business Media <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-8176-4684-4" title="Berezi:BookSources/978-0-8176-4684-4">978-0-8176-4684-4</a>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RTLRBK-wj6wC&pg=PR1">jatorrizkotik</a> artxibatua (artxibatze data: March 7, 2023).</li> <li> Kline, Morris. (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalthou00klin">Mathematical Thought from Ancient to Modern Times. </a> Oxford University Press <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-19-506135-2" title="Berezi:BookSources/978-0-19-506135-2">978-0-19-506135-2</a>..</li> <li> Monastyrsky, Michael. (2001). <a rel="nofollow" class="external text" href="http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf">Some Trends in Modern Mathematics and the Fields Medal. </a> Canadian Mathematical Society <a rel="nofollow" class="external text" href="http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf">jatorrizkotik</a> artxibatua (artxibatze data: August 13, 2006).</li> <li> Oakley, Barbara. (2014). <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780399165245">A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra). </a> Penguin Random House <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-399-16524-5" title="Berezi:BookSources/978-0-399-16524-5">978-0-399-16524-5</a>..</li> <li> Peirce, Benjamin. (1881). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=De0GAAAAYAAJ&q=Peirce+Benjamin+Linear+Associative+Algebra+&pg=PA1">Linear associative algebra. </a> (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph. argitaraldia)  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F2369153">10.2307/2369153</a>. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221">Eprint</a> and as an extract, D. Van Nostrand, 1882, Google <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_De0GAAAAYAAJ">Eprint</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=De0GAAAAYAAJ&q=Peirce+Benjamin+Linear+Associative+Algebra+&pg=PA1">jatorrizkotik</a> artxibatua (artxibatze data: March 31, 2021)..</li> <li> Peterson, Ivars. (2001). Mathematical Tourist, New and Updatad Snapshots of Modern Mathematics. Owl Books <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-8050-7159-7" title="Berezi:BookSources/978-0-8050-7159-7">978-0-8050-7159-7</a>..</li> <li> Popper, Karl R.. (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/insearchofbetter00karl">«On knowledge»</a> In Search of a Better World: Lectures and Essays from Thirty urteas. Routledge <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-415-13548-1" title="Berezi:BookSources/978-0-415-13548-1">978-0-415-13548-1</a>. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>: <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1992sbwl.book.....P">1992sbwl.book.....P</a>..</li> <li> Riehm, Carl. (August 2002). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200207/comm-riehm.pdf">The Early History of the Fields Medal. </a> <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200207/comm-riehm.pdf">jatorrizkotik</a> artxibatua (artxibatze data: October 26, 2006).</li> <li> Sevryuk, Mikhail B.. (January 2006). <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf">Book Reviews. </a>  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0273-0979-05-01069-4">10.1090/S0273-0979-05-01069-4</a>. <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf">jatorrizkotik</a> artxibatua (artxibatze data: July 23, 2006).</li> <li> Whittle, Peter. (1994). <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair">«Almost home»</a> Probability, statistics and optimisation: A Tribute to Peter Whittle. (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)". argitaraldia) John Wiley <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-471-94829-2" title="Berezi:BookSources/978-0-471-94829-2">978-0-471-94829-2</a>. <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair">jatorrizkotik</a> artxibatua (artxibatze data: December 19, 2013).</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2>[<a href="/w/index.php?title=Matematika&veaction=edit&section=20" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor">aldatu</a> | <a href="/w/index.php?title=Matematika&action=edit&section=20" title="Edit section's source code: Kanpo estekak">aldatu iturburu kodea</a>]</div> <div style="clear:both;"></div><style 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