Komplexní číslo – Wikipedie

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vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Zápis a související pojmy</span> </button> <ul id="toc-Zápis_a_související_pojmy-sublist" class="vector-toc-list"> <li id="toc-Značení" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Značení"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Značení</span> </div> </a> <ul id="toc-Značení-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Příklad" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Příklad"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Příklad</span> </div> </a> <ul id="toc-Příklad-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Důvody_pro_zavedení_komplexních_čísel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Důvody_pro_zavedení_komplexních_čísel"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Důvody pro zavedení komplexních čísel</span> </div> </a> <button aria-controls="toc-Důvody_pro_zavedení_komplexních_čísel-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Důvody pro zavedení komplexních čísel</span> </button> <ul id="toc-Důvody_pro_zavedení_komplexních_čísel-sublist" class="vector-toc-list"> <li id="toc-Historie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historie"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Historie</span> </div> </a> <ul id="toc-Historie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matematická_motivace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matematická_motivace"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Matematická motivace</span> </div> </a> <ul id="toc-Matematická_motivace-sublist" class="vector-toc-list"> <li id="toc-Příklad_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Příklad_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Příklad</span> </div> </a> <ul id="toc-Příklad_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Technické_aplikace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Technické_aplikace"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Technické aplikace</span> </div> </a> <ul id="toc-Technické_aplikace-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operace_s_komplexními_čísly" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operace_s_komplexními_čísly"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operace s komplexními čísly</span> </div> </a> <button aria-controls="toc-Operace_s_komplexními_čísly-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Operace s komplexními čísly</span> </button> <ul id="toc-Operace_s_komplexními_čísly-sublist" class="vector-toc-list"> <li id="toc-Algebraický_tvar_komplexních_čísel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraický_tvar_komplexních_čísel"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Algebraický tvar komplexních čísel</span> </div> </a> <ul id="toc-Algebraický_tvar_komplexních_čísel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometrické_znázornění_komplexních_čísel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometrické_znázornění_komplexních_čísel"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Geometrické znázornění komplexních čísel</span> </div> </a> <ul id="toc-Geometrické_znázornění_komplexních_čísel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Goniometrický_tvar_komplexních_čísel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Goniometrický_tvar_komplexních_čísel"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Goniometrický tvar komplexních čísel</span> </div> </a> <ul id="toc-Goniometrický_tvar_komplexních_čísel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Komplexní_funkce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Komplexní_funkce"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Komplexní funkce</span> </div> </a> <ul id="toc-Komplexní_funkce-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Základní_vlastnosti_tělesa_komplexních_čísel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Základní_vlastnosti_tělesa_komplexních_čísel"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Základní vlastnosti tělesa komplexních čísel</span> </div> </a> <ul id="toc-Základní_vlastnosti_tělesa_komplexních_čísel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definice_pomocí_uspořádaných_dvojic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definice_pomocí_uspořádaných_dvojic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Definice pomocí uspořádaných dvojic</span> </div> </a> <ul id="toc-Definice_pomocí_uspořádaných_dvojic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reprezentace_maticí" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reprezentace_maticí"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Reprezentace maticí</span> </div> </a> <ul id="toc-Reprezentace_maticí-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Literatura" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Literatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Literatura</span> </div> </a> <ul id="toc-Literatura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-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="Obsah" 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="Přepnout obsah" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Přepnout obsah</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Komplexní číslo</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Přejděte k článku v jiném jazyce. Je dostupný v 132 jazycích" > <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-132" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">132 jazyků</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Komplekse_getal" title="Komplekse getal – afrikánština" lang="af" hreflang="af" data-title="Komplekse getal" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Komplexe Zahl" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%A0%E1%89%85%E1%8C%A3%E1%8C%AB_%E1%89%81%E1%8C%A5%E1%88%AD" title="የአቅጣጫ ቁጥር – amharština" lang="am" hreflang="am" data-title="የአቅጣጫ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="amharština" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_complexo" title="Numero complexo – aragonština" lang="an" hreflang="an" data-title="Numero complexo" data-language-autonym="Aragonés" data-language-local-name="aragonština" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – angika" lang="anp" hreflang="anp" data-title="समिश्र संख्या" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%B1%D9%83%D8%A8" title="عدد مركب – arabština" lang="ar" hreflang="ar" data-title="عدد مركب" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – ásámština" lang="as" hreflang="as" data-title="জটিল সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="ásámština" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_complexu" title="Númberu complexu – asturština" lang="ast" hreflang="ast" data-title="Númberu complexu" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kompleks_%C9%99d%C9%99dl%C9%99r" title="Kompleks ədədlər – ázerbájdžánština" lang="az" hreflang="az" data-title="Kompleks ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D9%85%D9%BE%D9%84%DA%A9%D8%B3_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="کومپلکس ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="کومپلکس ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D2%BB%D0%B0%D0%BD" title="Комплекслы һан – baškirština" lang="ba" hreflang="ba" data-title="Комплекслы һан" data-language-autonym="Башҡортса" data-language-local-name="baškirština" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Kuompleks%C4%97nis_skaitlios" title="Kuompleksėnis skaitlios – žemaitština" lang="sgs" hreflang="sgs" data-title="Kuompleksėnis skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="žemaitština" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – běloruština" lang="be" hreflang="be" data-title="Камплексны лік" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Камплексны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексно число – bulharština" lang="bg" hreflang="bg" data-title="Комплексно число" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – bengálština" lang="bn" hreflang="bn" data-title="জটিল সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – bosenština" lang="bs" hreflang="bs" data-title="Kompleksan broj" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Комплекс тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_complex" title="Nombre complex – katalánština" lang="ca" hreflang="ca" data-title="Nombre complex" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%A6%D8%A7%D9%88%DB%8E%D8%AA%DB%95" title="ژمارەی ئاوێتە – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="ژمارەی ئاوێتە" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Комплекслă хисеп – čuvaština" lang="cv" hreflang="cv" data-title="Комплекслă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cymhlyg" title="Rhif cymhlyg – velština" lang="cy" hreflang="cy" data-title="Rhif cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="velština" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="doporučený článek"><a href="https://da.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – dánština" lang="da" hreflang="da" data-title="Komplekse tal" data-language-autonym="Dansk" data-language-local-name="dánština" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – němčina" lang="de" hreflang="de" data-title="Komplexe Zahl" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_kompleks" title="Amaro kompleks – Zazaki" lang="diq" hreflang="diq" data-title="Amaro kompleks" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Μιγαδικός αριθμός – řečtina" lang="el" hreflang="el" data-title="Μιγαδικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_cumpl%C3%AAs" title="Nómmer cumplês – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer cumplês" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complex_number" title="Complex number – angličtina" lang="en" hreflang="en" data-title="Complex number" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_nombro" title="Kompleksa nombro – esperanto" lang="eo" hreflang="eo" data-title="Kompleksa nombro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_complejo" title="Número complejo – španělština" lang="es" hreflang="es" data-title="Número complejo" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – estonština" lang="et" hreflang="et" data-title="Kompleksarv" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_konplexu" title="Zenbaki konplexu – baskičtina" lang="eu" hreflang="eu" data-title="Zenbaki konplexu" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%AE%D8%AA%D9%84%D8%B7" title="عدد مختلط – perština" lang="fa" hreflang="fa" data-title="عدد مختلط" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kompleksiluku" title="Kompleksiluku – finština" lang="fi" hreflang="fi" data-title="Kompleksiluku" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – võruština" lang="vro" hreflang="vro" data-title="Kompleksarv" data-language-autonym="Võro" data-language-local-name="võruština" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Komplekst_tal" title="Komplekst tal – faerština" lang="fo" hreflang="fo" data-title="Komplekst tal" data-language-autonym="Føroyskt" data-language-local-name="faerština" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_complexe" title="Nombre complexe – francouzština" lang="fr" hreflang="fr" data-title="Nombre complexe" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kompleks_taal" title="Kompleks taal – fríština (severní)" lang="frr" hreflang="frr" data-title="Kompleks taal" data-language-autonym="Nordfriisk" data-language-local-name="fríština (severní)" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kompleks_getal" title="Kompleks getal – fríština (západní)" lang="fy" hreflang="fy" data-title="Kompleks getal" data-language-autonym="Frysk" data-language-local-name="fríština (západní)" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_choimpl%C3%A9ascach" title="Uimhir choimpléascach – irština" lang="ga" hreflang="ga" data-title="Uimhir choimpléascach" data-language-autonym="Gaeilge" data-language-local-name="irština" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – čínština (dialekty Gan)" lang="gan" hreflang="gan" data-title="複數" data-language-autonym="贛語" data-language-local-name="čínština (dialekty Gan)" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_kompleks" title="Nonm kompleks – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm kompleks" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – galicijština" lang="gl" hreflang="gl" data-title="Número complexo" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapy_rypy%27%C5%A9" title="Papapy rypy'ũ – guaranština" lang="gn" hreflang="gn" data-title="Papapy rypy'ũ" data-language-autonym="Avañe'ẽ" data-language-local-name="guaranština" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%95%E0%AA%B0_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE%E0%AA%93" title="સંકર સંખ્યાઓ – gudžarátština" lang="gu" hreflang="gu" data-title="સંકર સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="gudžarátština" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A8%D7%95%D7%9B%D7%91" title="מספר מרוכב – hebrejština" lang="he" hreflang="he" data-title="מספר מרוכב" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – hindština" lang="hi" hreflang="hi" data-title="समिश्र संख्या" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Jatil_ginti" title="Jatil ginti – hindština (Fidži)" lang="hif" hreflang="hif" data-title="Jatil ginti" data-language-autonym="Fiji Hindi" data-language-local-name="hindština (Fidži)" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kompleksni_broj" title="Kompleksni broj – chorvatština" lang="hr" hreflang="hr" data-title="Kompleksni broj" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Komplex_sz%C3%A1mok" title="Komplex számok – maďarština" lang="hu" hreflang="hu" data-title="Komplex számok" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A9%D5%AB%D5%BE" title="Կոմպլեքս թիվ – arménština" lang="hy" hreflang="hy" data-title="Կոմպլեքս թիվ" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_complexe" title="Numero complexe – interlingua" lang="ia" hreflang="ia" data-title="Numero complexe" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_kompleks" title="Lumur kompleks – ibanština" lang="iba" hreflang="iba" data-title="Lumur kompleks" data-language-autonym="Jaku Iban" data-language-local-name="ibanština" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_kompleks" title="Bilangan kompleks – indonéština" lang="id" hreflang="id" data-title="Bilangan kompleks" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Komplexa_nombro" title="Komplexa nombro – ido" lang="io" hreflang="io" data-title="Komplexa nombro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tvinnt%C3%B6lur" title="Tvinntölur – islandština" lang="is" hreflang="is" data-title="Tvinntölur" data-language-autonym="Íslenska" data-language-local-name="islandština" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_complesso" title="Numero complesso – italština" lang="it" hreflang="it" data-title="Numero complesso" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E6%95%B0" title="複素数 – japonština" lang="ja" hreflang="ja" data-title="複素数" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Komplex_nomba" title="Komplex nomba – jamajská kreolština" lang="jam" hreflang="jam" data-title="Komplex nomba" data-language-autonym="Patois" data-language-local-name="jamajská kreolština" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/relcimdyna%27u" title="relcimdyna'u – lojban" lang="jbo" hreflang="jbo" data-title="relcimdyna'u" data-language-autonym="La .lojban." data-language-local-name="lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A1%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="კომპლექსური რიცხვი – gruzínština" lang="ka" hreflang="ka" data-title="კომპლექსური რიცხვი" data-language-autonym="ქართული" data-language-local-name="gruzínština" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Am%E1%B8%8Dan_asemlal" title="Amḍan asemlal – kabylština" lang="kab" hreflang="kab" data-title="Amḍan asemlal" data-language-autonym="Taqbaylit" data-language-local-name="kabylština" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Nd%C9%A9_nd%C9%A9_%C3%B1%CA%8A%C5%8B" title="Ndɩ ndɩ ñʊŋ – Kabiye" lang="kbp" hreflang="kbp" data-title="Ndɩ ndɩ ñʊŋ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%88%D0%B5%D0%BD_%D1%81%D0%B0%D0%BD" title="Кешен сан – kazaština" lang="kk" hreflang="kk" data-title="Кешен сан" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%80%E1%9E%BB%E1%9F%86%E1%9E%95%E1%9F%92%E1%9E%9B%E1%9E%B7%E1%9E%85" title="ចំនួនកុំផ្លិច – khmérština" lang="km" hreflang="km" data-title="ចំនួនកុំផ្លិច" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmérština" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수 – korejština" lang="ko" hreflang="ko" data-title="복소수" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niver_kompleth" title="Niver kompleth – kornština" lang="kw" hreflang="kw" data-title="Niver kompleth" data-language-autonym="Kernowek" data-language-local-name="kornština" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D1%82%D2%AF%D2%AF_%D1%81%D0%B0%D0%BD" title="Комплекстүү сан – kyrgyzština" lang="ky" hreflang="ky" data-title="Комплекстүү сан" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_complexus" title="Numerus complexus – latina" lang="la" hreflang="la" data-title="Numerus complexus" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Complex_getal" title="Complex getal – limburština" lang="li" hreflang="li" data-title="Complex getal" data-language-autonym="Limburgs" data-language-local-name="limburština" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_compless" title="Numer compless – lombardština" lang="lmo" hreflang="lmo" data-title="Numer compless" data-language-autonym="Lombard" data-language-local-name="lombardština" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AA%E0%BA%BB%E0%BA%99" title="ຈຳນວນສົນ – laoština" lang="lo" hreflang="lo" data-title="ຈຳນວນສົນ" data-language-autonym="ລາວ" data-language-local-name="laoština" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kompleksinis_skai%C4%8Dius" title="Kompleksinis skaičius – litevština" lang="lt" hreflang="lt" data-title="Kompleksinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komplekss_skaitlis" title="Komplekss skaitlis – lotyština" lang="lv" hreflang="lv" data-title="Komplekss skaitlis" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_haro" title="Isa haro – malgaština" lang="mg" hreflang="mg" data-title="Isa haro" data-language-autonym="Malagasy" data-language-local-name="malgaština" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексен број – makedonština" lang="mk" hreflang="mk" data-title="Комплексен број" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B0%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="മിശ്രസംഖ്യ – malajálamština" lang="ml" hreflang="ml" data-title="മിശ്രസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="malajálamština" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – mongolština" lang="mn" hreflang="mn" data-title="Комплекс тоо" data-language-autonym="Монгол" data-language-local-name="mongolština" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संमिश्र संख्या – maráthština" lang="mr" hreflang="mr" data-title="संमिश्र संख्या" data-language-autonym="मराठी" data-language-local-name="maráthština" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_kompleks" title="Nombor kompleks – malajština" lang="ms" hreflang="ms" data-title="Nombor kompleks" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%94%E1%80%BA%E1%80%95%E1%80%9C%E1%80%80%E1%80%BA%E1%80%85%E1%80%BA%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="ကွန်ပလက်စ်ကိန်း – barmština" lang="my" hreflang="my" data-title="ကွန်ပလက်စ်ကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="barmština" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Komplexe_Tall" title="Komplexe Tall – dolnoněmčina" lang="nds" hreflang="nds" data-title="Komplexe Tall" data-language-autonym="Plattdüütsch" data-language-local-name="dolnoněmčina" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Complex_getal" title="Complex getal – nizozemština" lang="nl" hreflang="nl" data-title="Complex getal" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Komplekse tal" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Komplekst_tall" title="Komplekst tall – norština (bokmål)" lang="nb" hreflang="nb" data-title="Komplekst tall" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_compl%C3%A8xe" title="Nombre complèxe – okcitánština" lang="oc" hreflang="oc" data-title="Nombre complèxe" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Xaxxamaa" title="Lakkoofsa Xaxxamaa – oromština" lang="om" hreflang="om" data-title="Lakkoofsa Xaxxamaa" data-language-autonym="Oromoo" data-language-local-name="oromština" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BE%D0%BD_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Комплексон нымæц – osetština" lang="os" hreflang="os" data-title="Комплексон нымæц" data-language-autonym="Ирон" data-language-local-name="osetština" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AA%E0%A8%B2%E0%A9%88%E0%A8%95%E0%A8%B8_%E0%A8%A8%E0%A9%B0%E0%A8%AC%E0%A8%B0" title="ਕੰਪਲੈਕਸ ਨੰਬਰ – paňdžábština" lang="pa" hreflang="pa" data-title="ਕੰਪਲੈਕਸ ਨੰਬਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="paňdžábština" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_zespolone" title="Liczby zespolone – polština" lang="pl" hreflang="pl" data-title="Liczby zespolone" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_compless" title="Nùmer compless – piemonština" lang="pms" hreflang="pms" data-title="Nùmer compless" data-language-autonym="Piemontèis" data-language-local-name="piemonština" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%85%D9%BE%D9%84%DB%8C%DA%A9%D8%B3_%D9%86%D9%85%D8%A8%D8%B1" title="کمپلیکس نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کمپلیکس نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – portugalština" lang="pt" hreflang="pt" data-title="Número complexo" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_complex" title="Număr complex – rumunština" lang="ro" hreflang="ro" data-title="Număr complex" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексное число – ruština" lang="ru" hreflang="ru" data-title="Комплексное число" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D1%96%D1%81%D0%BB%D0%BE" title="Комплексне чісло – rusínština" lang="rue" hreflang="rue" data-title="Комплексне чісло" data-language-autonym="Русиньскый" data-language-local-name="rusínština" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D0%B0%D1%85%D1%81%D0%B0%D0%B0%D0%BD" title="Комплекс ахсаан – jakutština" lang="sah" hreflang="sah" data-title="Комплекс ахсаан" data-language-autonym="Саха тыла" data-language-local-name="jakutština" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_cumplessu" title="Nùmmuru cumplessu – sicilština" lang="scn" hreflang="scn" data-title="Nùmmuru cumplessu" data-language-autonym="Sicilianu" data-language-local-name="sicilština" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Complex_nummer" title="Complex nummer – skotština" lang="sco" hreflang="sco" data-title="Complex nummer" data-language-autonym="Scots" data-language-local-name="skotština" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – srbochorvatština" lang="sh" hreflang="sh" data-title="Kompleksan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9A%E0%B7%93%E0%B6%BB%E0%B7%8A%E0%B6%AB_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="සංකීර්ණ සංඛ්‍යා – sinhálština" lang="si" hreflang="si" data-title="සංකීර්ණ සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="sinhálština" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Complex_number" title="Complex number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Complex number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Komplexn%C3%A9_%C4%8D%C3%ADslo" title="Komplexné číslo – slovenština" lang="sk" hreflang="sk" data-title="Komplexné číslo" data-language-autonym="Slovenčina" data-language-local-name="slovenština" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kompleksno_%C5%A1tevilo" title="Kompleksno število – slovinština" lang="sl" hreflang="sl" data-title="Kompleksno število" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kompleksloho" title="Kompleksloho – sámština (inarijská)" lang="smn" hreflang="smn" data-title="Kompleksloho" data-language-autonym="Anarâškielâ" data-language-local-name="sámština (inarijská)" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_kakan" title="Thiin kakan – somálština" lang="so" hreflang="so" data-title="Thiin kakan" data-language-autonym="Soomaaliga" data-language-local-name="somálština" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_kompleks%C3%AB" title="Numrat kompleksë – albánština" lang="sq" hreflang="sq" data-title="Numrat kompleksë" data-language-autonym="Shqip" data-language-local-name="albánština" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексан број – srbština" lang="sr" hreflang="sr" data-title="Комплексан број" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komplexa_tal" title="Komplexa tal – švédština" lang="sv" hreflang="sv" data-title="Komplexa tal" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_changamano" title="Namba changamano – svahilština" lang="sw" hreflang="sw" data-title="Namba changamano" data-language-autonym="Kiswahili" data-language-local-name="svahilština" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%86%E0%AE%A3%E0%AF%8D" title="சிக்கலெண் – tamilština" lang="ta" hreflang="ta" data-title="சிக்கலெண்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%82%E0%B0%95%E0%B1%80%E0%B0%B0%E0%B1%8D%E0%B0%A3_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="సంకీర్ణ సంఖ్యలు – telugština" lang="te" hreflang="te" data-title="సంకీర్ణ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="telugština" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D3%A3" title="Адади комплексӣ – tádžičtina" lang="tg" hreflang="tg" data-title="Адади комплексӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tádžičtina" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99" title="จำนวนเชิงซ้อน – thajština" lang="th" hreflang="th" data-title="จำนวนเชิงซ้อน" data-language-autonym="ไทย" data-language-local-name="thajština" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Komplikadong_bilang" title="Komplikadong bilang – tagalog" lang="tl" hreflang="tl" data-title="Komplikadong bilang" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karma%C5%9F%C4%B1k_say%C4%B1" title="Karmaşık sayı – turečtina" lang="tr" hreflang="tr" data-title="Karmaşık sayı" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%81%D0%B0%D0%BD" title="Комплекс сан – tatarština" lang="tt" hreflang="tt" data-title="Комплекс сан" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarština" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексне число – ukrajinština" lang="uk" hreflang="uk" data-title="Комплексне число" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AE%D9%84%D9%88%D8%B7_%D8%B9%D8%AF%D8%AF" title="مخلوط عدد – urdština" lang="ur" hreflang="ur" data-title="مخلوط عدد" data-language-autonym="اردو" data-language-local-name="urdština" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kompleks_sonlar" title="Kompleks sonlar – uzbečtina" lang="uz" hreflang="uz" data-title="Kompleks sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Numaro_conpleso" title="Numaro conpleso – benátština" lang="vec" hreflang="vec" data-title="Numaro conpleso" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ph%E1%BB%A9c" title="Số phức – vietnamština" lang="vi" hreflang="vi" data-title="Số phức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Complexe_getalln" title="Complexe getalln – vlámština (západní)" lang="vls" hreflang="vls" data-title="Complexe getalln" data-language-autonym="West-Vlams" data-language-local-name="vlámština (západní)" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Komplikado_nga_ihap" title="Komplikado nga ihap – warajština" lang="war" hreflang="war" data-title="Komplikado nga ihap" data-language-autonym="Winaray" data-language-local-name="warajština" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="复数（数学） – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="复数（数学）" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B8%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Комплексин тойг – kalmyčtina" lang="xal" hreflang="xal" data-title="Комплексин тойг" data-language-autonym="Хальмг" data-language-local-name="kalmyčtina" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%9E%D7%A4%D7%9C%D7%A2%D7%A7%D7%A1%D7%A2_%D7%A6%D7%90%D7%9C" title="קאמפלעקסע צאל – jidiš" lang="yi" hreflang="yi" data-title="קאמפלעקסע צאל" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_t%C3%B3%E1%B9%A3%C3%B2ro" title="Nọ́mbà tóṣòro – jorubština" lang="yo" hreflang="yo" data-title="Nọ́mbà tóṣòro" data-language-autonym="Yorùbá" data-language-local-name="jorubština" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学) – čínština" lang="zh" hreflang="zh" data-title="复数 (数学)" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – čínština (klasická)" lang="lzh" hreflang="lzh" data-title="複數" data-language-autonym="文言" data-language-local-name="čínština (klasická)" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ho%CC%8Dk-cha%CC%8Dp-s%C3%B2%CD%98" title="Ho̍k-cha̍p-sò͘ – čínština (dialekty Minnan)" lang="nan" hreflang="nan" data-title="Ho̍k-cha̍p-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="čínština (dialekty Minnan)" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – kantonština" lang="yue" hreflang="yue" data-title="複數" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11567#sitelinks-wikipedia" title="Editovat mezijazykové odkazy" class="wbc-editpage">Upravit odkazy</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Jmenné prostory"> <div id="p-associated-pages" 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<div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhled"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Vzhled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skrýt</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Complex_conjugate_picture.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/170px-Complex_conjugate_picture.svg.png" decoding="async" width="170" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/255px-Complex_conjugate_picture.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/340px-Complex_conjugate_picture.svg.png 2x" data-file-width="300" data-file-height="422" /></a><figcaption>Znázornění komplexního čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+\mathrm {i} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+\mathrm {i} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f034e7d7b0ff90492fb520f9af4687b508f8e9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.159ex; height:2.509ex;" alt="{\displaystyle z=x+\mathrm {i} y}"></span> a čísla k němu <a href="/wiki/Komplexn%C4%9B_sdru%C5%BEen%C3%A9_%C4%8D%C3%ADslo" title="Komplexně sdružené číslo">komplexně sdruženého</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=x-\mathrm {i} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=x-\mathrm {i} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a67770928e5f5907f8c155a33704db082337965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.367ex; height:2.509ex;" alt="{\displaystyle {\bar {z}}=x-\mathrm {i} y}"></span> v <a href="/wiki/Komplexn%C3%AD_rovina" title="Komplexní rovina">komplexní rovině</a>. <i>r</i> je <a href="/wiki/Absolutn%C3%AD_hodnota" title="Absolutní hodnota">absolutní hodnota</a> (norma), φ je <a class="mw-selflink-fragment" href="#Goniometrický_tvar_komplexních_čísel">argument</a>.</figcaption></figure> <p><b>Komplexní čísla</b> (z latinského <i>complexus</i>, složený) jsou rozšířením oboru <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálných čísel</a>, které má vlastnost, že v něm každá algebraická rovnice má příslušný počet řešení podle <a href="/wiki/Z%C3%A1kladn%C3%AD_v%C4%9Bta_algebry" title="Základní věta algebry">základní věty algebry</a>. Například <a href="/wiki/Kvadratick%C3%A1_rovnice" title="Kvadratická rovnice">kvadratická rovnice</a> <i>x</i><sup>2</sup> + 1 = 0 nemá v oboru reálných čísel řešení, protože její <a href="/wiki/Diskriminant" title="Diskriminant">diskriminant</a> (−4) je záporný a jeho odmocnina zde není definována. Komplexní číslo má dvě složky, reálnou a imaginární, a zapisuje se nejčastěji jako <i>a</i> + <i>b</i>i, přičemž i znamená <a href="/wiki/Imagin%C3%A1rn%C3%AD_jednotka" title="Imaginární jednotka">imaginární jednotku</a>, pro kterou platí vztah i<sup>2</sup> = −1. Zmíněná rovnice pak má dvě řešení, ± i. Pro operace s komplexními čísly platí pravidla pro počítání s dvojčleny. Množinu všech komplexních čísel obvykle značíme ℂ. </p><p>Komplexní čísla lze interpretovat geometricky. Zde je příklad v <a href="/wiki/Kart%C3%A9zsk%C3%A1_soustava_sou%C5%99adnic" title="Kartézská soustava souřadnic">kartézských souřadnicích</a>. Jako se reálná čísla zobrazují na reálné ose <i><b>Re</b></i>, budou imaginární čísla zobrazena na kolmé imaginární ose <i><b>Im</b></i> a každé komplexní číslo se zobrazí jako bod v rovině se souřadnicemi [<i>x</i>, <i>y</i>]. Číslo tvaru [<i>x</i>, 0] je reálné, číslo tvaru [0, <i>y</i>] je ryze imaginární. Absolutní hodnota komplexního čísla je pak vzdálenost bodu [<i>x</i>, <i>y</i>] od počátku souřadnic a číslo komplexně sdružené (tj. číslo [<i>x</i>, −<i>y</i>]) je zrcadlovým obrazem bodu [<i>x</i>, <i>y</i>] podle reálné osy x, tedy <i><b>Re</b></i>. </p><p>Komplexní čísla jsou významná nejen v matematice, ale také ve fyzice, především v elektrotechnice, optice a hydrodynamice. </p> <meta property="mw:PageProp/toc" /> <p><span id="Imaginární_číslo"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Zápis_a_související_pojmy"><span id="Z.C3.A1pis_a_souvisej.C3.ADc.C3.AD_pojmy"></span>Zápis a související pojmy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=1" title="Editace sekce: Zápis a související pojmy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=1" title="Editovat zdrojový kód sekce Zápis a související pojmy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Komplexním číslem</b> nazveme číslo tvaru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2726285fe73f29de7d337140ded1b4f3978f8cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:6.102ex; height:2.343ex;" alt="{\displaystyle a+b\mathrm {i} \,\!}"></span>, kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1dfcc2fcd791aacf48286147257b507b009e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.385ex; height:2.176ex;" alt="{\displaystyle b\,\!}"></span> jsou <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálná čísla</a>. Tento tvar komplexního čísla se nazývá <b>algebraický</b>. Písmeno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b12be33ba6cc8a9472c4fd9ea86928f9b5168c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.034ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} \,\!}"></span> značí <b>imaginární jednotku</b>, která se neformálně zavádí jako číslo splňující rovnici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} ^{2}+1=0\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}+1=0\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4630c2609ff15370281cbe485fdd48194b83b923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.999ex; height:3.009ex;" alt="{\displaystyle \mathrm {i} ^{2}+1=0\,,}"></span> což je intuitivně <a href="/wiki/Odmocnina" title="Odmocnina">odmocnina</a> z −1, která v reálných číslech neexistuje. Formálně správnější je např. <a class="mw-selflink-fragment" href="#Definice_pomocí_uspořádaných_dvojic">definice pomocí uspořádaných dvojic</a>, která je ale poněkud techničtější. </p><p><span id="Ryze_imaginární_číslo"></span> Reálné číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span> se nazývá <b>reálnou částí</b> tohoto komplexního čísla a reálné číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1dfcc2fcd791aacf48286147257b507b009e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.385ex; height:2.176ex;" alt="{\displaystyle b\,\!}"></span> jeho <b>imaginární částí</b>. Pokud je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69dbb9d932cad84dad034e4a881d4dc133eceb20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.646ex; height:2.176ex;" alt="{\displaystyle b=0\,\!}"></span>, je dotyčné číslo reálným číslem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span>, tj. reálná čísla tvoří podmnožinu čísel komplexních. Pokud je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916725854445eb0914114934985f95d94b9cc384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.878ex; height:2.176ex;" alt="{\displaystyle a=0\,\!}"></span>, mluvíme o <b>(ryze) imaginárním číslu</b>. Někteří autoři totiž pojmem <a href="/wiki/Imagin%C3%A1rn%C3%AD_%C4%8D%C3%ADslo" class="mw-redirect" title="Imaginární číslo">imaginární číslo</a> rozumí jakékoli komplexní číslo. </p><p>Někdy se imaginární jednotka značí též <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {j} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97685153a7b89e72bba63e2903f6d5a7663fe734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.127ex; width:0.838ex; height:2.509ex;" alt="{\displaystyle \mathrm {j} }"></span>, zejména ve vědeckotechnických oborech, kde se písmeno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f0f09f6fc40e634d34aed6e205ac0f7a40e062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }"></span> používá pro jiné účely. Lze se setkat též se symbolem 𝕚. </p><p>Na pořadí imaginární části a imaginární jednotky v zápisu imaginárního čísla nezáleží (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\mathrm {i} =\mathrm {i} b\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\mathrm {i} =\mathrm {i} b\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da057770cdf9929d36b00fe3c3c42eef11d02203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:6.774ex; height:2.176ex;" alt="{\displaystyle b\mathrm {i} =\mathrm {i} b\,\!}"></span>). </p> <div class="mw-heading mw-heading3"><h3 id="Značení"><span id="Zna.C4.8Den.C3.AD"></span>Značení</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=2" title="Editace sekce: Značení" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=2" title="Editovat zdrojový kód sekce Značení"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Množina všech komplexních čísel se značí obvykle písmenem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. </p><p>Potřebujeme-li pracovat pouze s reálnou, resp. imaginární částí komplexního čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6167f6c1390c0dd538c4a91d85446c6e566bfc95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,\!}"></span>, používáme zápis </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\mathrm {Re} (z)=\Re (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\mathrm {Re} (z)=\Re (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1969545d1c5f971ad08f60a6a01c932d95c29b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.889ex; height:2.843ex;" alt="{\displaystyle a=\mathrm {Re} (z)=\Re (z)}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\mathrm {Im} (z)=\Im (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">ℑ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=\mathrm {Im} (z)=\Im (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b99f598cb7513c5014150c9cf1afaa11e0a99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.053ex; height:2.843ex;" alt="{\displaystyle b=\mathrm {Im} (z)=\Im (z)}"></span>,</dd></dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197d543b33d1faa3bac737522a991e3575730668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.648ex; height:2.509ex;" alt="{\displaystyle a,b\,\!}"></span> jsou reálná čísla. Komplexní číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6167f6c1390c0dd538c4a91d85446c6e566bfc95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,\!}"></span> lze tedy také vyjádřit některým z následujících zápisů: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+\mathrm {i} b=\mathrm {Re} (z)+\mathrm {i} \mathrm {Im} (z)=\Re (z)+\mathrm {i} \Im (z)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi mathvariant="normal">ℑ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+\mathrm {i} b=\mathrm {Re} (z)+\mathrm {i} \mathrm {Im} (z)=\Re (z)+\mathrm {i} \Im (z)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9b093d9193129d1631bc3e57ed529b21fd66dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:43.78ex; height:2.843ex;" alt="{\displaystyle z=a+\mathrm {i} b=\mathrm {Re} (z)+\mathrm {i} \mathrm {Im} (z)=\Re (z)+\mathrm {i} \Im (z)\,\!}"></span></dd></dl> <p>S imaginární jednotkou se zachází jako s každým jiným číslem, proto je možné používat následujících zkrácených zápisů: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+x\cdot \mathrm {i} =x\cdot \mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+x\cdot \mathrm {i} =x\cdot \mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d079467f798d8e2030849c829160e074c187016f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:14.8ex; height:2.343ex;" alt="{\displaystyle 0+x\cdot \mathrm {i} =x\cdot \mathrm {i} \,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+0\cdot \mathrm {i} =x\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+0\cdot \mathrm {i} =x\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91bb7ca96b6f8b2c5b76592026b375afa3d22467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:12.474ex; height:2.343ex;" alt="{\displaystyle x+0\cdot \mathrm {i} =x\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot \mathrm {i} =\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot \mathrm {i} =\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921decd4eda10e8bc7715635fa917a089e7b1c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:7.621ex; height:2.176ex;" alt="{\displaystyle 1\cdot \mathrm {i} =\mathrm {i} \,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\cdot \mathrm {i} =-\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\cdot \mathrm {i} =-\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7319ea5cb942902ef404e07465083c8363f23738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:11.237ex; height:2.343ex;" alt="{\displaystyle -1\cdot \mathrm {i} =-\mathrm {i} \,\!}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Příklad"><span id="P.C5.99.C3.ADklad"></span>Příklad</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=3" title="Editace sekce: Příklad" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=3" title="Editovat zdrojový kód sekce Příklad"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=3+2\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=3+2\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eabb392eb42418cf1881044646cc1d2223c6b0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:10.386ex; height:2.343ex;" alt="{\displaystyle z=3+2\mathrm {i} \,\!}"></span> má reálnou část <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} (z)=3\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} (z)=3\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f51fd7b91f70675e34ab11c2855830c351be19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.288ex; height:2.843ex;" alt="{\displaystyle \mathrm {Re} (z)=3\,\!}"></span> a imaginární část <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Im} (z)=2\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} (z)=2\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38814b3d7456f83e5902109a3380e40e3b92c8a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.321ex; height:2.843ex;" alt="{\displaystyle \mathrm {Im} (z)=2\,\!}"></span>. Nejedná se ani o reálné, ani o ryze imaginární číslo. </p> <div class="mw-heading mw-heading2"><h2 id="Důvody_pro_zavedení_komplexních_čísel"><span id="D.C5.AFvody_pro_zaveden.C3.AD_komplexn.C3.ADch_.C4.8D.C3.ADsel"></span>Důvody pro zavedení komplexních čísel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=4" title="Editace sekce: Důvody pro zavedení komplexních čísel" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=4" title="Editovat zdrojový kód sekce Důvody pro zavedení komplexních čísel"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Historie">Historie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=5" title="Editace sekce: Historie" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=5" title="Editovat zdrojový kód sekce Historie"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Už perský matematik <a href="/wiki/Al-Chorezm%C3%AD" title="Al-Chorezmí">Al-Chorezmí</a> (asi 820) poznamenal, že některé kvadratické rovnice nemají reálné řešení, čehož si patrně byli vědomi i jeho předchůdci z Indie. Ačkoliv z dnešního pohledu se takové rovnice považují za řešitelné v komplexním oboru, toto samo o sobě, jako motivace pro zavedení komplexních čísel, nestačilo. Prvními, kdo z dnešního pohledu použili komplexní čísla byli <a href="/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a> a <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> (kolem 1530), kteří nezávisle na sobě navrhli metodu na řešení <a href="/wiki/Kubick%C3%A1_rovnice" title="Kubická rovnice">kubické rovnice</a>, která, ačkoliv je stále zajímala pouze reálná řešení, vyžaduje jako mezivýpočet použití komplexních čísel. Tartaglia metodu nejprve držel v tajnosti, ale podělil se o ni později, pod slibem mlčenlivosti, s italským matematikem <a href="/wiki/Girolamo_Cardano" class="mw-redirect" title="Girolamo Cardano">Gerolamem Cardanem</a>. Ten ji spolu s metodou pro řešení <a href="/wiki/Kvartick%C3%A1_rovnice" title="Kvartická rovnice">kvartické rovnice</a>, objevenou jeho žákem <a href="/wiki/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovicem Ferrarim</a>, též využívající komplexní čísla, publikoval v knize <a href="/wiki/Ars_Magna" title="Ars Magna">Ars Magna</a> (1545), přičemž uvedl, že del Ferro řešení nalezl dříve, než Tartaglia. <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> zavedl <a href="/wiki/1637" title="1637">1637</a> označení reálné a imaginární číslo a z jeho práce plyne geometrická interpretace komplexních čísel. Zajímavé výsledky zkoumání těchto „neskutečných“ čísel ukázal <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> a komplexní čísla rigorózně zavedl francouzský matematik <a href="/wiki/Augustin_Louis_Cauchy" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a> (1821) a nezávisle na něm <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1831). </p> <div class="mw-heading mw-heading3"><h3 id="Matematická_motivace"><span id="Matematick.C3.A1_motivace"></span>Matematická motivace</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=6" title="Editace sekce: Matematická motivace" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=6" title="Editovat zdrojový kód sekce Matematická motivace"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Obor reálných čísel, který vyjadřuje dostatečně dobře jakoukoliv kvantitu (množství), se tedy rozšiřuje do oboru komplexních čísel, jejichž význam není intuitivně příliš zřejmý, a to především proto, že v reálném oboru neleží řešení (kořeny) některých algebraických rovnic, čili obor reálných čísel není vzhledem k nim uzavřený. </p><p>V oboru reálných čísel existují <a href="/wiki/Polynom" title="Polynom">polynomy</a> (s reálnými <a href="/wiki/Koeficient" title="Koeficient">koeficienty</a> a nezápornými celočíselnými <a href="/wiki/Exponent_(matematika)" class="mw-redirect" title="Exponent (matematika)">exponenty</a>), které nemají v oboru reálných čísel žádný <a href="/wiki/Ko%C5%99en_(matematika)" title="Kořen (matematika)">kořen</a>, případně je počet jejich reálných kořenů nižší než stupeň polynomu. </p><p>Obor komplexních čísel je uzavřený nejen na výše uvedené kořeny polynomů s reálnými koeficienty, ale i na kořeny polynomů s komplexními koeficienty. Tuto uzavřenost vyjadřuje <a href="/wiki/Z%C3%A1kladn%C3%AD_v%C4%9Bta_algebry" title="Základní věta algebry">základní věta algebry</a>, která říká, že polynom <i>n</i>-tého stupně má v oboru komplexních čísel n kořenů (pokud počítáme jejich násobnost – polynom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-2x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-2x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/922c8025d0061c6a832d27e86cc999778924833c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.719ex; height:2.843ex;" alt="{\displaystyle x^{2}-2x+1}"></span> má dvojnásobný kořen <i>x</i>=1, protože jej lze rozložit na <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-1).(x-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1).(x-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a266239b1a7e3fbb530f8b73fbd4675c1964a90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.318ex; height:2.843ex;" alt="{\displaystyle (x-1).(x-1)}"></span>). </p><p>V dnešní době je <a href="/wiki/Komplexn%C3%AD_anal%C3%BDza" title="Komplexní analýza">komplexní analýza</a> důležitým matematickým prostředkem s četnými aplikacemi v různých jiných odvětvích matematiky, včetně například <a href="/wiki/Teorie_%C4%8D%C3%ADsel" title="Teorie čísel">teorie čísel</a>, vedoucí k výsledkům, které jsou bez použití komplexních čísel zcela nedostupné, nebo obtížněji dostupné. </p> <div class="mw-heading mw-heading4"><h4 id="Příklad_2"><span id="P.C5.99.C3.ADklad_2"></span>Příklad</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=7" title="Editace sekce: Příklad" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=7" title="Editovat zdrojový kód sekce Příklad"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polynom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5615a4b9c08ac5304fba8fa72bb594df2ebf80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:6.774ex; height:2.843ex;" alt="{\displaystyle x^{2}+1\,\!}"></span> nemá v oboru reálných čísel žádný kořen. V oboru komplexních čísel jeho kořeny jsou čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b12be33ba6cc8a9472c4fd9ea86928f9b5168c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.034ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} \,\!}"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb957ed6dfed5f943d706ccb71231b0011611567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:2.842ex; height:2.343ex;" alt="{\displaystyle -\mathrm {i} \,\!}"></span>, protože: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} ^{2}+1=-1+1=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}+1=-1+1=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c96882d644feabdc1ce39f5dc79873b92408de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:20.424ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} ^{2}+1=-1+1=0\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\mathrm {i} )^{2}+1=-1+1=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\mathrm {i} )^{2}+1=-1+1=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be2c8f6aadf908446f4a2981f32ef23d44a2bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:24.041ex; height:3.176ex;" alt="{\displaystyle (-\mathrm {i} )^{2}+1=-1+1=0\,\!}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Technické_aplikace"><span id="Technick.C3.A9_aplikace"></span>Technické aplikace</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=8" title="Editace sekce: Technické aplikace" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=8" title="Editovat zdrojový kód sekce Technické aplikace"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I samotné značení vyplývající z použití komplexních čísel může často zjednodušit a zpřehlednit zápisy a výpočty v některých případech, kde není zcela nutné, jako například <a href="/wiki/Fourierova_%C5%99ada" title="Fourierova řada">Fourierovy řady</a>. To má technické aplikace ve <a href="/wiki/Zpracov%C3%A1n%C3%AD_sign%C3%A1lu" title="Zpracování signálu">zpracování signálu</a> a výpočtu střídavých elektrických obvodů. Aparát komplexních čísel hojně využívá teorie <a href="/wiki/Kvantov%C3%A1_fyzika" title="Kvantová fyzika">kvantové fyziky</a>, kde <a href="/wiki/Vlnov%C3%A1_funkce" title="Vlnová funkce">vlnová funkce</a> nabývá hodnot v komplexním oboru. </p> <div class="mw-heading mw-heading2"><h2 id="Operace_s_komplexními_čísly"><span id="Operace_s_komplexn.C3.ADmi_.C4.8D.C3.ADsly"></span>Operace s komplexními čísly</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=9" title="Editace sekce: Operace s komplexními čísly" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=9" title="Editovat zdrojový kód sekce Operace s komplexními čísly"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algebraický_tvar_komplexních_čísel"><span id="Algebraick.C3.BD_tvar_komplexn.C3.ADch_.C4.8D.C3.ADsel"></span>Algebraický tvar komplexních čísel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=10" title="Editace sekce: Algebraický tvar komplexních čísel" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=10" title="Editovat zdrojový kód sekce Algebraický tvar komplexních čísel"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pro čísla v algebraickém tvaru lze jednoduchými algebraickými úpravami odvodit vztahy pro <a href="/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD" title="Sčítání">součet</a>, <a href="/wiki/Od%C4%8D%C3%ADt%C3%A1n%C3%AD" title="Odčítání">rozdíl</a> a <a href="/wiki/N%C3%A1soben%C3%AD" title="Násobení">součin</a> dvou komplexních čísel: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+\mathrm {i} b)+(c+\mathrm {i} d)=(a+c)+\mathrm {i} (b+d)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+\mathrm {i} b)+(c+\mathrm {i} d)=(a+c)+\mathrm {i} (b+d)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2442fa9e0c1071fe9a33e3f860fe6ba0f78f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:38.606ex; height:2.843ex;" alt="{\displaystyle (a+\mathrm {i} b)+(c+\mathrm {i} d)=(a+c)+\mathrm {i} (b+d)\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+\mathrm {i} b)-(c+\mathrm {i} d)=(a-c)+\mathrm {i} (b-d)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+\mathrm {i} b)-(c+\mathrm {i} d)=(a-c)+\mathrm {i} (b-d)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4379d21b1f32edfbc5cd3d9804b7caecb7a1b81f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:38.606ex; height:2.843ex;" alt="{\displaystyle (a+\mathrm {i} b)-(c+\mathrm {i} d)=(a-c)+\mathrm {i} (b-d)\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+\mathrm {i} b)\cdot (c+\mathrm {i} d)=(ac-bd)+\mathrm {i} (ad+bc)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+\mathrm {i} b)\cdot (c+\mathrm {i} d)=(ac-bd)+\mathrm {i} (ad+bc)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5e3f2231a7817bad730d8da21fa98cf09edc8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:41.895ex; height:2.843ex;" alt="{\displaystyle (a+\mathrm {i} b)\cdot (c+\mathrm {i} d)=(ac-bd)+\mathrm {i} (ad+bc)\,\!}"></span></dd></dl> <p><a href="/wiki/D%C4%9Blen%C3%AD" title="Dělení">Podíl</a> dvou komplexních čísel lze vyjádřit takto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a+\mathrm {i} b \over c+\mathrm {i} d}={(a+\mathrm {i} b)(c-\mathrm {i} d) \over (c+\mathrm {i} d)(c-\mathrm {i} d)}={(ac+bd)+\mathrm {i} (bc-ad) \over c^{2}+d^{2}}=\left({ac+bd \over c^{2}+d^{2}}\right)+\mathrm {i} \left({bc-ad \over c^{2}+d^{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a+\mathrm {i} b \over c+\mathrm {i} d}={(a+\mathrm {i} b)(c-\mathrm {i} d) \over (c+\mathrm {i} d)(c-\mathrm {i} d)}={(ac+bd)+\mathrm {i} (bc-ad) \over c^{2}+d^{2}}=\left({ac+bd \over c^{2}+d^{2}}\right)+\mathrm {i} \left({bc-ad \over c^{2}+d^{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebe2f7fcc46978a34cdb48f2316a331d6d4d3d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:82.252ex; height:6.509ex;" alt="{\displaystyle {a+\mathrm {i} b \over c+\mathrm {i} d}={(a+\mathrm {i} b)(c-\mathrm {i} d) \over (c+\mathrm {i} d)(c-\mathrm {i} d)}={(ac+bd)+\mathrm {i} (bc-ad) \over c^{2}+d^{2}}=\left({ac+bd \over c^{2}+d^{2}}\right)+\mathrm {i} \left({bc-ad \over c^{2}+d^{2}}\right).}"></span></dd></dl> <p>Pro komplexní číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+b\mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7f54052b27c21d6073ea59a31e499ea689970f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.901ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} }"></span> je definována <b>konjugace</b> (<a href="/wiki/Komplexn%C4%9B_sdru%C5%BEen%C3%A9_%C4%8D%C3%ADslo" title="Komplexně sdružené číslo">komplexně sdružené číslo</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}:=a-b\mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}:=a-b\mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71df7a58bb96c2f52a2e9b65346935cca3ae7bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.756ex; height:2.343ex;" alt="{\displaystyle {\bar {z}}:=a-b\mathrm {i} }"></span>. Jejich součin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z{\bar {z}}=a^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z{\bar {z}}=a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ea4049271016a6783990482e28295cee164f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.659ex; height:2.843ex;" alt="{\displaystyle z{\bar {z}}=a^{2}+b^{2}}"></span> je vždy reálný a nezáporný a je roven nule, pouze když <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"></span>. Pak můžeme psát pro inverzi stručně <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{-1}={\bar {z}}/(z{\bar {z}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{-1}={\bar {z}}/(z{\bar {z}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f7a606c0010d39b7378c5076955faaaf531851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.174ex; height:3.176ex;" alt="{\displaystyle z^{-1}={\bar {z}}/(z{\bar {z}})}"></span> pro <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8b7eb2d2a30057811a7835502717d3d6ece962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}"></span>. </p><p><b>Norma</b> (též <b>absolutní hodnota</b> nebo <b>modul</b>) komplexního čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+b\mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7f54052b27c21d6073ea59a31e499ea689970f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.901ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} }"></span> je definována jako <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|:={\sqrt {z{\bar {z}}}}={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|:={\sqrt {z{\bar {z}}}}={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3feaaea36d763483e42590946d23316e044f2d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.046ex; height:3.509ex;" alt="{\displaystyle |z|:={\sqrt {z{\bar {z}}}}={\sqrt {a^{2}+b^{2}}}}"></span>. Platí, že pro libovolná komplexní čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61184bc010f1c8b20a465ae5c41b013fe1a22abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.009ex;" alt="{\displaystyle z,w}"></span> je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |zw|=|z||w|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |zw|=|z||w|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df220deb42cd816f43478c2c5a17f7ceecb6650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.484ex; height:2.843ex;" alt="{\displaystyle |zw|=|z||w|}"></span>, tj. norma součinu je součin norem. </p> <div class="mw-heading mw-heading3"><h3 id="Geometrické_znázornění_komplexních_čísel"><span id="Geometrick.C3.A9_zn.C3.A1zorn.C4.9Bn.C3.AD_komplexn.C3.ADch_.C4.8D.C3.ADsel"></span>Geometrické znázornění komplexních čísel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=11" title="Editace sekce: Geometrické znázornění komplexních čísel" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=11" title="Editovat zdrojový kód sekce Geometrické znázornění komplexních čísel"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplexní čísla se zobrazují v komplexní (Gaussově) rovině jako <a href="/wiki/Bod" title="Bod">body</a> se souřadnicemi <i>x,y</i>; <i>x</i> je reálná část komplexního čísla, <i>y</i> imaginární část. Na <a href="/wiki/Osa" title="Osa">ose</a> <i>x</i> leží reálná čísla, ose <i>y</i> ryze imaginární čísla. Kombinací těchto dvou složek (reálné a imaginární) dostaneme množinu všech komplexních čísel, tj. <a href="/wiki/Gaussova_rovina" class="mw-redirect" title="Gaussova rovina">Gaussova rovina</a>. </p><p>Alternativně se pro znázornění množiny <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \cup \{\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb61866ff76f7d364846c91f6840957a946d6525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.909ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \cup \{\infty \}}"></span> používá jako model tzv. <a href="/wiki/Riemannova_sf%C3%A9ra" title="Riemannova sféra">Riemannova sféra</a>, kdy komplexní rovinu stereograficky promítneme na sféru tak, že nula je jižní pól, komplexní jednotky tvoří rovník a na severním pólu se nachází komplexní nekonečno. Toto rozšíření ℂ o nevlastní bod je někdy užitečné v <a href="/wiki/Komplexn%C3%AD_anal%C3%BDza" title="Komplexní analýza">komplexní analýze</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Goniometrický_tvar_komplexních_čísel"><span id="Goniometrick.C3.BD_tvar_komplexn.C3.ADch_.C4.8D.C3.ADsel"></span>Goniometrický tvar komplexních čísel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=12" title="Editace sekce: Goniometrický tvar komplexních čísel" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=12" title="Editovat zdrojový kód sekce Goniometrický tvar komplexních čísel"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Každé komplexní číslo <i>z</i> různé od nuly je možné jednoznačně vyjádřit v <a href="/wiki/Goniometrie" title="Goniometrie">goniometrickém</a> tvaru. Pokud si v komplexní rovině zvolíme <a href="/wiki/Pol%C3%A1rn%C3%AD_soustava_sou%C5%99adnic" title="Polární soustava souřadnic">polární</a> <a href="/wiki/Soustava_sou%C5%99adnic" title="Soustava souřadnic">souřadnicový systém</a>, vzdálenost od počátku označíme <i>|z|</i> (<a href="/wiki/Absolutn%C3%AD_hodnota" title="Absolutní hodnota">absolutní hodnota</a>, také nazývaná norma nebo modul) a orientovaný <a href="/wiki/%C3%9Ahel" title="Úhel">úhel</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\mathrm {IOZ} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\mathrm {IOZ} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec3d07e44a83aa6b5c1c04bbf6c130364f0f1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.687ex; height:2.676ex;" alt="{\displaystyle \varphi =\mathrm {IOZ} }"></span> (argument), kde I=[1;0]. O je počátkem soustavy a Z=[<i>a</i>;<i>b</i>] je obraz komplexního čísla z=<i>a</i> + <i>b</i>i, platí: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|(\cos \varphi +\mathrm {i} \cdot \sin \varphi )=|z|\cdot e^{\mathrm {i} \varphi }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>φ</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \varphi +\mathrm {i} \cdot \sin \varphi )=|z|\cdot e^{\mathrm {i} \varphi }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c90e81676e5e52cce986dda1ed28f81196a2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.72ex; height:3.176ex;" alt="{\displaystyle z=|z|(\cos \varphi +\mathrm {i} \cdot \sin \varphi )=|z|\cdot e^{\mathrm {i} \varphi }\,}"></span>.</dd></dl> <p>Modul lze z algebraického tvaru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+b\mathrm {i} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+b\mathrm {i} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90aa1c7bdba26a3f6301053d64e8c92215c18f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:10.288ex; height:2.343ex;" alt="{\displaystyle z=a+b\mathrm {i} \,\!}"></span> určit ze vztahu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe94d0c3b0c3704e8771d0932fff6f983ef0082b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.98ex; height:3.509ex;" alt="{\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}"></span>. Při zobrazení v komplexní rovině je to délka úsečky <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {OZ} |\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {OZ} |\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d4e5bcff3cf07644d6130117c6cffe855a3853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.909ex; height:2.843ex;" alt="{\displaystyle |\mathrm {OZ} |\,}"></span>.</dd></dl> <p>Argument lze vyjádřit ze vztahů: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arg(z)=\arg(a+b\mathrm {i} )=\operatorname {atan2} (b,a)={\begin{cases}\arctan({\frac {b}{a}})&{\mbox{jestliže }}a>0\\\arctan({\frac {b}{a}})+\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b\geq 0\\\arctan({\frac {b}{a}})-\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b<0\\{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b>0\\-{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b<0\\{\mbox{libovolné }}&{\mbox{jestliže }}a=0{\mbox{ a }}b=0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> a </mtext> </mstyle> </mrow> <mi>b</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> a </mtext> </mstyle> </mrow> <mi>b</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> a </mtext> </mstyle> </mrow> <mi>b</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> a </mtext> </mstyle> </mrow> <mi>b</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>libovolné </mtext> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>jestliže </mtext> </mstyle> </mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> a </mtext> </mstyle> </mrow> <mi>b</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arg(z)=\arg(a+b\mathrm {i} )=\operatorname {atan2} (b,a)={\begin{cases}\arctan({\frac {b}{a}})&{\mbox{jestliže }}a>0\\\arctan({\frac {b}{a}})+\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b\geq 0\\\arctan({\frac {b}{a}})-\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b<0\\{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b>0\\-{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b<0\\{\mbox{libovolné }}&{\mbox{jestliže }}a=0{\mbox{ a }}b=0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765a30e795c970fe178fc561255ca589615b0f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:82.465ex; height:21.509ex;" alt="{\displaystyle \varphi =\arg(z)=\arg(a+b\mathrm {i} )=\operatorname {atan2} (b,a)={\begin{cases}\arctan({\frac {b}{a}})&{\mbox{jestliže }}a>0\\\arctan({\frac {b}{a}})+\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b\geq 0\\\arctan({\frac {b}{a}})-\pi &{\mbox{jestliže }}a<0{\mbox{ a }}b<0\\{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b>0\\-{\frac {\pi }{2}}&{\mbox{jestliže }}a=0{\mbox{ a }}b<0\\{\mbox{libovolné }}&{\mbox{jestliže }}a=0{\mbox{ a }}b=0.\end{cases}}}"></span></dd></dl> <p>Aby byla hodnota argumentu jednoznačná, je nutné ji omezit na nějaký polootevřený <a href="/wiki/Interval_(matematika)" title="Interval (matematika)">interval</a> délky 2π, většinou se volí <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ;\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>;</mo> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ;\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b72fcd9b209ab8b64ea846cf870f148a5c5091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.315ex; height:2.843ex;" alt="{\displaystyle (-\pi ;\pi \rangle }"></span> nebo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0;2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>;</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0;2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab0278b12c3c7145e6bfb4db53e77c993fe26f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle \langle 0;2\pi )}"></span>. Funkce <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\operatorname {Arg} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>Arg</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\operatorname {Arg} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeb963dea27d22be2c5dc9fc46f9c938afea34c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.911ex; height:2.676ex;" alt="{\displaystyle \varphi =\operatorname {Arg} z}"></span> má tedy v odpovídajících bodech skok velikosti 2π. Z tohoto důvodu se například argument součinu dvou komplexních čísel může lišit od součtu jejich argumentů o násobek 2π. </p><p>Pro násobení, dělení a <a href="/wiki/Umoc%C5%88ov%C3%A1n%C3%AD" title="Umocňování">umocňování</a> komplexních čísel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}=|z_{1}|\cdot e^{\mathrm {i} \varphi _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}=|z_{1}|\cdot e^{\mathrm {i} \varphi _{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b24a1faa9c85f19bd5182f8b434d16c54eb615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.022ex; height:3.176ex;" alt="{\displaystyle z_{1}=|z_{1}|\cdot e^{\mathrm {i} \varphi _{1}}}"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}=|z_{2}|\cdot e^{\mathrm {i} \varphi _{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}=|z_{2}|\cdot e^{\mathrm {i} \varphi _{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f811e3d5540f3f8ecacbd46caf5ac59840dffd9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.022ex; height:3.176ex;" alt="{\displaystyle z_{2}=|z_{2}|\cdot e^{\mathrm {i} \varphi _{2}}}"></span> platí následující rovnice: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}\cdot z_{2}=|z_{1}|\cdot |z_{2}|\cdot e^{\mathrm {i} (\varphi _{1}+\varphi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}\cdot z_{2}=|z_{1}|\cdot |z_{2}|\cdot e^{\mathrm {i} (\varphi _{1}+\varphi _{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1d7e08143b7b717c981a75fc9a1cf2d2a0670d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.409ex; height:3.343ex;" alt="{\displaystyle z_{1}\cdot z_{2}=|z_{1}|\cdot |z_{2}|\cdot e^{\mathrm {i} (\varphi _{1}+\varphi _{2})}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {|z_{1}|}{|z_{2}|}}e^{\mathrm {i} (\varphi _{1}-\varphi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {|z_{1}|}{|z_{2}|}}e^{\mathrm {i} (\varphi _{1}-\varphi _{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314aada987fa0e463a293ff943b691d951809b28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.479ex; height:6.509ex;" alt="{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {|z_{1}|}{|z_{2}|}}e^{\mathrm {i} (\varphi _{1}-\varphi _{2})}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}=|z|^{n}e^{\mathrm {i} n\varphi }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>n</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=|z|^{n}e^{\mathrm {i} n\varphi }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b7d12963b0de779dedfe72d163a63d073be063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.229ex; height:3.176ex;" alt="{\displaystyle z^{n}=|z|^{n}e^{\mathrm {i} n\varphi }\,}"></span> (viz <a href="/wiki/Moivreova_v%C4%9Bta" title="Moivreova věta">Moivreovu větu</a>)</dd></dl> <p>Pro převod komplexních čísel z goniometrického tvaru na algebraický stačí zjistit hodnotu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/544fef63b011ca474c1f9e4ffad26719f1068213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.018ex; height:2.176ex;" alt="{\displaystyle \cos \varphi }"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5476adfa219ffe9cc3186d8226214b0f4734dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.763ex; height:2.676ex;" alt="{\displaystyle \sin \varphi }"></span> a roznásobit závorku jako při práci s klasickým mnohočlenem: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|\cos \varphi +\mathrm {i} \cdot |z|\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|\cos \varphi +\mathrm {i} \cdot |z|\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456d297947c7c79b52b2a5b4abc51fc8e3203cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.672ex; height:2.843ex;" alt="{\displaystyle z=|z|\cos \varphi +\mathrm {i} \cdot |z|\sin \varphi }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Komplexní_funkce"><span id="Komplexn.C3.AD_funkce"></span>Komplexní funkce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=13" title="Editace sekce: Komplexní funkce" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=13" title="Editovat zdrojový kód sekce Komplexní funkce"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplexní funkce reálné proměnné je <a href="/wiki/Funkce_(matematika)" title="Funkce (matematika)">funkce</a>, jejímž <a href="/wiki/Defini%C4%8Dn%C3%AD_obor" title="Definiční obor">definičním oborem</a> jsou reálná čísla a <a href="/wiki/Obor_hodnot" title="Obor hodnot">oborem hodnot</a> jsou komplexní čísla. Platí: <i>h</i>(<i>x</i>) = <i>f</i>(<i>x</i>) + i<i>g</i>(<i>x</i>) kde <i>f</i> je reálná část a <i>g</i> imaginární část komplexní funkce <i>h</i>. Obrazem takovéto funkce v Gaussově rovině je křivka, jejíž geometrický obraz je množina všech bodů <i>X</i> = [<i>f</i>(<i>x</i>),<i>g</i>(<i>x</i>)], kde <i>x</i> je z definičního oboru funkce. </p><p>Širším pojmem je funkce komplexní proměnné, jejímž definičním oborem jsou komplexní čísla. Studiem těchto funkcí se zabývá <a href="/wiki/Komplexn%C3%AD_anal%C3%BDza" title="Komplexní analýza">komplexní analýza</a>. V tomto oboru se podařilo odhalit mnohé souvislosti mezi rozdílnými funkcemi reálné proměnné. Příkladem je <a href="/wiki/Euler%C5%AFv_vzorec" title="Eulerův vzorec">Eulerův vzorec</a>, často využívaný při práci s komplexními čísly, ze kterého vyplývá i vztah mezi základními matematickými <a href="/wiki/Konstanta" title="Konstanta">konstantami</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\mathrm {i} \pi }+1=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\mathrm {i} \pi }+1=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c92f43b77e95881506df3f709be00a82d85483d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.366ex; height:2.843ex;" alt="{\displaystyle e^{\mathrm {i} \pi }+1=0\,}"></span>,</dd></dl> <p>oblíbený nejen mezi matematiky. </p><p>Komplexní analýza nabídla nové nástroje i reálné analýze, např. pro výpočet integrálů (<a href="/wiki/Cauchyho_vzorec" class="mw-redirect" title="Cauchyho vzorec">Cauchyho vzorec</a>, <a href="/wiki/Reziduov%C3%A1_v%C4%9Bta" class="mw-redirect" title="Reziduová věta">reziduová věta</a>) a našla široké uplatnění ve fyzice a technických aplikacích, např. při výpočtech fyzikálních polí a matematickém modelování proudění tekutin v hydrodynamice a aerodynamice. </p> <div class="mw-heading mw-heading2"><h2 id="Základní_vlastnosti_tělesa_komplexních_čísel"><span id="Z.C3.A1kladn.C3.AD_vlastnosti_t.C4.9Blesa_komplexn.C3.ADch_.C4.8D.C3.ADsel"></span>Základní vlastnosti tělesa komplexních čísel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=14" title="Editace sekce: Základní vlastnosti tělesa komplexních čísel" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=14" title="Editovat zdrojový kód sekce Základní vlastnosti tělesa komplexních čísel"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplexní čísla s operacemi sčítání a násobení tvoří komutativní <a href="/wiki/T%C4%9Bleso_(algebra)" title="Těleso (algebra)">těleso</a>. Je to největší komutativní algebraické nadtěleso (konečného stupně rozšíření) tělesa reálných čísel a <a href="/wiki/Algebraick%C3%BD_uz%C3%A1v%C4%9Br" title="Algebraický uzávěr">algebraický uzávěr</a> tělesa reálných čísel. Toto těleso nelze okruhově uspořádat, protože <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} ^{2}=-1<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}=-1<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13a1b22da61c2d95bb9b0b045cf5b2c6e3efedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.031ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} ^{2}=-1<0}"></span>. </p><p>Komplexní čísla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> je možno chápat jako dvoudimenzionální normovanou podílovou algebru nad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. Existují právě dva <a href="/wiki/Automorfizmus" class="mw-redirect" title="Automorfizmus">automorfizmy</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> jakožto algebry nad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>: identita a konjugace. </p><p>Je zajímavé, že existuje nekonečně mnoho automorfizmů <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> jako tělesa (ovšem jsou velmi nespojité a nezachovávají <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \subset \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \subset \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05052f2b40ab1ec93320942d62e4525e8087810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.455ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \subset \mathbb {C} }"></span>, což znamená, že reálná a čistě imaginární čísla nejsou určena samotnou strukturou tělesa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> – porovnej s <a href="/wiki/Kvaternion" title="Kvaternion">kvaterniony</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Definice_pomocí_uspořádaných_dvojic"><span id="Definice_pomoc.C3.AD_uspo.C5.99.C3.A1dan.C3.BDch_dvojic"></span>Definice pomocí uspořádaných dvojic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=15" title="Editace sekce: Definice pomocí uspořádaných dvojic" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=15" title="Editovat zdrojový kód sekce Definice pomocí uspořádaných dvojic"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplexní čísla formálně zaváděna jako všechny uspořádané dvojice <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálných čísel</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> s definovanými operacemi sčítání a násobení: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4831924a9066bf2a688d1ac5fa210269f8bbda01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.437ex; height:2.843ex;" alt="{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\cdot (c,d)=(ac-bd,ad+bc)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo>,</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\cdot (c,d)=(ac-bd,ad+bc)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c95ec2d5fe8aa062574815475fe3a05631ae94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.725ex; height:2.843ex;" alt="{\displaystyle (a,b)\cdot (c,d)=(ac-bd,ad+bc)\,}"></span></dd></dl> <p>Znaménko <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> u násobení obvykle vynecháváme. </p><p>Takováto definice je matematicky čistší, než výše uvedené neformální definice, protože pokud pouze postulujeme existenci nějaké množiny <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> a hodnoty <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> s vlastnostmi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \mathbb {C} \supsetneq \mathbb {R} .(\mathbb {C} \;{\mbox{je těleso}}\land \exists i\in \mathbb {C} .(i^{2}=-1))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>⊋</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>je těleso</mtext> </mstyle> </mrow> <mo>∧</mo> <mi mathvariant="normal">∃</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> <mo stretchy="false">(</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \mathbb {C} \supsetneq \mathbb {R} .(\mathbb {C} \;{\mbox{je těleso}}\land \exists i\in \mathbb {C} .(i^{2}=-1))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54bccc213d35cfc282da5101574a48ee8fc77c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.283ex; height:3.509ex;" alt="{\displaystyle \exists \mathbb {C} \supsetneq \mathbb {R} .(\mathbb {C} \;{\mbox{je těleso}}\land \exists i\in \mathbb {C} .(i^{2}=-1))}"></span>, tak přidání takového nového axiomu do teorie vyvolává otázku jeho bezespornosti se zbytkem teorie, což je dosti složitý problém. Definice pomocí uspořádaných dvojic tento problém obchází tím, že <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> s požadovanými vlastnostmi nepostuluje, nýbrž zkonstruuje z reálných čísel. </p><p>Použitím algebraických vlastností <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálných čísel</a> dostaneme následující tvrzení: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (a_{1},a_{2}),(b_{1},b_{2}),(c_{1},c_{2})\in {\mathbb {C} }:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (a_{1},a_{2}),(b_{1},b_{2}),(c_{1},c_{2})\in {\mathbb {C} }:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329772c43075297ce1ad341529e283c80ceea613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.495ex; height:2.843ex;" alt="{\displaystyle \forall (a_{1},a_{2}),(b_{1},b_{2}),(c_{1},c_{2})\in {\mathbb {C} }:}"></span> </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})+(b_{1},b_{2})=(b_{1},b_{2})+(a_{1},a_{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})+(b_{1},b_{2})=(b_{1},b_{2})+(a_{1},a_{2})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a178b8bdcc3ee3fd08ef9e76cd4b3b4fc12b36bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.883ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})+(b_{1},b_{2})=(b_{1},b_{2})+(a_{1},a_{2})\,}"></span> (komutativita sčítání) <br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})+{\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})+(b_{1},b_{2}){\big )}+(c_{1},c_{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})+{\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})+(b_{1},b_{2}){\big )}+(c_{1},c_{2})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce67e3f0658ab23f6f74246ccf28c49e83ccff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.754ex; height:3.176ex;" alt="{\displaystyle (a_{1},a_{2})+{\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})+(b_{1},b_{2}){\big )}+(c_{1},c_{2})\,}"></span> (asociativita sčítání) <br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})+(0,0)=(a_{1},a_{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})+(0,0)=(a_{1},a_{2})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b824c0b5246896398c6cd02379d8c3221a91972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.317ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})+(0,0)=(a_{1},a_{2})\,}"></span> (neutralita nuly vůči sčítání)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})+(-a_{1},-a_{2})=(0,0)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})+(-a_{1},-a_{2})=(0,0)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbda130c27eaf55cd8a107c5edab98eaa33a8828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.933ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})+(-a_{1},-a_{2})=(0,0)\,}"></span> (existence inverzního prvku vůči sčítání; pravidlo odčítání)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=(b_{1},b_{2})\cdot (a_{1},a_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=(b_{1},b_{2})\cdot (a_{1},a_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251f16c2a85d96047d0e6288e6b1c66696d96058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.173ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=(b_{1},b_{2})\cdot (a_{1},a_{2})}"></span> (komutativita násobení)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})\cdot (c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})\cdot (b_{1},b_{2}){\big )}\cdot (c_{1},c_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})\cdot (c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})\cdot (b_{1},b_{2}){\big )}\cdot (c_{1},c_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325e84482fb769bca7548de0beb6d19084a9418b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.722ex; height:3.176ex;" alt="{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})\cdot (c_{1},c_{2}){\big )}={\big (}(a_{1},a_{2})\cdot (b_{1},b_{2}){\big )}\cdot (c_{1},c_{2})}"></span> (asociativita násobení)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\cdot (1,0)=(a_{1},a_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\cdot (1,0)=(a_{1},a_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b59ca5acd036b440805658a9ccdc61ceefea10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.768ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})\cdot (1,0)=(a_{1},a_{2})}"></span> (neutralita jedničky vůči násobení)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\neq (0,0)\implies (a_{1},a_{2})\cdot \left({a_{1} \over a_{1}^{2}+a_{2}^{2}},{-a_{2} \over a_{1}^{2}+a_{2}^{2}}\right)=(1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\neq (0,0)\implies (a_{1},a_{2})\cdot \left({a_{1} \over a_{1}^{2}+a_{2}^{2}},{-a_{2} \over a_{1}^{2}+a_{2}^{2}}\right)=(1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980b38714163e55f6737c3cc389b60246a881a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.626ex; height:7.509ex;" alt="{\displaystyle (a_{1},a_{2})\neq (0,0)\implies (a_{1},a_{2})\cdot \left({a_{1} \over a_{1}^{2}+a_{2}^{2}},{-a_{2} \over a_{1}^{2}+a_{2}^{2}}\right)=(1,0)}"></span> (existence inverzního prvku vůči násobení; pravidlo dělení)<br /></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}=(a_{1},a_{2})\cdot (b_{1},b_{2})+(a_{1},a_{2})\cdot (c_{1},c_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}=(a_{1},a_{2})\cdot (b_{1},b_{2})+(a_{1},a_{2})\cdot (c_{1},c_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0e78d1b66c858f030f87bbbbd14b264aeabb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.005ex; height:3.176ex;" alt="{\displaystyle (a_{1},a_{2})\cdot {\big (}(b_{1},b_{2})+(c_{1},c_{2}){\big )}=(a_{1},a_{2})\cdot (b_{1},b_{2})+(a_{1},a_{2})\cdot (c_{1},c_{2})}"></span> (distributivita sčítání přes násobení)</li></ol> <p>Tyto vlastnosti dokládají, že množina komplexních čísel, spolu s takto definovaným sčítáním a násobením, tvoří <a href="/wiki/T%C4%9Bleso_(algebra)" title="Těleso (algebra)">těleso</a>. Neformálně řečeno, výše uvedené vlastnosti nás opravňují takto definované objekty nazývat čísly. </p><p>Prvek tvaru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7e272c21fb886ae2ab806119c8cd35c2a52754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle (a,0)}"></span> jednoznačně odpovídá reálnému číslu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, zavedené operace jsou rozšířením operací v reálném oboru (mají stejné výsledky). To nás opravňuje ke zkrácenému značení, kdy místo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7e272c21fb886ae2ab806119c8cd35c2a52754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle (a,0)}"></span> píšeme pouze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Prvek <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> pak nazveme <a href="/wiki/Imagin%C3%A1rn%C3%AD_jednotka" title="Imaginární jednotka">imaginární jednotkou</a> (zapisujeme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} =(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} =(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b403c65d0ada186abf24f1180513b4a75b03bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.913ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} =(0,1)}"></span>). Tím získáme obvyklé značení <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)=a+\mathrm {i} b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)=a+\mathrm {i} b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbf03e2c870a25959dcaecb64472eb436440d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.884ex; height:2.843ex;" alt="{\displaystyle (a,b)=a+\mathrm {i} b}"></span>. Pro číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f0f09f6fc40e634d34aed6e205ac0f7a40e062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }"></span> pak z definice platí očekávaný vztah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} ^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1903180d4ecaf05f4aef81a8ad389025f89c9d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.77ex; height:2.843ex;" alt="{\displaystyle \mathrm {i} ^{2}=-1}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Reprezentace_maticí"><span id="Reprezentace_matic.C3.AD"></span>Reprezentace maticí</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=16" title="Editace sekce: Reprezentace maticí" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=16" title="Editovat zdrojový kód sekce Reprezentace maticí"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplexní číslo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}"></span> můžeme reprezentovat <a href="/wiki/%C4%8Ctvercov%C3%A1_matice" title="Čtvercová matice">čtvercovou maticí</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span> reálných čísel ve formě: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874762a513e504f9a2b5854c093217705dc57b43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.531ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}"></span> </p><p>Potom platí následující vlastnosti: </p> <ol><li>Sčítání matic odpovídá sčítání komplexních čísel</li> <li><a href="/wiki/N%C3%A1soben%C3%AD_matic" title="Násobení matic">Násobení matic</a> odpovídá násobení komplexních čísel</li> <li><a href="/wiki/Determinant" title="Determinant">Determinant</a> matice odpovídá kvadrátu absolutní hodnoty komplexního čísla</li> <li><a href="/wiki/Transpozice_matice" title="Transpozice matice">Transpozice matice</a> odpovídá operaci komplexního sdružení</li> <li><a href="/wiki/Inverzn%C3%AD_matice" title="Inverzní matice">Inverzní matice</a> odpovídá převrácené hodnotě komplexního čísla</li></ol> <div class="mw-heading mw-heading2"><h2 id="Literatura">Literatura</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=17" title="Editace sekce: Literatura" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=17" title="Editovat zdrojový kód sekce Literatura"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Miloš Ráb: <i>Komplexní čísla v elementární matematice</i>, Masarykova univerzita, Brno, 1997, <span style="white-space:nowrap"><a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/80-210-1475-X" title="Speciální:Zdroje knih/80-210-1475-X"><span class="ISBN">80-210-1475-X</span></a></span></li> <li>Abramowitz and Stegun, <i>Handbook of Mathematical functions with formulas, graphs, and mathematical tables</i>. United States Department of Commerce, National Bureau of Standards, 1972.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=18" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=18" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Komplexn%C4%9B_sdru%C5%BEen%C3%A9_%C4%8D%C3%ADslo" title="Komplexně sdružené číslo">Komplexně sdružené číslo</a></li> <li><a href="/wiki/Komplexn%C3%AD_rovina" title="Komplexní rovina">Komplexní rovina</a></li> <li><a href="/wiki/Riemannova_sf%C3%A9ra" title="Riemannova sféra">Riemannova sféra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&veaction=edit&section=19" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Komplexn%C3%AD_%C4%8D%C3%ADslo&action=edit&section=19" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons"><img alt="Logo Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://commons.wikimedia.org/wiki/Category:Complex_numbers" class="extiw" title="c:Category:Complex numbers">komplexní číslo</a></span> na <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></span></span></span><i> </i></li> <li><a rel="nofollow" class="external text" href="http://www.karlin.mff.cuni.cz/~robova/stranky/silarova/index.html">Komplexní čísla ve výuce matematiky na střední škole s využitím internetu, Lenka Šilarová, diplomová práce MFF UK</a></li> <li><a rel="nofollow" class="external text" href="http://artemis.osu.cz/mmmat/txt/sm/kxo.htm">Repetitorium středoškolské matematiky (Ostravská univerzita) – Komplexní čísla</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output .navbox2{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox2 .navbox2{margin-top:0}.mw-parser-output .navbox2+.navbox2{margin-top:-1px}.mw-parser-output .navbox2-inner,.mw-parser-output .navbox2-subgroup{width:100%}.mw-parser-output .navbox2-group,.mw-parser-output .navbox2-title,.mw-parser-output .navbox2-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output th.navbox2-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox2,.mw-parser-output .navbox2-subgroup{background-color:#fdfdfd}.mw-parser-output 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href="/wiki/Kategorie:Komplexn%C3%AD_%C4%8D%C3%ADsla" title="Kategorie:Komplexní čísla">Komplexní čísla</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Skryté kategorie: <ul><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_NKC" title="Kategorie:Monitoring:Články s identifikátorem NKC">Monitoring:Články s identifikátorem NKC</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_PSH" title="Kategorie:Monitoring:Články s identifikátorem PSH">Monitoring:Články s identifikátorem PSH</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_BNF" title="Kategorie:Monitoring:Články s identifikátorem BNF">Monitoring:Články s identifikátorem BNF</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_GND" title="Kategorie:Monitoring:Články s identifikátorem GND">Monitoring:Články s identifikátorem GND</a></li><li><a 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