Número complexo – Wikipédia, a enciclopédia livre

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</a> </li> <li id="toc-História" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#História"> <div class="vector-toc-text"> 1 História </div> </a> <ul id="toc-História-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definições" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definições"> <div class="vector-toc-text"> 2 Definições </div> </a> <button aria-controls="toc-Definições-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Alternar a subsecção Definições </button> <ul id="toc-Definições-sublist" class="vector-toc-list"> <li id="toc-Plano_complexo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plano_complexo"> <div class="vector-toc-text"> 2.1 Plano complexo </div> </a> <ul id="toc-Plano_complexo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operações_elementares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operações_elementares"> <div class="vector-toc-text"> 2.2 Operações elementares </div> </a> <ul id="toc-Operações_elementares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_módulo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#O_módulo"> <div class="vector-toc-text"> 2.3 O módulo </div> </a> <ul id="toc-O_módulo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriedades_algébricas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propriedades_algébricas"> <div class="vector-toc-text"> 3 Propriedades algébricas </div> </a> <button aria-controls="toc-Propriedades_algébricas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Alternar a subsecção Propriedades algébricas </button> <ul id="toc-Propriedades_algébricas-sublist" class="vector-toc-list"> <li id="toc-Radical_algébrico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radical_algébrico"> <div class="vector-toc-text"> 3.1 Radical algébrico </div> </a> <ul id="toc-Radical_algébrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estrutura_de_campo[8]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estrutura_de_campo[8]"> <div class="vector-toc-text"> 3.2 Estrutura de campo[8] </div> </a> <ul id="toc-Estrutura_de_campo[8]-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriedades_topológicas_e_analíticas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propriedades_topológicas_e_analíticas"> <div class="vector-toc-text"> 4 Propriedades topológicas e analíticas </div> </a> <button aria-controls="toc-Propriedades_topológicas_e_analíticas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Alternar a subsecção Propriedades topológicas e analíticas </button> <ul id="toc-Propriedades_topológicas_e_analíticas-sublist" class="vector-toc-list"> <li id="toc-Convergência_nos_complexos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergência_nos_complexos"> <div class="vector-toc-text"> 4.1 Convergência nos complexos </div> </a> <ul id="toc-Convergência_nos_complexos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-O_conjunto_dos_números_complexos_como_extensão_algébrica" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#O_conjunto_dos_números_complexos_como_extensão_algébrica"> <div class="vector-toc-text"> 5 O conjunto dos números complexos como extensão algébrica </div> </a> <button aria-controls="toc-O_conjunto_dos_números_complexos_como_extensão_algébrica-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Alternar a subsecção O conjunto dos números complexos como extensão algébrica </button> <ul id="toc-O_conjunto_dos_números_complexos_como_extensão_algébrica-sublist" class="vector-toc-list"> <li id="toc-Logaritmos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logaritmos"> <div class="vector-toc-text"> 5.1 Logaritmos </div> </a> <ul id="toc-Logaritmos-sublist" class="vector-toc-list"> <li id="toc-Função_logarítmica_natural" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Função_logarítmica_natural"> <div class="vector-toc-text"> 5.1.1 Função logarítmica natural </div> </a> <ul id="toc-Função_logarítmica_natural-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Função_logarítmica_decimal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Função_logarítmica_decimal"> <div class="vector-toc-text"> 5.1.2 Função logarítmica decimal </div> </a> <ul id="toc-Função_logarítmica_decimal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Gráficos_de_funções_complexas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Gráficos_de_funções_complexas"> <div class="vector-toc-text"> 6 Gráficos de funções complexas </div> </a> <ul id="toc-Gráficos_de_funções_complexas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forma_trigonométrica_dos_números_complexos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Forma_trigonométrica_dos_números_complexos"> <div class="vector-toc-text"> 7 Forma trigonométrica dos números complexos </div> </a> <button aria-controls="toc-Forma_trigonométrica_dos_números_complexos-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Alternar a subsecção Forma trigonométrica dos números complexos </button> <ul id="toc-Forma_trigonométrica_dos_números_complexos-sublist" class="vector-toc-list"> <li id="toc-Representação_trigonométrica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representação_trigonométrica"> <div class="vector-toc-text"> 7.1 Representação trigonométrica </div> </a> <ul id="toc-Representação_trigonométrica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Igualdade_de_números_complexos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Igualdade_de_números_complexos"> <div class="vector-toc-text"> 7.2 Igualdade de números complexos </div> </a> <ul id="toc-Igualdade_de_números_complexos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simétrico_de_um_número_complexo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simétrico_de_um_número_complexo"> <div class="vector-toc-text"> 7.3 Simétrico de um número complexo </div> </a> <ul id="toc-Simétrico_de_um_número_complexo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjugado_de_um_número_complexo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conjugado_de_um_número_complexo"> <div class="vector-toc-text"> 7.4 Conjugado de um número complexo </div> </a> <ul id="toc-Conjugado_de_um_número_complexo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Produto_dos_números_complexos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Produto_dos_números_complexos"> <div class="vector-toc-text"> 7.5 Produto dos números complexos </div> </a> <ul id="toc-Produto_dos_números_complexos-sublist" class="vector-toc-list"> <li id="toc-O_produto_de_um_número_complexo_Z_por_um_número_real_K" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#O_produto_de_um_número_complexo_Z_por_um_número_real_K"> <div class="vector-toc-text"> 7.5.1 O produto de um número complexo Z por um número real K </div> </a> <ul id="toc-O_produto_de_um_número_complexo_Z_por_um_número_real_K-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_produto_de_um_número_complexo_Z_por_um_imaginário_puro" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#O_produto_de_um_número_complexo_Z_por_um_imaginário_puro"> <div class="vector-toc-text"> 7.5.2 O produto de um número complexo Z por um imaginário puro </div> </a> <ul id="toc-O_produto_de_um_número_complexo_Z_por_um_imaginário_puro-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_produto_de_um_número_complexo_genérico_Z_por_um_outro_número_complexo_W" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#O_produto_de_um_número_complexo_genérico_Z_por_um_outro_número_complexo_W"> <div class="vector-toc-text"> 7.5.3 O produto de um número complexo genérico Z por um outro número complexo W </div> </a> <ul id="toc-O_produto_de_um_número_complexo_genérico_Z_por_um_outro_número_complexo_W-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Soma_dos_números_complexos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Soma_dos_números_complexos"> <div class="vector-toc-text"> 7.6 Soma dos números complexos </div> </a> <ul id="toc-Soma_dos_números_complexos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> 8 Ver também </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> 9 Referências </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> 10 Ligações externas </div> </a> <ul id="toc-Ligações_externas-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="Conteúdo" 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="Alternar o índice" > <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" > Alternar o índice </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">Número complexo</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="Ir para um artigo noutra língua. Disponível em 132 línguas" > <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" > 132 línguas </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 — africanês" lang="af" hreflang="af" data-title="Komplekse getal" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl — alemão suíço" lang="gsw" hreflang="gsw" data-title="Komplexe Zahl" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target">Alemannisch</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 — amárico" lang="am" hreflang="am" data-title="የአቅጣጫ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_complexo" title="Numero complexo — aragonês" lang="an" hreflang="an" data-title="Numero complexo" data-language-autonym="Aragonés" data-language-local-name="aragonês" class="interlanguage-link-target">Aragonés</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">अंगिका</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 — árabe" lang="ar" hreflang="ar" data-title="عدد مركب" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target">العربية</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 — assamês" lang="as" hreflang="as" data-title="জটিল সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="assamês" class="interlanguage-link-target">অসমীয়া</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 — asturiano" lang="ast" hreflang="ast" data-title="Númberu complexu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target">Asturianu</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 — azerbaijano" lang="az" hreflang="az" data-title="Kompleks ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target">Azərbaycanca</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">تۆرکجه</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 — bashkir" lang="ba" hreflang="ba" data-title="Комплекслы һан" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target">Башҡортса</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 — Samogitian" lang="sgs" hreflang="sgs" data-title="Kuompleksėnis skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</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 — bielorrusso" lang="be" hreflang="be" data-title="Камплексны лік" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%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">Беларуская (тарашкевіца)</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 — búlgaro" lang="bg" hreflang="bg" data-title="Комплексно число" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target">Български</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 — bengalês" lang="bn" hreflang="bn" data-title="জটিল সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj — bósnio" lang="bs" hreflang="bs" data-title="Kompleksan broj" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target">Bosanski</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">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_complex" title="Nombre complex — catalão" lang="ca" hreflang="ca" data-title="Nombre complex" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target">Català</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 — curdo central" lang="ckb" hreflang="ckb" data-title="ژمارەی ئاوێتە" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Komplexn%C3%AD_%C4%8D%C3%ADslo" title="Komplexní číslo — checo" lang="cs" hreflang="cs" data-title="Komplexní číslo" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target">Čeština</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 — chuvash" lang="cv" hreflang="cv" data-title="Комплекслă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cymhlyg" title="Rhif cymhlyg — galês" lang="cy" hreflang="cy" data-title="Rhif cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="galês" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="artigo recomendado"><a href="https://da.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal — dinamarquês" lang="da" hreflang="da" data-title="Komplekse tal" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl — alemão" lang="de" hreflang="de" data-title="Komplexe Zahl" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target">Deutsch</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">Zazaki</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 — grego" lang="el" hreflang="el" data-title="Μιγαδικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target">Ελληνικά</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">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complex_number" title="Complex number — inglês" lang="en" hreflang="en" data-title="Complex number" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target">English</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">Esperanto</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 — espanhol" lang="es" hreflang="es" data-title="Número complejo" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv — estónio" lang="et" hreflang="et" data-title="Kompleksarv" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_konplexu" title="Zenbaki konplexu — basco" lang="eu" hreflang="eu" data-title="Zenbaki konplexu" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target">Euskara</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 — persa" lang="fa" hreflang="fa" data-title="عدد مختلط" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kompleksiluku" title="Kompleksiluku — finlandês" lang="fi" hreflang="fi" data-title="Kompleksiluku" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv — Võro" lang="vro" hreflang="vro" data-title="Kompleksarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Komplekst_tal" title="Komplekst tal — feroês" lang="fo" hreflang="fo" data-title="Komplekst tal" data-language-autonym="Føroyskt" data-language-local-name="feroês" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_complexe" title="Nombre complexe — francês" lang="fr" hreflang="fr" data-title="Nombre complexe" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kompleks_taal" title="Kompleks taal — frísio setentrional" lang="frr" hreflang="frr" data-title="Kompleks taal" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kompleks_getal" title="Kompleks getal — frísico ocidental" lang="fy" hreflang="fy" data-title="Kompleks getal" data-language-autonym="Frysk" data-language-local-name="frísico ocidental" class="interlanguage-link-target">Frysk</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 — irlandês" lang="ga" hreflang="ga" data-title="Uimhir choimpléascach" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target">Gaeilge</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 — gan" lang="gan" hreflang="gan" data-title="複數" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/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">Kriyòl gwiyannen</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 — galego" lang="gl" hreflang="gl" data-title="Número complexo" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target">Galego</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 — guarani" lang="gn" hreflang="gn" data-title="Papapy rypy'ũ" data-language-autonym="Avañe'ẽ" data-language-local-name="guarani" class="interlanguage-link-target">Avañe'ẽ</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 — guzerate" lang="gu" hreflang="gu" data-title="સંકર સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="guzerate" class="interlanguage-link-target">ગુજરાતી</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 — hebraico" lang="he" hreflang="he" data-title="מספר מרוכב" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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 — hindi" lang="hi" hreflang="hi" data-title="समिश्र संख्या" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Jatil_ginti" title="Jatil ginti — Fiji Hindi" lang="hif" hreflang="hif" data-title="Jatil ginti" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kompleksni_broj" title="Kompleksni broj — croata" lang="hr" hreflang="hr" data-title="Kompleksni broj" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target">Hrvatski</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 — húngaro" lang="hu" hreflang="hu" data-title="Komplex számok" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target">Magyar</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 — arménio" lang="hy" hreflang="hy" data-title="Կոմպլեքս թիվ" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_complexe" title="Numero complexe — interlíngua" lang="ia" hreflang="ia" data-title="Numero complexe" data-language-autonym="Interlingua" data-language-local-name="interlíngua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_kompleks" title="Lumur kompleks — iban" lang="iba" hreflang="iba" data-title="Lumur kompleks" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target">Jaku Iban</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_kompleks" title="Bilangan kompleks — indonésio" lang="id" hreflang="id" data-title="Bilangan kompleks" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target">Bahasa Indonesia</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">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tvinnt%C3%B6lur" title="Tvinntölur — islandês" lang="is" hreflang="is" data-title="Tvinntölur" data-language-autonym="Íslenska" data-language-local-name="islandês" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_complesso" title="Numero complesso — italiano" lang="it" hreflang="it" data-title="Numero complesso" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target">Italiano</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 — japonês" lang="ja" hreflang="ja" data-title="複素数" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Komplex_nomba" title="Komplex nomba — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Komplex nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/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">La .lojban.</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 — georgiano" lang="ka" hreflang="ka" data-title="კომპლექსური რიცხვი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target">ქართული</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 — kabyle" lang="kab" hreflang="kab" data-title="Amḍan asemlal" data-language-autonym="Taqbaylit" data-language-local-name="kabyle" class="interlanguage-link-target">Taqbaylit</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">Kabɩyɛ</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 — cazaque" lang="kk" hreflang="kk" data-title="Кешен сан" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target">Қазақша</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 — khmer" lang="km" hreflang="km" data-title="ចំនួនកុំផ្លិច" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target">ភាសាខ្មែរ</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 — coreano" lang="ko" hreflang="ko" data-title="복소수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niver_kompleth" title="Niver kompleth — córnico" lang="kw" hreflang="kw" data-title="Niver kompleth" data-language-autonym="Kernowek" data-language-local-name="córnico" class="interlanguage-link-target">Kernowek</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 — quirguiz" lang="ky" hreflang="ky" data-title="Комплекстүү сан" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_complexus" title="Numerus complexus — latim" lang="la" hreflang="la" data-title="Numerus complexus" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Complex_getal" title="Complex getal — limburguês" lang="li" hreflang="li" data-title="Complex getal" data-language-autonym="Limburgs" data-language-local-name="limburguês" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_compless" title="Numer compless — lombardo" lang="lmo" hreflang="lmo" data-title="Numer compless" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AA%E0%BA%BB%E0%BA%99" title="ຈຳນວນສົນ — laosiano" lang="lo" hreflang="lo" data-title="ຈຳນວນສົນ" data-language-autonym="ລາວ" data-language-local-name="laosiano" class="interlanguage-link-target">ລາວ</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 — lituano" lang="lt" hreflang="lt" data-title="Kompleksinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komplekss_skaitlis" title="Komplekss skaitlis — letão" lang="lv" hreflang="lv" data-title="Komplekss skaitlis" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_haro" title="Isa haro — malgaxe" lang="mg" hreflang="mg" data-title="Isa haro" data-language-autonym="Malagasy" data-language-local-name="malgaxe" class="interlanguage-link-target">Malagasy</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 — macedónio" lang="mk" hreflang="mk" data-title="Комплексен број" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%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 — malaiala" lang="ml" hreflang="ml" data-title="മിശ്രസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target">മലയാളം</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 — mongol" lang="mn" hreflang="mn" data-title="Комплекс тоо" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target">Монгол</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 — marata" lang="mr" hreflang="mr" data-title="संमिश्र संख्या" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_kompleks" title="Nombor kompleks — malaio" lang="ms" hreflang="ms" data-title="Nombor kompleks" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target">Bahasa Melayu</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 — birmanês" lang="my" hreflang="my" data-title="ကွန်ပလက်စ်ကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanês" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Komplexe_Tall" title="Komplexe Tall — baixo-alemão" lang="nds" hreflang="nds" data-title="Komplexe Tall" data-language-autonym="Plattdüütsch" data-language-local-name="baixo-alemão" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Complex_getal" title="Complex getal — neerlandês" lang="nl" hreflang="nl" data-title="Complex getal" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Komplekse tal" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Komplekst_tall" title="Komplekst tall — norueguês bokmål" lang="nb" hreflang="nb" data-title="Komplekst tall" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target">Norsk bokmål</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 — occitano" lang="oc" hreflang="oc" data-title="Nombre complèxe" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Xaxxamaa" title="Lakkoofsa Xaxxamaa — oromo" lang="om" hreflang="om" data-title="Lakkoofsa Xaxxamaa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target">Oromoo</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 — ossético" lang="os" hreflang="os" data-title="Комплексон нымæц" data-language-autonym="Ирон" data-language-local-name="ossético" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%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 — panjabi" lang="pa" hreflang="pa" data-title="ਕੰਪਲੈਕਸ ਨੰਬਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_zespolone" title="Liczby zespolone — polaco" lang="pl" hreflang="pl" data-title="Liczby zespolone" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target">Polski</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 — Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer compless" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%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">پنجابی</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 — romeno" lang="ro" hreflang="ro" data-title="Număr complex" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="artigo destacado"><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 — russo" lang="ru" hreflang="ru" data-title="Комплексное число" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target">Русский</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 — Rusyn" lang="rue" hreflang="rue" data-title="Комплексне чісло" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</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 — sakha" lang="sah" hreflang="sah" data-title="Комплекс ахсаан" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target">Саха тыла</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 — siciliano" lang="scn" hreflang="scn" data-title="Nùmmuru cumplessu" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Complex_nummer" title="Complex nummer — scots" lang="sco" hreflang="sco" data-title="Complex nummer" data-language-autonym="Scots" data-language-local-name="scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj — servo-croata" lang="sh" hreflang="sh" data-title="Kompleksan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</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 — cingalês" lang="si" hreflang="si" data-title="සංකීර්ණ සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="cingalês" class="interlanguage-link-target">සිංහල</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">Simple English</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 — eslovaco" lang="sk" hreflang="sk" data-title="Komplexné číslo" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kompleksno_%C5%A1tevilo" title="Kompleksno število — esloveno" lang="sl" hreflang="sl" data-title="Kompleksno število" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kompleksloho" title="Kompleksloho — inari sami" lang="smn" hreflang="smn" data-title="Kompleksloho" data-language-autonym="Anarâškielâ" data-language-local-name="inari sami" class="interlanguage-link-target">Anarâškielâ</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_kakan" title="Thiin kakan — somali" lang="so" hreflang="so" data-title="Thiin kakan" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target">Soomaaliga</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 — albanês" lang="sq" hreflang="sq" data-title="Numrat kompleksë" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target">Shqip</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 — sérvio" lang="sr" hreflang="sr" data-title="Комплексан број" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komplexa_tal" title="Komplexa tal — sueco" lang="sv" hreflang="sv" data-title="Komplexa tal" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_changamano" title="Namba changamano — suaíli" lang="sw" hreflang="sw" data-title="Namba changamano" data-language-autonym="Kiswahili" data-language-local-name="suaíli" class="interlanguage-link-target">Kiswahili</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 — tâmil" lang="ta" hreflang="ta" data-title="சிக்கலெண்" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target">தமிழ்</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 — telugu" lang="te" hreflang="te" data-title="సంకీర్ణ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target">తెలుగు</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 — tajique" lang="tg" hreflang="tg" data-title="Адади комплексӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%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 — tailandês" lang="th" hreflang="th" data-title="จำนวนเชิงซ้อน" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Komplikadong_bilang" title="Komplikadong bilang — tagalo" lang="tl" hreflang="tl" data-title="Komplikadong bilang" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target">Tagalog</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 — turco" lang="tr" hreflang="tr" data-title="Karmaşık sayı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target">Türkçe</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 — tatar" lang="tt" hreflang="tt" data-title="Комплекс сан" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target">Татарча / tatarça</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 — ucraniano" lang="uk" hreflang="uk" data-title="Комплексне число" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target">Українська</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 — urdu" lang="ur" hreflang="ur" data-title="مخلوط عدد" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kompleks_sonlar" title="Kompleks sonlar — usbeque" lang="uz" hreflang="uz" data-title="Kompleks sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Numaro_conpleso" title="Numaro conpleso — Venetian" lang="vec" hreflang="vec" data-title="Numaro conpleso" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</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 — vietnamita" lang="vi" hreflang="vi" data-title="Số phức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Complexe_getalln" title="Complexe getalln — West Flemish" lang="vls" hreflang="vls" data-title="Complexe getalln" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</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 — waray" lang="war" hreflang="war" data-title="Komplikado nga ihap" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target">Winaray</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） — wu" lang="wuu" hreflang="wuu" data-title="复数（数学）" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%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 — kalmyk" lang="xal" hreflang="xal" data-title="Комплексин тойг" data-language-autonym="Хальмг" data-language-local-name="kalmyk" class="interlanguage-link-target">Хальмг</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 — iídiche" lang="yi" hreflang="yi" data-title="קאמפלעקסע צאל" data-language-autonym="ייִדיש" data-language-local-name="iídiche" class="interlanguage-link-target">ייִדיש</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 — ioruba" lang="yo" hreflang="yo" data-title="Nọ́mbà tóṣòro" data-language-autonym="Yorùbá" data-language-local-name="ioruba" 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Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (Encontre fontes: <a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  • <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  • <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=N%C3%BAmero+complexo">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=N%C3%BAmero+complexo&oq=N%C3%BAmero+complexo">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=N%C3%BAmero+complexo">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=N%C3%BAmero+complexo">A</a>)). (Abril de 2021)</div></td></tr></tbody></table> <table style="float: right; background:white; margin: 0 0 0.5em 1em; width: 250px;" id="toc"> <tbody><tr> <td> </td></tr> <tr style="text-align: center; background:#ccccff;"> <td>Conjuntos de números </td></tr> <tr style="text-align: center;"> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>⊂</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732e95003e763760b0f84db054b6dc6e46c82801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.515ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>⊂</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9feb18f9b160bf251490ff2711ec5e9b72e4d5fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.186ex; height:2.176ex;" alt="{\displaystyle \mathbb {I} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \cdots }"> <div class="plainlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/N%C3%BAmero_natural" title="Número natural">Naturais</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></li> <li><a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro">Inteiros</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></li> <li><a href="/wiki/N%C3%BAmero_racional" title="Número racional">Racionais</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></li> <li><a href="/wiki/N%C3%BAmero_irracional" title="Número irracional">Irracionais</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8205f06e0d279689ed04a1ac04a3d9c249c637df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle \mathbb {I} }"> (Este conjunto não faz parte dos conjuntos anteriores mas está contido em <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">)</li> <li><a href="/wiki/N%C3%BAmero_real" title="Número real">Reais</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></li> <li><a href="/wiki/N%C3%BAmero_imagin%C3%A1rio" title="Número imaginário">Imaginários</a></li> <li><a class="mw-selflink selflink">Complexos</a> <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><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} }"></li> <li><a href="/wiki/N%C3%BAmero_hiper-real" title="Número hiper-real">Números hiper-reais</a></li> <li><a href="/wiki/N%C3%BAmero_hipercomplexo" title="Número hipercomplexo">Números hipercomplexos</a></li></ul> </div> </td></tr> <tr style="text-align: center;"> <td> <div class="plainlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/Quaterni%C3%A3o" title="Quaternião">Quaterniões</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></li> <li><a href="/wiki/Octoni%C3%A3o" title="Octonião">Octoniões</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></li> <li><a href="/wiki/Sedeni%C3%A3o" title="Sedenião">Sedeniões</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></li> <li><a href="/wiki/N%C3%BAmero_complexo_hiperb%C3%B3lico" title="Número complexo hiperbólico">Complexos hiperbólicos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1,1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1,1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7352b456c6586e1cb3b7e46ecd29047709daa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.012ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1,1}}"></li> <li><a href="/wiki/Quaterni%C3%A3o_hiperb%C3%B3lico" title="Quaternião hiperbólico">Quaterniões hiperbólicos</a></li> <li><a href="/wiki/N%C3%BAmero_bicomplexo" title="Número bicomplexo">Bicomplexos</a></li> <li><a href="/w/index.php?title=Biquaterni%C3%B5es&action=edit&redlink=1" class="new" title="Biquaterniões (página não existe)">Biquaterniões</a></li> <li><a href="/w/index.php?title=Coquaterni%C3%B5es&action=edit&redlink=1" class="new" title="Coquaterniões (página não existe)">Coquaterniões</a></li></ul> </div> </td></tr></tbody></table> Em <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a>, um número complexo é um elemento de um <a href="/wiki/Sistema_num%C3%A9rico" class="mw-redirect" title="Sistema numérico">sistema numérico</a> que contém os <a href="/wiki/N%C3%BAmeros_reais" class="mw-redirect" title="Números reais">números reais</a> e um elemento específico denotado i, chamado de unidade imaginária, e que satisfaz a equação i2 = −1. O fato de um <a href="/wiki/N%C3%BAmero_negativo" title="Número negativo">número negativo</a> não ter <a href="/wiki/Raiz_quadrada" title="Raiz quadrada">raiz quadrada</a> parece ter sido claro para os matemáticos que se depararam com esta questão, até a concepção do modelo dos números complexos.<a href="#cite_note-PRINC-1">[1]</a><a href="#cite_note-INTRO-2">[2]</a> Um número complexo é um <a href="/wiki/N%C3%BAmero" title="Número">número</a> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> que pode ser escrito na forma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+yi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+yi}">, sendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> números reais e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> denota a <a href="/wiki/Unidade_imagin%C3%A1ria" title="Unidade imaginária">unidade imaginária</a>. Esta tem a propriedade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a48123ed64e9456cbbd8f995aeb8893f18a2d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.573ex; height:3.009ex;" alt="{\displaystyle i^{2}=-1,}"> sendo que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> são chamados respectivamente parte real e parte imaginária de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">.<a href="#cite_note-3">[3]</a><a href="#cite_note-GIEZZI-4">[4]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Illustration_of_a_complex_number.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Illustration_of_a_complex_number.svg/220px-Illustration_of_a_complex_number.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Illustration_of_a_complex_number.svg/330px-Illustration_of_a_complex_number.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Illustration_of_a_complex_number.svg/440px-Illustration_of_a_complex_number.svg.png 2x" data-file-width="832" data-file-height="754" /></a><figcaption></figcaption></figure> O conjunto dos números complexos, denotado por <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><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} }">, contém o conjunto dos <a href="/wiki/N%C3%BAmeros_reais" class="mw-redirect" title="Números reais">números reais</a>. Munido de operações de adição e multiplicação obtidas por extensão das operações de mesma denominação nos números reais, adquire uma <a href="/wiki/Estrutura_alg%C3%A9brica" title="Estrutura algébrica">estrutura algébrica</a> denominada <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a> <a href="/wiki/Corpo_algebricamente_fechado" title="Corpo algebricamente fechado">algebricamente fechado</a>, sendo que esse fechamento consiste na propriedade que tem o <a href="/wiki/Conjunto" title="Conjunto">conjunto</a> de possuir todas as soluções de qualquer <a href="/wiki/Equa%C3%A7%C3%A3o_polinomial" title="Equação polinomial">equação polinomial</a> com coeficientes naquele mesmo conjunto (no caso, o conjunto dos complexos). O conjunto dos números complexos também pode ser entendido por seu <a href="/wiki/Isomorfismo" title="Isomorfismo">isomorfismo</a> com um <a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">espaço vetorial</a> sobre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, o conjunto dos reais.<a href="#cite_note-5">[5]</a> Além disso, a cada número complexo podemos atribuir um número real positivo chamado <a href="/wiki/M%C3%B3dulo_(%C3%A1lgebra)" title="Módulo (álgebra)">módulo</a>, dado por: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|={\sqrt {x^{2}+y^{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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bd44a6d60e8a02c0646ab894fd7b9743eab576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.89ex; height:4.843ex;" alt="{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}"></dd></dl> O módulo de z, visto como uma <a href="/wiki/Norma_(matem%C3%A1tica)" title="Norma (matemática)">norma</a> no espaço vetorial, conduz a um <a href="/wiki/Espa%C3%A7o_normado" class="mw-redirect" title="Espaço normado">espaço normado</a> topologicamente <a href="/wiki/Espa%C3%A7o_completo" title="Espaço completo">completo</a>.[<a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">carece de fontes</a>?] Os números complexos são representados geometricamente no plano complexo. Nele, representa-se a parte real, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> no eixo horizontal e a parte imaginária, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd9829c9ef4adcb0f9f5d53b27463a873a8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y,}"> no eixo vertical. Os números complexos são utilizados em várias áreas do conhecimento, tais como <a href="/wiki/Engenharia" title="Engenharia">engenharia</a>, <a href="/wiki/Eletromagnetismo" title="Eletromagnetismo">eletromagnetismo</a>, <a href="/wiki/F%C3%ADsica_qu%C3%A2ntica" class="mw-redirect" title="Física quântica">física quântica</a>, <a href="/wiki/Teoria_do_caos" title="Teoria do caos">teoria do caos</a>, <a href="/wiki/Processamento_de_sinal" title="Processamento de sinal">processamento de sinais</a>, teoria de controle, dinâmica de fluidos, <a href="/wiki/Cartografia" title="Cartografia">cartografia</a>, análise de vibração, além da própria matemática, em que são estudadas análise complexa, álgebra linear complexa, álgebra de Lie complexa, com aplicações em resolução de <a href="/wiki/Equa%C3%A7%C3%A3o_alg%C3%A9brica" title="Equação algébrica">equações algébricas</a> e <a href="/wiki/Equa%C3%A7%C3%A3o_diferencial" title="Equação diferencial">equações diferenciais</a>.<a href="#cite_note-6">[6]</a> Em algumas situações, é comum a troca da letra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> pela letra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcff3a7481f4597bfd961687562f495e266d2b32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.632ex; height:2.509ex;" alt="{\displaystyle j,}"> devido ao frequente uso da primeira como indicação de <a href="/wiki/Corrente_el%C3%A9trica" title="Corrente elétrica">corrente elétrica</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="História">História</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=1" title="Editar secção: História" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=1" title="Editar código-fonte da secção: História">editar código-fonte</a>]</div> O conceito de número complexo teve um desenvolvimento gradual. Começaram a ser utilizados formalmente no século XVI em fórmulas de resolução de <a href="/wiki/Equa%C3%A7%C3%A3o_do_terceiro_grau" class="mw-redirect" title="Equação do terceiro grau">equações de terceiro</a> e quarto graus.<a href="#cite_note-USP-7">[7]</a> Os primeiros que conseguiram dar soluções a equações cúbicas foram <a href="/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a> e <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" title="Niccolò Fontana Tartaglia">Tartaglia</a>. Este último, depois de ter sido alvo de muita insistência, passou os resultados que tinha obtido a <a href="/wiki/Girolamo_Cardano" title="Girolamo Cardano">Girolamo Cardano</a>, que prometeu não divulgá-los. Cardano, depois de conferir a exatidão das resoluções de Tartaglia, não honrou sua promessa e publicou os resultados, mencionando o autor, em sua obra <a href="/wiki/Ars_Magna" title="Ars Magna">Ars Magna</a> de 1545, iniciando uma enorme inimizade. A fórmula deduzida por Tartaglia afirmava que a solução da equação <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}+px+q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>x</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}+px+q=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a4d20e12d57f5645758aec0e916f800a81de47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.894ex; height:3.009ex;" alt="{\displaystyle x^{3}+px+q=0}"> era dada por <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}+{\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}+{\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301a9f5e6bb5c6f7efc67fde59b80a5051c73159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.513ex; height:7.676ex;" alt="{\displaystyle x={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}+{\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {\left({\frac {q}{2}}\right)^{2}+\left({\frac {p}{3}}\right)^{3}}}}}.}"></dd></dl> Um problema inquietante percebido na época foi que algumas equações (as equações que têm três raízes reais, chamadas de <a href="/wiki/Casus_irreducibilis" title="Casus irreducibilis">casus irreducibilis</a>) levavam a raízes quadradas de números negativos. Por exemplo, a equação: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-15x-4=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-15x-4=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f93d3cbfd324e74c862e1c511c39e510d1c8a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.143ex; height:2.843ex;" alt="{\displaystyle x^{3}-15x-4=0}"></dd></dl> tem três raízes reais, como se pode observar facilmente ou pelo gráfico da função: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{3}-15x-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{3}-15x-4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d258a66d65aa2092a892b6818c6b4253a3a3a343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.398ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{3}-15x-4}"></dd></dl> ou por fatoração: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-15x-4=(x-4)(x^{2}+4x+1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-15x-4=(x-4)(x^{2}+4x+1)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f128c578d6d45226fde4053443bc16d6250af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.912ex; height:3.176ex;" alt="{\displaystyle x^{3}-15x-4=(x-4)(x^{2}+4x+1)=0}"></dd></dl> se e somente se: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=4;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=4;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2914e6a124436d58776dde0ff47fbf4e85a2e6b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x=4;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-2-{\sqrt {3}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-2-{\sqrt {3}};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2ce302eb1ac9391b0d4fa63fb442dd41af95b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.984ex; height:2.843ex;" alt="{\displaystyle x=-2-{\sqrt {3}};}"></dd></dl> ou: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-2+{\sqrt {3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-2+{\sqrt {3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37faa34f4f0b89347c5ce56a45b18b5c0202098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.984ex; height:2.843ex;" alt="{\displaystyle x=-2+{\sqrt {3}}.}"></dd></dl> Entretanto, usando-se a fórmula de Tartaglia, chega-se a: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573262f780972f7c7b785df859855e4894ca6709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:34.385ex; height:4.843ex;" alt="{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}}"></dd></dl> Essa questão evidenciou o fato de que havia mais a se investigar e a se aprender sobre os números. <a href="/wiki/Rafael_Bombelli" title="Rafael Bombelli">Rafael Bombelli</a> experimentou escrever as expressões: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{2+{\sqrt {-121}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{2+{\sqrt {-121}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe799503f04f86e713ce2aa1f4487aad28f10be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.558ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{3}]{2+{\sqrt {-121}}}}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{2-{\sqrt {-121}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{2-{\sqrt {-121}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4729e6e25d4d9121ba028ea4ec7d8fcdd5f8d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.558ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{3}]{2-{\sqrt {-121}}}}}"></dd></dl> na forma: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {-b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477a1a9626d9b0205a2f36eed0212f9f25e0ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {-b}}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-{\sqrt {-b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2539d6759a6672a746c787105d86807b0c9110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a-{\sqrt {-b}}}"></dd></dl> respectivamente. Admitindo válidas as propriedades usuais das operações tais como comutativa, distributiva etc., usou-as nas expressões obtidas, obtendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4208bf5a67fc2ceb3a3bcd75aebb1d74fbb531bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=2}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab9941ee6a108649ad03bdc40b14181a636ce3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.905ex; height:2.176ex;" alt="{\displaystyle b=1.}"> Com isso, chegou a: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}=(2+{\sqrt {-1}})+(2-{\sqrt {-1}})=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>121</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}=(2+{\sqrt {-1}})+(2-{\sqrt {-1}})=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec0f92ce0823113f0d15544b7803ed40a50228b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:66.022ex; height:4.843ex;" alt="{\displaystyle x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}=(2+{\sqrt {-1}})+(2-{\sqrt {-1}})=4}"></dd></dl> No início, os números complexos não eram vistos como números, mas sim como um artifício algébrico útil para se resolver equações. <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a>, no século XVII, os chamou de números imaginários. <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> e <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>, no século XVIII começaram a estabelecer uma estrutura algébrica para os números complexos. Em particular, Euler denotou a raiz quadrada de -1 por <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcf9ad7ad44f04fa43c5b604b4801e089981cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}"> Ainda no <a href="/wiki/S%C3%A9culo_XVIII" title="Século XVIII">século XVIII</a> os números complexos passaram a ser interpretados como pontos do plano (<a href="/wiki/Plano_de_Argand-Gauss" class="mw-redirect" title="Plano de Argand-Gauss">plano de Argand-Gauss</a>), o que permitiu a escrita de um número complexo na <a href="/wiki/Coordenadas_polares" title="Coordenadas polares">forma polar</a>. Com isso, conseguiu-se calcular potências e raízes de modo eficiente e claro. Ainda no século XVIII, Gauss demonstrou o <a href="/wiki/Teorema_Fundamental_da_%C3%81lgebra" class="mw-redirect" title="Teorema Fundamental da Álgebra">Teorema Fundamental da Álgebra</a>. <div class="mw-heading mw-heading2"><h2 id="Definições">Definições</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=2" title="Editar secção: Definições" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=2" title="Editar código-fonte da secção: Definições">editar código-fonte</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Plano_complexo">Plano complexo</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=3" title="Editar secção: Plano complexo" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=3" title="Editar código-fonte da secção: Plano complexo">editar código-fonte</a>]</div> <figure typeof="mw:File/Frame"><a href="/wiki/Ficheiro:Argandgaussplane.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d6/Argandgaussplane.png" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a><figcaption>No plano de Argand-Gauss, parte real é representada pela reta das abscissas (x, horizontal) e a parte imaginária pela reta das ordenadas (y, vertical).</figcaption></figure> <div class="hatnote"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" />Ver artigo principal: <a href="/wiki/Plano_complexo" title="Plano complexo">Plano complexo</a></div> O <a href="/wiki/Plano_complexo" title="Plano complexo">plano complexo</a>, também chamado de plano de Argand-Gauss, é uma representação geométrica do conjunto dos números complexos. Da mesma forma, como cada ponto da reta está associado a um <a href="/wiki/N%C3%BAmero_real" title="Número real">número real</a>, o plano complexo associa <a href="/wiki/Fun%C3%A7%C3%A3o_bijetora" class="mw-redirect" title="Função bijetora">biunivocamente</a> o ponto ( x , y ) do <a href="/wiki/Plano_(geometria)" title="Plano (geometria)">plano</a> ao número complexo x + yi. Esta associação conduz a pelo menos duas formas de representar um número complexo: <ul><li>Forma retangular ou cartesiana:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=(x,y)=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=(x,y)=x+yi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e649bfe49ba5bb5d0c11485f318f6d282a3e127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.334ex; height:2.843ex;" alt="{\displaystyle Z=(x,y)=x+yi}"></dd></dl></dd></dl> representa o número Z em <a href="/wiki/Coordenadas_cartesianas" class="mw-redirect" title="Coordenadas cartesianas">coordenadas cartesianas</a> separando a parte real da parte imaginária. <ul><li>Forma polar:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=r(\cos \theta +i\operatorname {sen} \theta )=r\mathrm {cis} (\theta )=re^{i\theta },\ \ \ |Z|=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <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> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=r(\cos \theta +i\operatorname {sen} \theta )=r\mathrm {cis} (\theta )=re^{i\theta },\ \ \ |Z|=r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738050d547159d8b22c0bacca690c5bbf9e16197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.315ex; height:3.176ex;" alt="{\displaystyle Z=r(\cos \theta +i\operatorname {sen} \theta )=r\mathrm {cis} (\theta )=re^{i\theta },\ \ \ |Z|=r}"></dd></dl></dd></dl> onde r é a <a href="/wiki/Dist%C3%A2ncia_euclidiana" title="Distância euclidiana">distância euclidiana</a> do ponto <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75dda3e1eeef9d3090265878710e513046691300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.107ex; height:2.843ex;" alt="{\displaystyle Z=(x,y)}"></dd></dl></dd></dl> até a origem do <a href="/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas">sistema de coordenadas</a>, chamada de módulo do número complexo. Este é denotado por: <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |Z|={\sqrt {x^{2}+y^{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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |Z|={\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c682ad4bc7e6291061f403814a6f0c3f6baea4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.835ex; height:4.843ex;" alt="{\displaystyle |Z|={\sqrt {x^{2}+y^{2}}}}"></dd></dl></dd></dl> Já <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> se trata do <a href="/wiki/%C3%82ngulo" title="Ângulo">ângulo</a> entre a <a href="/wiki/Semi-reta" class="mw-redirect" title="Semi-reta">semi-reta</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {OZ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>Z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {OZ}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4bda2c8f1bdad7406f7aebd3abbe666db5bdc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.629ex; height:3.009ex;" alt="{\displaystyle {\overline {OZ}}}"> e o semi-eixo real, chamado de argumento do número complexo Z e denotado por <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg(Z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg(Z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138dceb16051c11b42fe956dfdc0d2e7b3de2185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.373ex; height:2.843ex;" alt="{\displaystyle \arg(Z).}"> Através da <a href="/wiki/Identidade_de_Euler" title="Identidade de Euler">identidade de Euler</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=\cos \theta +i\operatorname {sen} \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=\cos \theta +i\operatorname {sen} \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e37dd0b03668f7d85e63721183008609bdb1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.737ex; height:2.843ex;" alt="{\displaystyle e^{i\theta }=\cos \theta +i\operatorname {sen} \theta .}"> A forma polar é equivalente à chamada forma exponencial: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=re^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=re^{i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd04f98d20919586cee0c960d0ed5c9986e09760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.482ex; height:2.676ex;" alt="{\displaystyle Z=re^{i\theta }}"> <div class="mw-heading mw-heading3"><h3 id="Operações_elementares">Operações elementares</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=4" title="Editar secção: Operações elementares" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=4" title="Editar código-fonte da secção: Operações elementares">editar código-fonte</a>]</div> O conjunto dos números complexos é um <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a>. Portanto, é fechado sobre as operações de <a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição">adição</a> e <a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">multiplicação</a>, além de possuir a propriedade de que todo elemento não-nulo do conjunto possui um <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">inverso multiplicativo</a>. Todas as operações do corpo podem ser performadas através das propriedades <a href="/wiki/Associatividade" title="Associatividade">associativa</a>, <a href="/wiki/Comutatividade" title="Comutatividade">comutativa</a> e <a href="/wiki/Distributividade" title="Distributividade">distributiva</a>, levando em consideração a identidade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <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 i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}"> Sejam z e w dois números complexos dados por <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b244679a222179f2892475d0544be0b8064a0e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle z=(a,b)}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=(c,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=(c,d)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460cb33b16c3d3254f02dc678913649f297c9fc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.829ex; height:2.843ex;" alt="{\displaystyle w=(c,d)}"> então definem-se as relações e operações elementares tal como segue: <ul><li>Identidade:</li></ul> <dl><dd><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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9019f53aafbd8139cbcd997ede4b94010afc40bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.851ex; height:1.676ex;" alt="{\displaystyle z=w}"> <a href="/wiki/Se_e_somente_se" title="Se e somente se">se e somente se</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb3f1b1ad87e99791ba713839204a88b27239a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.676ex;" alt="{\displaystyle a=c}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=d.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=d.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817f52a91b02049a8c0c26feae29cd32ffddea33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.959ex; height:2.176ex;" alt="{\displaystyle b=d.}"></dd></dl> <ul><li>Soma:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+w=w+z=(a+bi)+(c+di)=(a+c)+(b+d)i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mi>w</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</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> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+w=w+z=(a+bi)+(c+di)=(a+c)+(b+d)i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9cf5e2819dbf58179dfa6d7840b7ba3ca1b818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.068ex; height:2.843ex;" alt="{\displaystyle z+w=w+z=(a+bi)+(c+di)=(a+c)+(b+d)i}"></dd></dl> <ul><li>Produto:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=wz=(a+bi)(c+di)=ac+adi+bci+bdi^{2}=ac+adi+bci-bd=(ac-bd)+(ad+bc)i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</mi> <mo>=</mo> <mi>w</mi> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mi>i</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <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> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=wz=(a+bi)(c+di)=ac+adi+bci+bdi^{2}=ac+adi+bci-bd=(ac-bd)+(ad+bc)i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0d0cdd527ddcd3ec0e0e9c6adeed9ac0525ddd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:98.103ex; height:3.176ex;" alt="{\displaystyle zw=wz=(a+bi)(c+di)=ac+adi+bci+bdi^{2}=ac+adi+bci-bd=(ac-bd)+(ad+bc)i}"></dd></dl> <ul><li>Conjugado:</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Complexo.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Complexo.png/220px-Complexo.png" decoding="async" width="220" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Complexo.png/330px-Complexo.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Complexo.png/440px-Complexo.png 2x" data-file-width="741" data-file-height="521" /></a><figcaption>Exemplo número complexo com módulo 2 e argumento 120°. Em vermelho o conjugado deste número em verde o oposto</figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}=a-bi,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}=a-bi,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf295cb556a788a869f3dccbba456f442540f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.823ex; height:2.676ex;" alt="{\displaystyle {\overline {z}}=a-bi,}"> onde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64281d029a1d4bef9545644f01821c713f876f76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle {\overline {z}}}"> denota o conjugado de z. Outra notação usada para o conjugado de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {z}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {z}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f561c174a2b01acddc31778f5cf93de83a13561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.343ex;" alt="{\displaystyle {z}^{*}.}"></dd></dl> O conjugado de um número complexo é seu simétrico no <a href="/wiki/Plano_complexo" title="Plano complexo">plano complexo</a> em relação ao <a href="/wiki/N%C3%BAmero_real" title="Número real">eixo real</a>. A soma e o produto de um número complexo com seu conjugado tem parte imaginária nula. <ul><li>Soma de um complexo por seu conjugado:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+{\overline {z}}=(a+bi)+(a-bi)=2a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+{\overline {z}}=(a+bi)+(a-bi)=2a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d305d5a58c06666084c182e632c159a75f63bb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.572ex; height:2.843ex;" alt="{\displaystyle z+{\overline {z}}=(a+bi)+(a-bi)=2a.}"></dd></dl> <ul><li>Produto de um complexo por seu conjugado:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\cdot {\overline {z}}=(a+bi)(a-bi)=a^{2}-abi+abi-b^{2}i^{2}=a^{2}-b^{2}i^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mi>i</mi> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mi>i</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\cdot {\overline {z}}=(a+bi)(a-bi)=a^{2}-abi+abi-b^{2}i^{2}=a^{2}-b^{2}i^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f5f79b485392b8996bca36418463cb22e1cf70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.082ex; height:3.176ex;" alt="{\displaystyle z\cdot {\overline {z}}=(a+bi)(a-bi)=a^{2}-abi+abi-b^{2}i^{2}=a^{2}-b^{2}i^{2}.}"> Como <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a48123ed64e9456cbbd8f995aeb8893f18a2d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.573ex; height:3.009ex;" alt="{\displaystyle i^{2}=-1,}"> temos que o produto de um Número Complexo <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><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}"> pelo seu Conjugado <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0440d3ed62463a7a0087db9dd21a9de6c6a9d7c2" 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}"> se dá por: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z){\overline {z}}=a^{2}+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z){\overline {z}}=a^{2}+b^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c731d3e1642109e69c647729edd1b90c931d8807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.027ex; height:3.176ex;" alt="{\displaystyle (z){\overline {z}}=a^{2}+b^{2}.}"></dd></dl> <ul><li>Módulo:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|z\right|=r={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mo>=</mo> <mi>r</mi> <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 \left|z\right|=r={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1433f25951ddb1b67a4b2b8e3fc92bbca85bfe1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.127ex; height:3.509ex;" alt="{\displaystyle \left|z\right|=r={\sqrt {a^{2}+b^{2}}}}"></dd></dl> <ul><li>Inverso multiplicativo (para <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><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}">):</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{z}}={\frac {1}{a+bi}}={\frac {a-bi}{(a+bi)(a-bi)}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {\overline {z}}{\left|z\right|^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> <msup> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{z}}={\frac {1}{a+bi}}={\frac {a-bi}{(a+bi)(a-bi)}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {\overline {z}}{\left|z\right|^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70622d6f4a64db5e029b26377e8b67a5adc2ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.225ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{z}}={\frac {1}{a+bi}}={\frac {a-bi}{(a+bi)(a-bi)}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {\overline {z}}{\left|z\right|^{2}}}.}"></dd></dl> As operações de <a href="/wiki/Subtra%C3%A7%C3%A3o" title="Subtração">subtração</a> e <a href="/wiki/Divis%C3%A3o" title="Divisão">divisão</a> são efetuadas transformando em adição com o <a href="/wiki/Oposto_aditivo" class="mw-redirect" title="Oposto aditivo">oposto aditivo</a> e em multiplicação com o <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">inverso multiplicativo</a>, respectivamente. Algumas operações são mais facilmente realizadas na forma polar: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+bi=r(\cos \varphi +i\mathrm {sen} \,\varphi )=re^{i\varphi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> </mrow> <mspace width="thinmathspace" /> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+bi=r(\cos \varphi +i\mathrm {sen} \,\varphi )=re^{i\varphi }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b48fde02d63fedb827dc013f7ddbefab9c57f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.575ex; height:3.176ex;" alt="{\displaystyle z=a+bi=r(\cos \varphi +i\mathrm {sen} \,\varphi )=re^{i\varphi }.}"></dd></dl> <ul><li>Produto:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\cdot w=r_{1}\,e^{i\varphi _{1}}\cdot r_{2}\,e^{i\varphi _{2}}=r_{1}\,r_{2}\,e^{i(\varphi _{1}+\varphi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>⋅</mo> <mi>w</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <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\cdot w=r_{1}\,e^{i\varphi _{1}}\cdot r_{2}\,e^{i\varphi _{2}}=r_{1}\,r_{2}\,e^{i(\varphi _{1}+\varphi _{2})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5adfdb3d2674a94eb0f98723a4c76cb38be8390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.101ex; height:3.176ex;" alt="{\displaystyle z\cdot w=r_{1}\,e^{i\varphi _{1}}\cdot r_{2}\,e^{i\varphi _{2}}=r_{1}\,r_{2}\,e^{i(\varphi _{1}+\varphi _{2})}}"></dd></dl> <ul><li>Inverso multiplicativo (para <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><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}">):</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{z}}={\frac {1}{r_{1}\,e^{i\varphi _{1}}}}={\frac {1}{r_{1}}}\cdot e^{-i(\varphi _{1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{z}}={\frac {1}{r_{1}\,e^{i\varphi _{1}}}}={\frac {1}{r_{1}}}\cdot e^{-i(\varphi _{1})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d5fedbe977f8e6454edbfa6d462b03e5da607f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.277ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{z}}={\frac {1}{r_{1}\,e^{i\varphi _{1}}}}={\frac {1}{r_{1}}}\cdot e^{-i(\varphi _{1})}}"></dd></dl> <ul><li>Divisão:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z}{w}}={\frac {r_{1}\,e^{i\varphi _{1}}}{r_{2}\,e^{i\varphi _{2}}}}={\frac {r_{1}}{r_{2}}}\,e^{i(\varphi _{1}-\varphi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>w</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <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}{w}}={\frac {r_{1}\,e^{i\varphi _{1}}}{r_{2}\,e^{i\varphi _{2}}}}={\frac {r_{1}}{r_{2}}}\,e^{i(\varphi _{1}-\varphi _{2})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a73997e4f0c0f8540f894a6eb320b68f6313ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.393ex; height:6.176ex;" alt="{\displaystyle {\frac {z}{w}}={\frac {r_{1}\,e^{i\varphi _{1}}}{r_{2}\,e^{i\varphi _{2}}}}={\frac {r_{1}}{r_{2}}}\,e^{i(\varphi _{1}-\varphi _{2})}}"></dd></dl> <ul><li>Potenciação:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}={\big (}r_{1}\,e^{i\varphi _{1}}{\big )}^{n}=r_{1}^{n}\,e^{in\varphi _{1}},~~n=0,1,2,3,\ldots }"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}={\big (}r_{1}\,e^{i\varphi _{1}}{\big )}^{n}=r_{1}^{n}\,e^{in\varphi _{1}},~~n=0,1,2,3,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3371b11aa543837989e6fa4243a4dec9644bee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.761ex; height:3.343ex;" alt="{\displaystyle z^{n}={\big (}r_{1}\,e^{i\varphi _{1}}{\big )}^{n}=r_{1}^{n}\,e^{in\varphi _{1}},~~n=0,1,2,3,\ldots }"></dd></dl> <ul><li>Conjugado:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}=r_{1}\,e^{-i\varphi _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}=r_{1}\,e^{-i\varphi _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d4645eae46514dfad8a2e873fb1e8072a31a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.864ex; height:3.009ex;" alt="{\displaystyle {\overline {z}}=r_{1}\,e^{-i\varphi _{1}}}"></dd></dl> O produto de um número complexo pelo seu conjugado é: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z{\overline {z}}=r_{1}\,e^{i\varphi _{1}}\cdot r_{1}\,e^{-i\varphi _{1}}=r_{1}\cdot r_{1}\,e^{i\varphi _{1}-i\varphi _{1}}=r_{1}^{2}\,e^{0}=r_{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z{\overline {z}}=r_{1}\,e^{i\varphi _{1}}\cdot r_{1}\,e^{-i\varphi _{1}}=r_{1}\cdot r_{1}\,e^{i\varphi _{1}-i\varphi _{1}}=r_{1}^{2}\,e^{0}=r_{1}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26809764f7e35e8de514067fcb4593b7df3fc343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.751ex; height:3.343ex;" alt="{\displaystyle z{\overline {z}}=r_{1}\,e^{i\varphi _{1}}\cdot r_{1}\,e^{-i\varphi _{1}}=r_{1}\cdot r_{1}\,e^{i\varphi _{1}-i\varphi _{1}}=r_{1}^{2}\,e^{0}=r_{1}^{2}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="O_módulo">O módulo</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=5" title="Editar secção: O módulo" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=5" title="Editar código-fonte da secção: O módulo">editar código-fonte</a>]</div> Sejam z e w dois números complexos dados por <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b244679a222179f2892475d0544be0b8064a0e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle z=(a,b)}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=(c,d),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=(c,d),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679691bbd5db8b801f4a112c03ad7cfb2a02ad6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle w=(c,d),}"> o módulo possui as seguintes propriedades: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcl}|z|&=&{\sqrt {a^{2}+b^{2}}};\\|{\overline {z}}|&=&|z|;\\|z\cdot w|&=&|z|\cdot |w|;\\|z+w|&\leq &|z|+|w|;\\|z|&=&0\Longleftrightarrow z=0.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <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> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>⋅</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <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"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mo>≤</mo> </mtd> <mtd> <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"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>0</mn> <mo stretchy="false">⟺</mo> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}|z|&=&{\sqrt {a^{2}+b^{2}}};\\|{\overline {z}}|&=&|z|;\\|z\cdot w|&=&|z|\cdot |w|;\\|z+w|&\leq &|z|+|w|;\\|z|&=&0\Longleftrightarrow z=0.\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1c79a05aa4878a5601ee738f462038c6ef84fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:26.856ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{lcl}|z|&=&{\sqrt {a^{2}+b^{2}}};\\|{\overline {z}}|&=&|z|;\\|z\cdot w|&=&|z|\cdot |w|;\\|z+w|&\leq &|z|+|w|;\\|z|&=&0\Longleftrightarrow z=0.\end{array}}}"> Cabe ressaltar, conforme exposto acima, que o número zero é o único número complexo cujo módulo de Z é igual a Z. A <a href="/wiki/Dist%C3%A2ncia" title="Distância">distância</a> entre dois números complexos é definida como: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>dist</mtext> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>,</mo> <mi>w</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e69e31d9131f3177c1bf29cf621aa4e8326e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.728ex; height:2.843ex;" alt="{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Propriedades_algébricas">Propriedades algébricas</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=6" title="Editar secção: Propriedades algébricas" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=6" title="Editar código-fonte da secção: Propriedades algébricas">editar código-fonte</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Carl_Friedrich_Gauss.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/220px-Carl_Friedrich_Gauss.jpg" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/330px-Carl_Friedrich_Gauss.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/440px-Carl_Friedrich_Gauss.jpg 2x" data-file-width="917" data-file-height="1180" /></a><figcaption>Gauss demonstrou que o conjunto dos números complexos é algebricamente fechado</figcaption></figure> O conjunto dos números complexos formam um <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a> <a href="/wiki/Fecho_alg%C3%A9brico" title="Fecho algébrico">algebricamente fechado</a>. Isso significa que toda <a href="/wiki/Equa%C3%A7%C3%A3o_alg%C3%A9brica" title="Equação algébrica">equação algébrica</a> de grau não nulo pode possuir como solução um número complexo. Mais formalmente, a seguinte equação <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=0,\quad \alpha _{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=0,\quad \alpha _{n}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effdda1fef39f80f5bb390fdfaf8eab922cd01ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.072ex; height:3.176ex;" alt="{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=0,\quad \alpha _{n}\neq 0}"></dd></dl> possui pelo menos uma solução complexa. Este resultado é conhecido como <a href="/wiki/Teorema_fundamental_da_%C3%A1lgebra" title="Teorema fundamental da álgebra">teorema fundamental da álgebra</a> e foi demonstrado primeiramente pelo matemático <a href="/wiki/Alemanha" title="Alemanha">alemão</a> <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>. Uma consequência deste teorema é que todo <a href="/wiki/Polin%C3%B4mio" class="mw-redirect" title="Polinômio">polinômio</a> de grau n pode ser decomposto em um produto de n fatores lineares complexos: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=\alpha _{n}\left(z-z_{1}\right)\left(z-z_{2}\right)\cdots \left(z-z_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=\alpha _{n}\left(z-z_{1}\right)\left(z-z_{2}\right)\cdots \left(z-z_{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d7b85f55449b7c3ec00fb2d28441b60546019d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.348ex; height:3.176ex;" alt="{\displaystyle \alpha _{n}z^{n}+\alpha _{n-1}z^{n-1}+\ldots +\alpha _{1}z+\alpha _{0}=\alpha _{n}\left(z-z_{1}\right)\left(z-z_{2}\right)\cdots \left(z-z_{n}\right)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Radical_algébrico">Radical algébrico</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=7" title="Editar secção: Radical algébrico" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=7" title="Editar código-fonte da secção: Radical algébrico">editar código-fonte</a>]</div> <div class="hatnote"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" />Ver artigo principal: <a href="/wiki/Raiz_da_unidade" title="Raiz da unidade">Raiz da unidade</a></div> O radical algébrico é definido no conjunto dos números complexos como uma <a href="/wiki/Fun%C3%A7%C3%A3o_multivalente" class="mw-redirect" title="Função multivalente">função multivalente</a>, devido ao fato que a <a href="/wiki/Equa%C3%A7%C3%A3o_alg%C3%A9brica" title="Equação algébrica">equação algébrica</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}=A}"> <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> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e79d500d19252cbe195f7c59a47f8e0f670f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.15ex; height:2.343ex;" alt="{\displaystyle z^{n}=A}"></dd></dl> possui n soluções distintas para cada <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1c7fe8961ff54be58571e3f20deda4d0f9e0f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:2.676ex;" alt="{\displaystyle A\neq 0,}"> que são dadas pela <a href="/wiki/F%C3%B3rmula_de_De_Moivre" title="Fórmula de De Moivre">fórmula de De Moivre</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}=|A|^{1/n}\left(e^{i(\theta +2k\pi )/n}\right),~~k=0,1,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{k}=|A|^{1/n}\left(e^{i(\theta +2k\pi )/n}\right),~~k=0,1,\ldots ,n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25211ebdbdab2a278d0a9ee5ab299adfbd96c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.797ex; height:4.843ex;" alt="{\displaystyle z_{k}=|A|^{1/n}\left(e^{i(\theta +2k\pi )/n}\right),~~k=0,1,\ldots ,n-1}"></dd></dl> onde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=|A|e^{i\theta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=|A|e^{i\theta }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06490d3e9314d8d8202fbd4cd6efdb9786feaebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.179ex; height:3.176ex;" alt="{\displaystyle A=|A|e^{i\theta }.}"> <div class="mw-heading mw-heading3"><h3 id="Estrutura_de_campo[8]">Estrutura de campo<a href="#cite_note-8">[8]</a></h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=8" title="Editar secção: Estrutura de campo[8]" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=8" title="Editar código-fonte da secção: Estrutura de campo[8]">editar código-fonte</a>]</div> O conjunto C de números complexos é um campo. Resumidamente, isso significa que os seguintes fatos são válidos: primeiro, quaisquer dois números complexos podem ser adicionados e multiplicados para produzir outro número complexo. Em segundo lugar, para qualquer número complexo <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, seu <a href="/wiki/Inverso_aditivo" title="Inverso aditivo">inverso aditivo</a> −<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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> também é um número complexo; e terceiro, todo número complexo diferente de zero tem um número complexo <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">recíproco</a>. Além disso, essas operações satisfazem uma série de leis, por exemplo, a lei da <a href="/wiki/Comutatividade" title="Comutatividade">comutatividade</a> de adição e multiplicação para quaisquer dois números complexos <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abf655fa14f7ea44ad0ca781b59ff59c5f49117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}+z_{2}=z_{2}+z_{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> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}+z_{2}=z_{2}+z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19adfdf8b2d796d07110bb5f0b751bfb5525e435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.321ex; height:2.343ex;" alt="{\displaystyle z_{1}+z_{2}=z_{2}+z_{1}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}z_{2}=z_{2}z_{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> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}z_{2}=z_{2}z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcf8316b7c96bb93f14d2a83a0151124e797658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.64ex; height:2.009ex;" alt="{\displaystyle z_{1}z_{2}=z_{2}z_{1}}"> Essas duas leis e os outros requisitos de um campo podem ser comprovados pelas fórmulas fornecidas acima, usando o fato de que os próprios números reais formam um campo. Ao contrário dos reais, C não é um campo ordenado, ou seja, não é possível definir uma relação <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3621e468231ab352b7caa30bcf0ce9b452241a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}"> que seja compatível com a adição e multiplicação. Na verdade, em qualquer campo ordenado, o quadrado de qualquer elemento é necessariamente positivo, então <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <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 i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}"> impede a existência de uma ordem em C. Quando o campo subjacente para um tópico ou construção matemática é o campo de números complexos, o nome do tópico geralmente é modificado para refletir esse fato. Por exemplo: <a href="/wiki/An%C3%A1lise_complexa" title="Análise complexa">análise complexa</a>, <a href="/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática)">matriz</a> complexa, <a href="/wiki/Polin%C3%B4mio" class="mw-redirect" title="Polinômio">polinômio</a> complexo e <a href="/wiki/%C3%81lgebra_de_Lie" title="Álgebra de Lie">álgebra de Lie</a> complexa. <div class="mw-heading mw-heading2"><h2 id="Propriedades_topológicas_e_analíticas">Propriedades topológicas e analíticas</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=9" title="Editar secção: Propriedades topológicas e analíticas" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=9" title="Editar código-fonte da secção: Propriedades topológicas e analíticas">editar código-fonte</a>]</div> O conjunto dos números complexos munido da distância <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>dist</mtext> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>,</mo> <mi>w</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e69e31d9131f3177c1bf29cf621aa4e8326e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.728ex; height:2.843ex;" alt="{\displaystyle {\hbox{dist}}\left(z,w\right)=|z-w|}"> forma um <a href="/wiki/Espa%C3%A7o_m%C3%A9trico_completo" class="mw-redirect" title="Espaço métrico completo">espaço métrico completo</a>. De fato, o módulo possui todas as características de uma <a href="/wiki/Norma_(matem%C3%A1tica)" title="Norma (matemática)">norma</a>. <div class="mw-heading mw-heading3"><h3 id="Convergência_nos_complexos">Convergência nos complexos</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=10" title="Editar secção: Convergência nos complexos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=10" title="Editar código-fonte da secção: Convergência nos complexos">editar código-fonte</a>]</div> Diz-se que uma sequência <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle z_{n}}"> de números complexos é convergente se existe um número complexo <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> tal que: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }|z-z_{n}|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }|z-z_{n}|=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e0314241d376f78dde5c9cb420c046c447d92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.442ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }|z-z_{n}|=0}"></dd></dl> neste caso, denota-se: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }z_{n}=z\quad {\hbox{ou}}\quad z_{n}\to z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>z</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>ou</mtext> </mstyle> </mrow> <mspace width="1em" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }z_{n}=z\quad {\hbox{ou}}\quad z_{n}\to z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c769af36026f92c313ad4ae082c0548a3c508e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.248ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }z_{n}=z\quad {\hbox{ou}}\quad z_{n}\to z}"></dd></dl> <ul><li>É fácil verificar que se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{n}=a_{n}+ib_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{n}=a_{n}+ib_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ca3bec110621a7206d97723d704e1b36531c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.352ex; height:2.509ex;" alt="{\displaystyle z_{n}=a_{n}+ib_{n},}"> então <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle z_{n}}"> converge para <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"> <a href="/wiki/Se_e_somente_se" title="Se e somente se">se e somente se</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"> converge para <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"> converge para <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef051eb30c89e5493d672f6479566c673b0890a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}"></li> <li>Do fato de que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left||z_{n}|-|z|\right|\leq |z_{n}-z|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left||z_{n}|-|z|\right|\leq |z_{n}-z|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03da98d1413c3a21ae8d902b3e9af54e4c9de51f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.376ex; height:2.843ex;" alt="{\displaystyle \left||z_{n}|-|z|\right|\leq |z_{n}-z|,}"> é válido que se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{n}\to z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{n}\to z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b0e950dfe52b9d6c8c0776975b12eb7f7b99d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.002ex; height:2.176ex;" alt="{\displaystyle z_{n}\to z}"> então <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z_{n}|\to |z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z_{n}|\to |z|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e70a3ec7bf631318cbf1c3fcefbbc967d1ad99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.589ex; height:2.843ex;" alt="{\displaystyle |z_{n}|\to |z|}"></li></ul> <div class="mw-heading mw-heading2"><h2 id="O_conjunto_dos_números_complexos_como_extensão_algébrica">O conjunto dos números complexos como extensão algébrica</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=11" title="Editar secção: O conjunto dos números complexos como extensão algébrica" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=11" title="Editar código-fonte da secção: O conjunto dos números complexos como extensão algébrica">editar código-fonte</a>]</div> Outra forma de entendermos os números complexos é através de polinômios com coeficientes reais. Ao usarmos a identidade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}+1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3aa8f20fda643e642824583ff5f7e420f2ab36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.12ex; height:2.843ex;" alt="{\displaystyle i^{2}+1=0}"> para simplificarmos identidades algébricas, estamos efetivamente encontrando o maior fator de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c45dfbe141c1088f8b20ef0ee1279cff0c7e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.86ex; height:2.843ex;" alt="{\displaystyle i^{2}+1}"> na expressão e o substituindo por zero. Por exemplo, a expressão <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{3}-i+5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{3}-i+5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3383c662beebf95a4a3c4563ec00fe43cbf8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.502ex; height:2.843ex;" alt="{\displaystyle i^{3}-i+5}"> pode ser escrita formalmente como <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i^{2}+1)i-2i+5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>i</mi> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i^{2}+1)i-2i+5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffef0f64f07aabee54ffe486250f8780df188fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.279ex; height:3.176ex;" alt="{\displaystyle (i^{2}+1)i-2i+5}"> e, substituindo a identidade, encontramos <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{3}-i+5=-2i+5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{3}-i+5=-2i+5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec339b6c722d20001519e6e64a404652869e8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.377ex; height:2.843ex;" alt="{\displaystyle i^{3}-i+5=-2i+5}">. Esse processo pode ser sistematizado da seguinte forma: dado um polinômio de coeficientes reais na variável <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> (isso é, um elemento de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [x]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/453d1013f9dd290be70d5fe534e0d3311b0a7c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.301ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [x]}">), podemos dividi-lo por <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}"> e trabalharmos somente com o seu resto (que deve, pelas propriedades da <a href="/wiki/Divis%C3%A3o_polinomial" title="Divisão polinomial">Divisão polinomial</a>) ter grau um, ou seja, ser da forma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6cf02d491319b79bce5b8d208c838e157a9367d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.397ex; height:2.343ex;" alt="{\displaystyle a+bx}">. Esse processo de adicionar a raiz de um polinômio a um corpo é conhecido, no campo da <a href="/wiki/%C3%81lgebra_abstrata" title="Álgebra abstrata">álgebra abstrata</a>, como extensão de corpos. O número <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> pode ser interpretado como o elemento que gera a <a href="/wiki/Extens%C3%A3o_alg%C3%A9brica" title="Extensão algébrica">extensão algébrica</a> dos números reais contendo a raiz do polinômio <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a57b24e88923ce4e4b43bc2876a32f175074f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.034ex; height:2.843ex;" alt="{\displaystyle x^{2}+1.}"> Isto é, o <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} =\mathbb {R} [i]}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} =\mathbb {R} [i]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9f76c21012ac42d0a7bc8b0b983f68531b830f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.551ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} =\mathbb {R} [i]}"> é isomorfo ao <a href="/wiki/Corpo_quociente" class="mw-redirect" title="Corpo quociente">corpo quociente</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x]/(x^{2}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [x]/(x^{2}+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0ade67281f83ef6b6b7f43bf783c081adb1fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.66ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} [x]/(x^{2}+1)}"> pela aplicação <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :\mathbb {C} \rightarrow \mathbb {R} [x]/(x^{2}+1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :\mathbb {C} \rightarrow \mathbb {R} [x]/(x^{2}+1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d2e250d5ad56146ed7623b1a3a3617fec6d6ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.922ex; height:3.176ex;" alt="{\displaystyle \phi :\mathbb {C} \rightarrow \mathbb {R} [x]/(x^{2}+1),}"> onde o <a href="/wiki/Homomorfismo_de_an%C3%A9is" title="Homomorfismo de anéis">homomorfismo de anéis</a> é tal que, restrito aos reais, é a aplicação identidade e leva <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> em <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (i)=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (i)=x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc20fc7729a7f0936168e6cfec262904d2279a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.072ex; height:2.843ex;" alt="{\displaystyle \phi (i)=x.}"> Por essa própria construção vemos que, como <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="{\displaystyle -i}"> também é raiz de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}">, os corpos obtidos adicionando qualquer uma dessas raízes devem ser isomorfos, isso é, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [i]\sim \mathbb {R} [-i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mo>−</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [i]\sim \mathbb {R} [-i]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db7ceff0c660a111cd2ef4e071c640544d322ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.455ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [i]\sim \mathbb {R} [-i]}">. Ao tratar desse ponto de vista algébrico, entendemos que a conjugação, por ser um isomorfismo de corpos, preserva somas e produtos. <div class="mw-heading mw-heading3"><h3 id="Logaritmos">Logaritmos</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=12" title="Editar secção: Logaritmos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=12" title="Editar código-fonte da secção: Logaritmos">editar código-fonte</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Função_logarítmica_natural">Função logarítmica natural</h4>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=13" title="Editar secção: Função logarítmica natural" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=13" title="Editar código-fonte da secção: Função logarítmica natural">editar código-fonte</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=re^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.889ex; height:2.676ex;" alt="{\displaystyle z=re^{i\theta }}"> , onde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> é o módulo e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> é o argumento medido em radianos do número complexo <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{w}=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{w}=z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a82920d2ab8f3f245d57006ac4744aa4fb5934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.679ex; height:2.343ex;" alt="{\displaystyle e^{w}=z}">, gostaríamos de escrever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(w)=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(w)=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de1d760a335ea4998f38c5d6b706f787ff421f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.595ex; height:2.843ex;" alt="{\displaystyle \ln(w)=e}">. Isso não é imediato no entanto, porque, diferente do caso real, a exponencial complexa não é uma <a href="/wiki/Fun%C3%A7%C3%A3o_injectiva" title="Função injectiva">Função injectiva</a>, de fato <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2i\pi n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>π</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2i\pi n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce31a24d566df50bdbe600c9d54db2c82155698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.894ex; height:2.676ex;" alt="{\displaystyle e^{2i\pi n}=1}"> para todo número inteiro <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. Não sendo injetiva, a <a href="/wiki/Fun%C3%A7%C3%A3o_exponencial" title="Função exponencial">função exponencial</a> com <a href="/wiki/Contradom%C3%ADnio" title="Contradomínio">Contradomínio</a> nos números complexos não admite <a href="/wiki/Fun%C3%A7%C3%A3o_inversa" title="Função inversa">Função inversa</a> no sentido usual. Essas considerações de lado, escreva <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b9499ca42aaff850433b57155142510d08247a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.633ex; height:2.343ex;" alt="{\displaystyle w=a+bi}"> então <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{w}=e^{a}e^{bi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{w}=e^{a}e^{bi}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115dc584545146e22d45a8cfed1d6a115b8d9fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.365ex; height:2.676ex;" alt="{\displaystyle e^{w}=e^{a}e^{bi}}"> e portanto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{a}e^{bi}=re^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{a}e^{bi}=re^{i\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d4eba86273d2a32cfb00b2841084821eef1b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.575ex; height:2.676ex;" alt="{\displaystyle e^{a}e^{bi}=re^{i\theta }}">. Da igualdade desses números complexos escritos na forma polar temos: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\ln r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\ln r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1e103d8fc94bd9959c19af585e5f28d4d00406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.703ex; height:2.176ex;" alt="{\displaystyle a=\ln r}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\theta +2k\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=\theta +2k\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b4aa1d8223eae6cd9426860b91afc43c3251bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.733ex; height:2.343ex;" alt="{\displaystyle b=\theta +2k\pi }"> onde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> representa algum <a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro">número inteiro</a>. Desse raciocínio, definimos a função logarítmica natural de uma variável complexa z pela equação: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a1668203dbe173e0ff29584427a18097ea1235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.415ex; height:2.176ex;" alt="{\displaystyle \ln z}"> = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50e7ced87dd3836226833e0ce3de84ae5e95cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.375ex; height:2.176ex;" alt="{\displaystyle \ln r}"> + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2k\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e882576cae16f4ca02e7b919cd26f144cc206e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.706ex; height:2.176ex;" alt="{\displaystyle 2k\pi }"> )</dd></dl> Assim, a função <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ln}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ln}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2234193f1d24ab0337abad752c47e059232e6fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.088ex; height:2.176ex;" alt="{\displaystyle ln}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é multivalente com infinitos valores - mesmo para números reais. Se alterarmos o domínio dessa função logaritmo para uma <a href="/wiki/Superf%C3%ADcie_de_Riemann" title="Superfície de Riemann">Superfície de Riemann</a> convenientemente definida, podemos transformar o logaritmo complexo em uma função de fato. Chamamos de valor principal de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ln}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ln}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2234193f1d24ab0337abad752c47e059232e6fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.088ex; height:2.176ex;" alt="{\displaystyle ln}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> o número definido por: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z=\ln r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z=\ln r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8ea817422d2c7a4346cd63724e4171064c4401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.888ex; height:2.176ex;" alt="{\displaystyle \ln z=\ln r}"> + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Função_logarítmica_decimal">Função logarítmica decimal</h4>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=14" title="Editar secção: Função logarítmica decimal" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=14" title="Editar código-fonte da secção: Função logarítmica decimal">editar código-fonte</a>]</div> Em termos de logaritmos decimais, podemos definir a função logarítmica anterior como: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lg z=\lg r+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lg</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mi>lg</mi> <mo>⁡</mo> <mi>r</mi> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lg z=\lg r+}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529532b5c0c0b26b1a5690529e5cc76463c406a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.436ex; height:2.509ex;" alt="{\displaystyle \lg z=\lg r+}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\lg e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>lg</mi> <mo>⁡</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\lg e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6699614269118a7d52a2422432b2d085593473d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.509ex;" alt="{\displaystyle i\lg e}"> ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> ± <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2k\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e882576cae16f4ca02e7b919cd26f144cc206e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.706ex; height:2.176ex;" alt="{\displaystyle 2k\pi }"> )</dd></dl> Essa função também é multivalente e têm seu valor principal quando <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274c79b3ac027e0d0cd04cf86ae43c15567ba0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.119ex; height:2.176ex;" alt="{\displaystyle k=0.}"> <div class="mw-heading mw-heading2"><h2 id="Gráficos_de_funções_complexas">Gráficos de funções complexas</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=15" title="Editar secção: Gráficos de funções complexas" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=15" title="Editar código-fonte da secção: Gráficos de funções complexas">editar código-fonte</a>]</div> A representação gráfica de uma função com domínio e imagem no campo dos complexos é impraticável, pois tal função reside na <a href="/wiki/Quarta_dimens%C3%A3o" title="Quarta dimensão">quarta dimensão</a>, ou seja, seria preciso um sistema de coordenadas com quatro eixos perpendiculares entre si para a construção da curva, a qual seria uma "superfície-2D" representada num "hiperespaço-4D". Todavia, existem diversas maneiras de se estudar o comportamento de tais funções sem sair de nosso <a href="/wiki/Espa%C3%A7o_euclidiano" title="Espaço euclidiano">espaço euclidiano</a> de três dimensões. Uma delas, pouco usual, é representar uma função complexa, por exemplo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=-z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=-z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2f88a9709b2aaf107a39d7f951fe38c609cb79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.171ex; height:2.843ex;" alt="{\displaystyle f(z)=-z}">, no próprio plano de Argand-Gauss, utilizando cores para representar o "jeito" da função. Este método denomina-se "Color Domain" ou Domínio de Cores. Temos então que para todo ponto do plano complexo está associada uma cor que corresponde à imagem da função neste ponto. Outra opção é representar apenas os valores da função que têm imagem real, como na figura da próxima seção. Esta secção da curva de uma função complexa irá resultar em uma nova curva unidimensional que está distribuída no espaço tridimensional. A representação dos valores reais da imagem da função complexa é interessante principalmente porque nos ajuda a compreender, por exemplo, as raízes complexas de um polinômio, como <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29abf07856399bb09ed7d4baa39b319fc4e0c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.37ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{2}+1}">, cujas raízes são <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="{\displaystyle -i}">. Observe que na figura que segue o plano X/Y corresponde ao <a href="/wiki/Plano_complexo" title="Plano complexo">plano de Argand-Gauss</a>, e o eixo Z de valores reais representa a imagem de apenas números complexos cuja transformação <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74261e1a2745545257965d1289a444de614e5419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.147ex; height:2.843ex;" alt="{\displaystyle z^{2}+1}"> possui parte imaginária nula. Isso não quer dizer que a função não tenha imagem no campo complexo, apenas que essa imagem não pode ser representada na figura. <div class="mw-heading mw-heading2"><h2 id="Forma_trigonométrica_dos_números_complexos">Forma trigonométrica dos números complexos</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=16" title="Editar secção: Forma trigonométrica dos números complexos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=16" title="Editar código-fonte da secção: Forma trigonométrica dos números complexos">editar código-fonte</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Representação_trigonométrica">Representação trigonométrica</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=17" title="Editar secção: Representação trigonométrica" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=17" title="Editar código-fonte da secção: Representação trigonométrica">editar código-fonte</a>]</div> Na representação trigonométrica, um número complexo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"> é determinado pelo módulo do vetor que o representa, e pelo ângulo que faz com o semieixo positivo das abscissas.<a href="#cite_note-9">[9]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Complexos.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Complexos.png/371px-Complexos.png" decoding="async" width="371" height="346" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Complexos.png/557px-Complexos.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/27/Complexos.png 2x" data-file-width="568" data-file-height="530" /></a><figcaption></figcaption></figure> Um vetor é representado por um segmento de reta orientado, e define grandezas que se caracterizam por: <ul><li>Módulo: exprime o comprimento do segmento;</li> <li>Direção: é dada pelo ângulo entre a reta suporte e a horizontal;</li> <li>Sentido: é dado pela seta.</li></ul> Quando <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}">: <ul><li>Argumento de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é o ângulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></li> <li>Módulo de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é o comprimento <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</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>=</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 r=|z|={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12537816d6f93568746382709987322394074c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.127ex; height:3.509ex;" alt="{\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}}"></li></ul> O argumento geral de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta +2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta +2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6148cb72bdcfcd987118010a50d52863dcefc34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.637ex; height:2.343ex;" alt="{\displaystyle \theta +2\pi k}"> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta +360^{\circ }k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>+</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta +360^{\circ }k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519d553b703b9dcc4fe6bbed74676d0f447bf59d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.684ex; height:2.509ex;" alt="{\displaystyle \theta +360^{\circ }k}">, o argumento principal é o valor de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> no intervalo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\theta \leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70d91280db2ab2fd5ef5d17654d46c869caaf23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.76ex; height:2.343ex;" alt="{\displaystyle -\pi <\theta \leq \pi }"> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -180^{\circ }<\theta \leq 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>θ</mi> <mo>≤</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -180^{\circ }<\theta \leq 180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e9f9b1cdf060724ce22757832b571d23091ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.179ex; height:2.509ex;" alt="{\displaystyle -180^{\circ }<\theta \leq 180^{\circ }}">. A partir das relações trigonométricas, obtêm-se: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={a \over |z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={a \over |z|}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b433ceaccbbde7a12e9ca0cd596840638a397b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.905ex; height:5.509ex;" alt="{\displaystyle \cos \theta ={a \over |z|}}">, isto é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=|z|\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</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>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=|z|\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5140759b63f1af71f41d0b0f5effdd212c954558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.686ex; height:2.843ex;" alt="{\displaystyle a=|z|\cos \theta }"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sen} \theta ={b \over |z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen} \theta ={b \over |z|}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5691c60ffad70e6cf87e382fed56652ca8c320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.035ex; height:6.176ex;" alt="{\displaystyle \operatorname {sen} \theta ={b \over |z|}}">, isto é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=|z|\operatorname {sen} \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</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>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=|z|\operatorname {sen} \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84986acf1e3389849b7662a3f5137539ab6ef46b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.584ex; height:2.843ex;" alt="{\displaystyle b=|z|\operatorname {sen} \theta }"></dd></dl> Portanto, para o número complexo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=(|z|\cos \theta )+i(|z|\operatorname {sen} \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</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>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</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>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=(|z|\cos \theta )+i(|z|\operatorname {sen} \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a7240727a77b5303ae9b06f77aa9ffb80a954f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.293ex; height:2.843ex;" alt="{\displaystyle z=(|z|\cos \theta )+i(|z|\operatorname {sen} \theta )}"> Exemplos: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">1) Se <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é um número real, com <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3749e5cd50ee274eb73aea2ade8441687140a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|=1}">, e o ponto P pertence à reta das abcissas, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Isto é: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0+2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0+2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b43292a5901984446d05fe6e085cef9de1e2a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.897ex; height:2.343ex;" alt="{\displaystyle \theta =0+2\pi k}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3749e5cd50ee274eb73aea2ade8441687140a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|=1}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" 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=1}"> na forma trigonométrica é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos 2\pi k+i\operatorname {sen} 2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos 2\pi k+i\operatorname {sen} 2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19df579e050d37d575d6c3a96c07007f6fe39d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.754ex; height:2.343ex;" alt="{\displaystyle z=\cos 2\pi k+i\operatorname {sen} 2\pi k}">, com <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Isso quer dizer que existem infinitas representações trigonométricas para <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, correspondentes a giros dados em torno da origem. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Neste caso, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" 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=1}"> pode ser representado por: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left(2\pi \cdot 0\right)+i\operatorname {sen} \left(2\pi \cdot 0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left(2\pi \cdot 0\right)+i\operatorname {sen} \left(2\pi \cdot 0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1294b5294b730a1581c84df3afea9cce16daaf15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.859ex; height:2.843ex;" alt="{\displaystyle z=\cos \left(2\pi \cdot 0\right)+i\operatorname {sen} \left(2\pi \cdot 0\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos 0+i\operatorname {sen} 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos 0+i\operatorname {sen} 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16150be82f92fce6c2feecab460dc9f5e42c9fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.668ex; height:2.343ex;" alt="{\displaystyle z=\cos 0+i\operatorname {sen} 0}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left(2\pi \cdot 1\right)+i\operatorname {sen} \left(2\pi \cdot 1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left(2\pi \cdot 1\right)+i\operatorname {sen} \left(2\pi \cdot 1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141428e72d8880b2ce99116650fc637015e5e173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.859ex; height:2.843ex;" alt="{\displaystyle z=\cos \left(2\pi \cdot 1\right)+i\operatorname {sen} \left(2\pi \cdot 1\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos 2\pi +i\operatorname {sen} 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos 2\pi +i\operatorname {sen} 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779f196482c836dde41e4333d6b1f8e4e4ddd7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.332ex; height:2.343ex;" alt="{\displaystyle z=\cos 2\pi +i\operatorname {sen} 2\pi }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left(2\pi \cdot 2\right)+i\operatorname {sen} \left(2\pi \cdot 2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left(2\pi \cdot 2\right)+i\operatorname {sen} \left(2\pi \cdot 2\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25c020da6653503953a274dfa7595d24b148dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.859ex; height:2.843ex;" alt="{\displaystyle z=\cos \left(2\pi \cdot 2\right)+i\operatorname {sen} \left(2\pi \cdot 2\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos 4\pi +i\operatorname {sen} 4\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>4</mn> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos 4\pi +i\operatorname {sen} 4\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c0c475b18838bf5351adc8050d86a840562edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.332ex; height:2.343ex;" alt="{\displaystyle z=\cos 4\pi +i\operatorname {sen} 4\pi }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Etc.. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">2) Se <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> é um <a href="/wiki/N%C3%BAmero_imagin%C3%A1rio" title="Número imaginário">número imaginário</a>, com <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3749e5cd50ee274eb73aea2ade8441687140a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|=1}">, e o ponto P pertence à reta das ordenadas, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Isto é: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {1}{2}}\pi +2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {1}{2}}\pi +2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab54500ba5830756a2250766f2f3503b60d5cdde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.066ex; height:5.176ex;" alt="{\displaystyle \theta ={\frac {1}{2}}\pi +2\pi k}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3749e5cd50ee274eb73aea2ade8441687140a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|=1}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef697db8a1de796f4d9b7fa0cf2abcb6c262dbfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.989ex; height:2.176ex;" alt="{\displaystyle z=i}"> na forma trigonométrica é <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi k\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi k\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi k\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi k\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69b68b15650a9a25ee36bedd68ee3fff85b918b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.165ex; height:6.176ex;" alt="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi k\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi k\right)}">, com <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Como no exemplo anterior, existem infinitas representações trigonométricas para <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, correspondentes a giros dados em torno da origem. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }">Neste caso, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef697db8a1de796f4d9b7fa0cf2abcb6c262dbfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.989ex; height:2.176ex;" alt="{\displaystyle z=i}"> pode ser representado por <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d678daead3b5bcefb6698f1e4e03168d556b3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.425ex; height:6.176ex;" alt="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 0\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos {\frac {1}{2}}\pi +i\operatorname {sen} {\frac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos {\frac {1}{2}}\pi +i\operatorname {sen} {\frac {1}{2}}\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6587ba35b6146cf63e06941508fe4994fa0f0330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.004ex; height:5.176ex;" alt="{\displaystyle z=\cos {\frac {1}{2}}\pi +i\operatorname {sen} {\frac {1}{2}}\pi }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e3857fc4aa418d2c3b710edb023d07726c3fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.425ex; height:6.176ex;" alt="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 1\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos {\frac {5}{2}}\pi +i\operatorname {sen} {\frac {5}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos {\frac {5}{2}}\pi +i\operatorname {sen} {\frac {5}{2}}\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6fc7c35c7fe9ae640e9c12fef578fdf0b45eefa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.004ex; height:5.176ex;" alt="{\displaystyle z=\cos {\frac {5}{2}}\pi +i\operatorname {sen} {\frac {5}{2}}\pi }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb766c6e9bea1fb262f7a956e394e934527cbf16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.425ex; height:6.176ex;" alt="{\displaystyle z=\cos \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)+i\operatorname {sen} \left({\frac {1}{2}}\pi +2\pi \cdot 2\right)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos {\frac {9}{2}}\pi +i\operatorname {sen} {\frac {9}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos {\frac {9}{2}}\pi +i\operatorname {sen} {\frac {9}{2}}\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db21620a9edc95059e866fed3ec6303d0e01091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.004ex; height:5.176ex;" alt="{\displaystyle z=\cos {\frac {9}{2}}\pi +i\operatorname {sen} {\frac {9}{2}}\pi }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a9ceb3f51a3855999d6bbee5f3b6a8d54ade22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:4.645ex; height:0.343ex;" alt="{\displaystyle \qquad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a9ceb3f51a3855999d6bbee5f3b6a8d54ade22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:4.645ex; height:0.343ex;" alt="{\displaystyle \qquad }">Etc.. <div class="mw-heading mw-heading3"><h3 id="Igualdade_de_números_complexos">Igualdade de números complexos</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=18" title="Editar secção: Igualdade de números complexos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=18" title="Editar código-fonte da secção: Igualdade de números complexos">editar código-fonte</a>]</div> Dados dois números complexos <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=c+id}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mi>c</mi> <mo>+</mo> <mi>i</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=c+id}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dfe3ac74e6eb5dea5405b418f78cc75d90f7d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.628ex; height:2.343ex;" alt="{\displaystyle w=c+id}"> têm-se, na forma trigonométrica, um argumento geral, sendo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|(\cos \left(\theta +2\pi k\right)+i\operatorname {sen} \left(\theta +2\pi k\right))}"> <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> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \left(\theta +2\pi k\right)+i\operatorname {sen} \left(\theta +2\pi k\right))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2066e97f51df5679ca2299ea36cd4cf8f45288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.652ex; height:2.843ex;" alt="{\displaystyle z=|z|(\cos \left(\theta +2\pi k\right)+i\operatorname {sen} \left(\theta +2\pi k\right))}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=|w|(\cos \left(\alpha +2\pi k\right)+i\operatorname {sen} \left(\alpha +2\pi k\right))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=|w|(\cos \left(\alpha +2\pi k\right)+i\operatorname {sen} \left(\alpha +2\pi k\right))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b440f9ebdfd0a76b3672a9db971306760fb85fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.598ex; height:2.843ex;" alt="{\displaystyle w=|w|(\cos \left(\alpha +2\pi k\right)+i\operatorname {sen} \left(\alpha +2\pi k\right))}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9019f53aafbd8139cbcd997ede4b94010afc40bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.851ex; height:1.676ex;" alt="{\displaystyle z=w}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\cos \theta =|w|\cos \alpha }"> <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> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\cos \theta =|w|\cos \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876c1dcc21b362c2045f4dc85ea4bb2152e84efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.787ex; height:2.843ex;" alt="{\displaystyle |z|\cos \theta =|w|\cos \alpha }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\operatorname {sen} \theta =|w|\operatorname {sen} \alpha }"> <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> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\operatorname {sen} \theta =|w|\operatorname {sen} \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8a72e0e2896b4c6f3eda3a4c68dd326295a95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.047ex; height:2.843ex;" alt="{\displaystyle |z|\operatorname {sen} \theta =|w|\operatorname {sen} \alpha }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <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 |z|=|w|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c48688643850ddefee2e685743b9b2420f2afb1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.438ex; height:2.843ex;" alt="{\displaystyle |z|=|w|}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\theta +2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\theta +2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c195bfee04160197aa92fd24771044cf90f516b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.223ex; height:2.343ex;" alt="{\displaystyle \alpha =\theta +2\pi k}"> A igualdade exige que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <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 |z|=|w|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c48688643850ddefee2e685743b9b2420f2afb1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.438ex; height:2.843ex;" alt="{\displaystyle |z|=|w|}"> mas não exige que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd1f9ffb4129df5da7289c5c31d5665ee8ae1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.677ex; height:2.176ex;" alt="{\displaystyle \theta =\alpha }">, mas sim que os vetores coincidam, na mesma direção, módulo e sentido. <div class="mw-heading mw-heading3"><h3 id="Simétrico_de_um_número_complexo">Simétrico de um número complexo</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=19" title="Editar secção: Simétrico de um número complexo" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=19" title="Editar código-fonte da secção: Simétrico de um número complexo">editar código-fonte</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Sim%C3%A9trico_de_um_Complexo.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Sim%C3%A9trico_de_um_Complexo.png/203px-Sim%C3%A9trico_de_um_Complexo.png" decoding="async" width="203" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Sim%C3%A9trico_de_um_Complexo.png/305px-Sim%C3%A9trico_de_um_Complexo.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Sim%C3%A9trico_de_um_Complexo.png/406px-Sim%C3%A9trico_de_um_Complexo.png 2x" data-file-width="654" data-file-height="567" /></a><figcaption></figcaption></figure> O simétrico de um número complexo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"> é o número <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -z=-(a+ib)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -z=-(a+ib)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37650ce72cbb4e3bb39f7dd76085a6d21a18f1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.482ex; height:2.843ex;" alt="{\displaystyle -z=-(a+ib)}">, ou seja <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -z=(-a)+i(-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -z=(-a)+i(-b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f4b80bd08f242d92d1b856a370063648eec148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.1ex; height:2.843ex;" alt="{\displaystyle -z=(-a)+i(-b)}">. Corresponde a uma rotação de 180° em torno da origem, a partir de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">. Em notação trigonométrica: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}"> <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> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f79ad9ac38760050e59b965c63e84885356666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.102ex; height:2.843ex;" alt="{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-z)=|z|\left(\cos \left(\theta +\pi \right)+i\operatorname {sen} \left(\theta +\pi \right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</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> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>+</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-z)=|z|\left(\cos \left(\theta +\pi \right)+i\operatorname {sen} \left(\theta +\pi \right)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9aa3f8deb1c9737e247a40d1ed3f4a842d79d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.909ex; height:2.843ex;" alt="{\displaystyle (-z)=|z|\left(\cos \left(\theta +\pi \right)+i\operatorname {sen} \left(\theta +\pi \right)\right)}"></dd></dl> Exemplo: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690ec2437b8420af2a8591da98e7b42a93702330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.471ex; height:4.843ex;" alt="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-z)=-1-i=|z|(\cos(\theta +\pi )+i\operatorname {sen}(\theta +\pi ))={\sqrt {2}}\left(\cos {\frac {5}{4}}\pi +i\operatorname {sen} {\frac {5}{4}}\pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>i</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> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-z)=-1-i=|z|(\cos(\theta +\pi )+i\operatorname {sen}(\theta +\pi ))={\sqrt {2}}\left(\cos {\frac {5}{4}}\pi +i\operatorname {sen} {\frac {5}{4}}\pi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0474569b78066921121e25737989ff5b124e7af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.056ex; height:6.176ex;" alt="{\displaystyle (-z)=-1-i=|z|(\cos(\theta +\pi )+i\operatorname {sen}(\theta +\pi ))={\sqrt {2}}\left(\cos {\frac {5}{4}}\pi +i\operatorname {sen} {\frac {5}{4}}\pi \right)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Conjugado_de_um_número_complexo">Conjugado de um número complexo</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=20" title="Editar secção: Conjugado de um número complexo" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=20" title="Editar código-fonte da secção: Conjugado de um número complexo">editar código-fonte</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Conjugado_de_um_N%C3%BAmero_Complexo.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Conjugado_de_um_N%C3%BAmero_Complexo.png/184px-Conjugado_de_um_N%C3%BAmero_Complexo.png" decoding="async" width="184" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Conjugado_de_um_N%C3%BAmero_Complexo.png/276px-Conjugado_de_um_N%C3%BAmero_Complexo.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Conjugado_de_um_N%C3%BAmero_Complexo.png/368px-Conjugado_de_um_N%C3%BAmero_Complexo.png 2x" data-file-width="603" data-file-height="581" /></a><figcaption></figcaption></figure> O conjugado de um número complexo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768804bf1e5f2b40378497f93bb04a10ce64098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}"> é o número <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=a-ib}"> <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>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=a-ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc3dacb24890c881da71d9cf816d7a69638a6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.265ex; height:2.343ex;" alt="{\displaystyle {\bar {z}}=a-ib}">. Corresponde a uma reflexão de <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> na reta das abcissas. Em notação trigonométrica: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baec6942aba12dd974a6b3bc9ee0fc03fb0f3690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.715ex; height:2.843ex;" alt="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=|z|(\cos(-\theta )+i\operatorname {sen}(-\theta ))}"> <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> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=|z|(\cos(-\theta )+i\operatorname {sen}(-\theta ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad33071459ddfbff0f495b065cf08d873506388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.384ex; height:2.843ex;" alt="{\displaystyle {\bar {z}}=|z|(\cos(-\theta )+i\operatorname {sen}(-\theta ))}"></dd></dl> Exemplo: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1+i=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</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> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1+i=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103968dacfc9e299e8b18c31ad7b4078cb86e89e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.858ex; height:4.843ex;" alt="{\displaystyle z=1+i=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)={\sqrt {2}}\left(\cos {\frac {\pi }{4}}+i\operatorname {sen} {\frac {\pi }{4}}\right)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=-1-i=|z|\left(\cos(-\theta )+i\operatorname {sen}(-\theta )\right)={\sqrt {2}}\left(\cos \left(-{\frac {5}{4}}\right)\pi +i\operatorname {sen} \left(-{\frac {5}{4}}\right)\pi \right)}"> <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> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>i</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> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>π</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=-1-i=|z|\left(\cos(-\theta )+i\operatorname {sen}(-\theta )\right)={\sqrt {2}}\left(\cos \left(-{\frac {5}{4}}\right)\pi +i\operatorname {sen} \left(-{\frac {5}{4}}\right)\pi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be47e11ecc61d056e08dfc09e33752a09bf3df1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.99ex; height:6.176ex;" alt="{\displaystyle {\bar {z}}=-1-i=|z|\left(\cos(-\theta )+i\operatorname {sen}(-\theta )\right)={\sqrt {2}}\left(\cos \left(-{\frac {5}{4}}\right)\pi +i\operatorname {sen} \left(-{\frac {5}{4}}\right)\pi \right)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Produto_dos_números_complexos">Produto dos números complexos</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=21" title="Editar secção: Produto dos números complexos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=21" title="Editar código-fonte da secção: Produto dos números complexos">editar código-fonte</a>]</div> Seja <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}"> <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> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f79ad9ac38760050e59b965c63e84885356666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.102ex; height:2.843ex;" alt="{\displaystyle z=|z|\left(\cos \theta +i\operatorname {sen} \theta \right)}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=|w|\left(\cos \alpha +i\operatorname {sen} \alpha \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=|w|\left(\cos \alpha +i\operatorname {sen} \alpha \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5ef5c872b436d254cdf7df738a3d3e506d73d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.048ex; height:2.843ex;" alt="{\displaystyle w=|w|\left(\cos \alpha +i\operatorname {sen} \alpha \right)}">, a interpretação geométrica do produto dos números complexos pode seguir os seguintes casos: <div class="mw-heading mw-heading4"><h4 id="O_produto_de_um_número_complexo_Z_por_um_número_real_K">O produto de um número complexo Z por um número real K</h4>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=22" title="Editar secção: O produto de um número complexo Z por um número real K" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=22" title="Editar código-fonte da secção: O produto de um número complexo Z por um número real K">editar código-fonte</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot z=k\cdot |z|(\cos \theta +i\operatorname {sen} \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <mi>z</mi> <mo>=</mo> <mi>k</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot z=k\cdot |z|(\cos \theta +i\operatorname {sen} \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3096db798391104a0cb5e4e0ab364e0fc91489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.496ex; height:2.843ex;" alt="{\displaystyle k\cdot z=k\cdot |z|(\cos \theta +i\operatorname {sen} \theta )}"> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:2z.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/2z.png/182px-2z.png" decoding="async" width="182" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/2z.png/273px-2z.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/2z.png/364px-2z.png 2x" data-file-width="624" data-file-height="561" /></a><figcaption></figcaption></figure> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>1}">, então o produto corresponde a uma ampliação do vetor <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> Exemplo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abedff32339aca0bfbf207af87dd88fc4745a794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.54ex; height:3.176ex;" alt="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2z=2+2i=2|z|(\cos \theta +i\operatorname {sen} \theta )=2{\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mo>=</mo> <mn>2</mn> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2z=2+2i=2|z|(\cos \theta +i\operatorname {sen} \theta )=2{\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64a43e147ee1967b86259bee09f59361e0280bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.189ex; height:3.176ex;" alt="{\displaystyle 2z=2+2i=2|z|(\cos \theta +i\operatorname {sen} \theta )=2{\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Meio_z.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Meio_z.png/190px-Meio_z.png" decoding="async" width="190" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Meio_z.png/285px-Meio_z.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Meio_z.png/380px-Meio_z.png 2x" data-file-width="616" data-file-height="569" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<k<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<k<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c548b9b4ff336614c8116ed1520e374cba8a2f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.733ex; height:2.176ex;" alt="{\displaystyle 0<k<1}">, então o produto corresponde a uma contração do vetor <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> Exemplo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abedff32339aca0bfbf207af87dd88fc4745a794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.54ex; height:3.176ex;" alt="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}z={\frac {1}{2}}+{\frac {1}{2}}i={\frac {1}{2}}|z|(\cos \theta +i\operatorname {sen} \theta )={\frac {\sqrt {2}}{2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}z={\frac {1}{2}}+{\frac {1}{2}}i={\frac {1}{2}}|z|(\cos \theta +i\operatorname {sen} \theta )={\frac {\sqrt {2}}{2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f49828326b0e37e6040cd6e5cd17ed81bbd840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.208ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{2}}z={\frac {1}{2}}+{\frac {1}{2}}i={\frac {1}{2}}|z|(\cos \theta +i\operatorname {sen} \theta )={\frac {\sqrt {2}}{2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:-z.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/-z.png/189px--z.png" decoding="async" width="189" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/-z.png/284px--z.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/-z.png/378px--z.png 2x" data-file-width="541" data-file-height="523" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59e54fad8568e90715f2b10521d3e39bc45fca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k<0}">, então o produto corresponde a uma ampliação ou contração do vetor <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, seguida de uma rotação de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d0431ce231935522dc0cb52df7f2b406cdadc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }}">, pois <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> passará para a semi-reta oposta, que contém <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c4571a4a3ceb7e7e55712372835ebe65d20f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.896ex; height:2.176ex;" alt="{\displaystyle -z}">. Exemplo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abedff32339aca0bfbf207af87dd88fc4745a794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.54ex; height:3.176ex;" alt="{\displaystyle z=1+i=|z|(\cos \theta +i\operatorname {sen} \theta )={\sqrt {2}}(\cos 45^{\circ }+i\operatorname {sen} 45^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)z=-(1+1i)=|z|(\cos(\theta +180^{\circ })+i\operatorname {sen}(\theta +180^{\circ }))={\sqrt {2}}(\cos 225^{\circ }+i\operatorname {sen} 225^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mi>i</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 stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>225</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>225</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)z=-(1+1i)=|z|(\cos(\theta +180^{\circ })+i\operatorname {sen}(\theta +180^{\circ }))={\sqrt {2}}(\cos 225^{\circ }+i\operatorname {sen} 225^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d57c870251b6180272d0765557bb21879ab823a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.033ex; height:3.176ex;" alt="{\displaystyle (-1)z=-(1+1i)=|z|(\cos(\theta +180^{\circ })+i\operatorname {sen}(\theta +180^{\circ }))={\sqrt {2}}(\cos 225^{\circ }+i\operatorname {sen} 225^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <div class="mw-heading mw-heading4"><h4 id="O_produto_de_um_número_complexo_Z_por_um_imaginário_puro">O produto de um número complexo Z por um imaginário puro</h4>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=23" title="Editar secção: O produto de um número complexo Z por um imaginário puro" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=23" title="Editar código-fonte da secção: O produto de um número complexo Z por um imaginário puro">editar código-fonte</a>]</div> Dados <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baec6942aba12dd974a6b3bc9ee0fc03fb0f3690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.715ex; height:2.843ex;" alt="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5351a110ef1a29de283f35353d1c40a03bc6734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.444ex; height:2.843ex;" alt="{\displaystyle w=|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e91b7636eedad2e2781f4f7ab67949818d6f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.061ex; height:2.843ex;" alt="{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos 90^{\circ }+i\operatorname {sen} 90^{\circ })}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef09104d61b4d4a828a2ec64041884477171920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.875ex; height:2.843ex;" alt="{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}"> A partir desta etapa, é necessário utilizar a expressão trigonométrica da soma dos ângulos dos senos e cossenos: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta +\alpha )=\cos \theta \cos \alpha -\operatorname {sen} \theta \operatorname {sen} \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>−</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta +\alpha )=\cos \theta \cos \alpha -\operatorname {sen} \theta \operatorname {sen} \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99adea7c498fb30b16476db3a4eca4c0fc4ca704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.461ex; height:2.843ex;" alt="{\displaystyle \cos(\theta +\alpha )=\cos \theta \cos \alpha -\operatorname {sen} \theta \operatorname {sen} \alpha }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sen}(\theta +\alpha )=\cos \theta \operatorname {sen} \alpha +\cos \alpha \operatorname {sen} \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen}(\theta +\alpha )=\cos \theta \operatorname {sen} \alpha +\cos \alpha \operatorname {sen} \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29eaad20889e91a29bcc8c64334ca208c08e888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.591ex; height:2.843ex;" alt="{\displaystyle \operatorname {sen}(\theta +\alpha )=\cos \theta \operatorname {sen} \alpha +\cos \alpha \operatorname {sen} \theta }"> Logo, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta +90^{\circ })=\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta +90^{\circ })=\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1affaa329889073aa42f79379ced84c1828c2e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.136ex; height:2.843ex;" alt="{\displaystyle \cos(\theta +90^{\circ })=\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ }}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sen}(\theta +90^{\circ })=\cos \theta \operatorname {sen} 90^{\circ }+\cos 90^{\circ }\operatorname {sen} \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen}(\theta +90^{\circ })=\cos \theta \operatorname {sen} 90^{\circ }+\cos 90^{\circ }\operatorname {sen} \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc885a47b5a522e7428643b554da7860dba65c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.266ex; height:2.843ex;" alt="{\displaystyle \operatorname {sen}(\theta +90^{\circ })=\cos \theta \operatorname {sen} 90^{\circ }+\cos 90^{\circ }\operatorname {sen} \theta }"> Então, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sen</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>+</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef09104d61b4d4a828a2ec64041884477171920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.875ex; height:2.843ex;" alt="{\displaystyle zw=|z||w|((\cos \theta \cos 90^{\circ }-\operatorname {sen} \theta \operatorname {sen} 90^{\circ })+i(\cos \theta \operatorname {sen} 90^{\circ }+\operatorname {sen} \theta \cos 90^{\circ }))}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =|z||w|(\cos(\theta +90^{\circ })+i\operatorname {sen}(\theta +90^{\circ }))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =|z||w|(\cos(\theta +90^{\circ })+i\operatorname {sen}(\theta +90^{\circ }))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0626f6661ea76b4bde2de7810a4daf5d22a78f97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.223ex; height:2.843ex;" alt="{\displaystyle =|z||w|(\cos(\theta +90^{\circ })+i\operatorname {sen}(\theta +90^{\circ }))}"> O produto de um número complexo por um número imaginário puro corresponde a uma ampliação ou contração do vetor, seguido de uma rotação de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"> no sentido anti-horário, em torno da origem do vetor obtido. <div class="mw-heading mw-heading4"><h4 id="O_produto_de_um_número_complexo_genérico_Z_por_um_outro_número_complexo_W">O produto de um número complexo genérico Z por um outro número complexo W</h4>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=24" title="Editar secção: O produto de um número complexo genérico Z por um outro número complexo W" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=24" title="Editar código-fonte da secção: O produto de um número complexo genérico Z por um outro número complexo W">editar código-fonte</a>]</div> Dados <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baec6942aba12dd974a6b3bc9ee0fc03fb0f3690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.715ex; height:2.843ex;" alt="{\displaystyle z=|z|(\cos \theta +i\operatorname {sen} \theta )}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=|w|(\cos \alpha +i\operatorname {sen} \alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=|w|(\cos \alpha +i\operatorname {sen} \alpha )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28d7b758e711abccc6c9a5330cf72efca86e671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.661ex; height:2.843ex;" alt="{\displaystyle w=|w|(\cos \alpha +i\operatorname {sen} \alpha )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos \alpha +i\operatorname {sen} \alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos \alpha +i\operatorname {sen} \alpha )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e920163eacb63bb7fde68fb8a2098de9ca0785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.278ex; height:2.843ex;" alt="{\displaystyle zw=|z|(\cos \theta +i\operatorname {sen} \theta )|w|(\cos \alpha +i\operatorname {sen} \alpha )}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z||w|((\cos \theta \cos \alpha -\operatorname {sen} \theta \cos \alpha )+i(\cos \theta \cos \alpha +\operatorname {sen} \theta \cos \alpha ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>−</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>sen</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z||w|((\cos \theta \cos \alpha -\operatorname {sen} \theta \cos \alpha )+i(\cos \theta \cos \alpha +\operatorname {sen} \theta \cos \alpha ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee10264919a3ba9d5bf92b55399ff4619f015f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.048ex; height:2.843ex;" alt="{\displaystyle zw=|z||w|((\cos \theta \cos \alpha -\operatorname {sen} \theta \cos \alpha )+i(\cos \theta \cos \alpha +\operatorname {sen} \theta \cos \alpha ))}"> Assim como no caso anterior, é necessário utilizar a soma dos angulos dos senos e cossenos. Logo, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw=|z||w|(\cos(\theta +\alpha )+i\operatorname {sen}(\theta +\alpha ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw=|z||w|(\cos(\theta +\alpha )+i\operatorname {sen}(\theta +\alpha ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c28eb6a4c6caf9e857b768e2b44305b5a23977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.837ex; height:2.843ex;" alt="{\displaystyle zw=|z||w|(\cos(\theta +\alpha )+i\operatorname {sen}(\theta +\alpha ))}"> O produto de um número complexo <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> por outro número complexo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"> corresponde a uma ampliação ou contração do vetor, seguido de uma rotação do ângulo igual ao argumento do vetor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"> no sentido anti-horário em torno da origem do vetor obtido. <div class="mw-heading mw-heading3"><h3 id="Soma_dos_números_complexos">Soma dos números complexos</h3>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=25" title="Editar secção: Soma dos números complexos" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=25" title="Editar código-fonte da secção: Soma dos números complexos">editar código-fonte</a>]</div> A soma de números complexos corresponde à soma dos vetores complexos associados a esses números. Dados quaisquer números reais <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> (de vetor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OA}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OA}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8f56dec8d7ea2a7c57e9ff32dae814da42670f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.516ex; height:3.676ex;" alt="{\displaystyle {\vec {OA}}}">) e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"> (de vetor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OB}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c17ebf9f2ea32fac3c1832a9e781cff4334ecab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.537ex; height:3.676ex;" alt="{\displaystyle {\vec {OB}}}">), a soma <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463532c6f3bbdf0dfab77e87217169457a81fe23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.593ex; height:2.176ex;" alt="{\displaystyle z+w}"> tem como representação vetorial o vetor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001a34507625971421c9e70411aa4582e5d6ec26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.54ex; height:3.676ex;" alt="{\displaystyle {\vec {OC}}}">, dado por <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {OA}}+{\vec {OB}}={\vec {OC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {OA}}+{\vec {OB}}={\vec {OC}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38bd4f15b40fd63bc7283ca1dc3ef7adf0c9b6ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.532ex; height:3.843ex;" alt="{\displaystyle {\vec {OA}}+{\vec {OB}}={\vec {OC}}}">. Exemplos: <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Soma1.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soma1.png/220px-Soma1.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soma1.png/330px-Soma1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soma1.png/440px-Soma1.png 2x" data-file-width="665" data-file-height="505" /></a><figcaption></figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Soma2.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Soma2.png/220px-Soma2.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Soma2.png/330px-Soma2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Soma2.png/440px-Soma2.png 2x" data-file-width="666" data-file-height="505" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\vec {OA}}=1+3i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\vec {OA}}=1+3i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e4813b127d1546efa4096d0d35305f48ca7589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.769ex; height:3.843ex;" alt="{\displaystyle z={\vec {OA}}=1+3i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\vec {OB}}=5+1i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mn>1</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w={\vec {OB}}=5+1i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b26c21537d56ea9e62df308eae889fb119bc9ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.366ex; height:3.843ex;" alt="{\displaystyle w={\vec {OB}}=5+1i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=6+4i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=6+4i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7948d36961dec1f4a780d0538cabb5818d8e8230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.29ex; height:3.843ex;" alt="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=6+4i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\vec {OA}}=-2+2i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\vec {OA}}=-2+2i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4db1eb70aab12ce311b3a0e6494e2c3c5c1fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.577ex; height:3.843ex;" alt="{\displaystyle z={\vec {OA}}=-2+2i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\vec {OB}}=4+2i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w={\vec {OB}}=4+2i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3695dfebebe3fc4ec5829049a24f436691d8adb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.366ex; height:3.843ex;" alt="{\displaystyle w={\vec {OB}}=4+2i}"> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Undefined" title="Undefined"><img resource="/wiki/Ficheiro:Soma3.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Soma3.png/220px-Soma3.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Soma3.png/330px-Soma3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Soma3.png/440px-Soma3.png 2x" data-file-width="666" data-file-height="424" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+4i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+4i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfcff73b96e25b57acaad990c35429b1addf9073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.29ex; height:3.843ex;" alt="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+4i}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; 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width:18.577ex; height:3.843ex;" alt="{\displaystyle z={\vec {OA}}=-2+2i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\vec {OB}}=4-1i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>1</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w={\vec {OB}}=4-1i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9324a657388722764451d529d3f2814ea446eb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.366ex; height:3.843ex;" alt="{\displaystyle w={\vec {OB}}=4-1i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+1i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+1i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0893894859c261a96a7a9f0619922c20190d8e85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.29ex; height:3.843ex;" alt="{\displaystyle z+w={\vec {OA}}+{\vec {OB}}={\vec {OC}}=2+1i}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47b8a82bd20d8116f80f819da4659c851f288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.323ex; height:0.343ex;" alt="{\displaystyle \quad }"> <div class="mw-heading mw-heading2"><h2 id="Ver_também">Ver também</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=26" title="Editar secção: Ver também" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=26" title="Editar código-fonte da secção: Ver também">editar código-fonte</a>]</div> <table class="infobox noprint" style="width:250px; line-height:2.2em; font-size:90%"> <tbody><tr style="line-height:1.3em"> <td colspan="2" style="text-align: center;">Outros projetos <a href="/wiki/Wikimedia" class="mw-redirect" title="Wikimedia">Wikimedia</a> também contêm material sobre este tema: </td></tr> <tr> <th><a href="https://pt.wikibooks.org/wiki/Special:Search/An%C3%A1lise_complexa/Introdu%C3%A7%C3%A3o" title="Wikilivros"><img alt="Wikilivros" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/21px-Wikibooks-logo.svg.png" decoding="async" width="21" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/42px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a> </th> <td><a href="https://pt.wikibooks.org/wiki/Special:Search/An%C3%A1lise_complexa/Introdu%C3%A7%C3%A3o" class="extiw" title="b:Special:Search/Análise complexa/Introdução">Livros e manuais</a> no <a href="https://pt.wikibooks.org/wiki/P%C3%A1gina_principal" class="extiw" title="b:Página principal">Wikilivros</a> </td></tr> </tbody></table><div id="interProject" style="display:none;"> <ul><li><a href="https://pt.wikibooks.org/wiki/Special:Search/An%C3%A1lise_complexa/Introdu%C3%A7%C3%A3o" class="extiw" title="b:Special:Search/Análise complexa/Introdução">Wikilivros</a></li></ul> </div> <ul><li><a href="/wiki/N%C3%BAmero_complexo_hiperb%C3%B3lico" title="Número complexo hiperbólico">Número complexo hiperbólico</a></li> <li><a href="/wiki/N%C3%BAmero_transfinito" title="Número transfinito">Número transfinito</a></li> <li><a href="/wiki/Quaterni%C3%B5es" class="mw-redirect" title="Quaterniões">Quaterniões</a></li> <li><a href="/wiki/Quaterni%C3%B5es_hiperb%C3%B3licos" class="mw-redirect" title="Quaterniões hiperbólicos">Quaterniões hiperbólicos</a></li></ul> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso.">Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-PRINC-1"><a href="#cite_ref-PRINC_1-0">↑</a> Whitehead, Alfred North & Russell, Bertrand: Principia Mathematica. 3 vols, Merchant Books, 2001, ISBN 978-A1603861823 (vol. 1), ISBN aw978-1603861830 (vol. 2), <a href="/wiki/Especial:Fontes_de_livros/9781603861847" class="internal mw-magiclink-isbn">ISBN 978-1603861847</a> (vol. 3) </li> <li id="cite_note-INTRO-2"><a href="#cite_ref-INTRO_2-0">↑</a> Russell, Bertrand (1919), Introduction to Mathematical Philosophy, George Allen and Unwin, London, UK. Reimpressão, John G. Slater (intro.), Routledge, London, UK, 1993 </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> Trigonometria e Números Complexos, por M. P. do Carmo, A. C. Morgado, E. Wagner; IMPA-VITAE, Brasil, 1992 </li> <li id="cite_note-GIEZZI-4"><a href="#cite_ref-GIEZZI_4-0">↑</a> <cite class="citation book">Gelson, Iezzi (1977). Fundamentos de Matemática elementar. 6 3 ed. São Paulo: Atual. p. 1-9</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.aufirst=Iezzi&rft.aulast=Gelson&rft.btitle=Fundamentos+de+Matem%C3%A1tica+elementar&rft.date=1977&rft.edition=3&rft.genre=book&rft.pages=1-9&rft.place=S%C3%A3o+Paulo&rft.pub=Atual&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> <cite class="citation web">PAULANTI, Cláudio (2014). <a rel="nofollow" class="external text" href="https://canal.cecierj.edu.br/012016/453f95169841f5f28c400aed94d56d18.pdf">«Conjunto dos números complexos»</a> (PDF). Fundação CECIERJ. Consultado em 30 de novembro de 2019</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.aufirst=Cl%C3%A1udio&rft.aulast=PAULANTI&rft.btitle=Conjunto+dos+n%C3%BAmeros+complexos&rft.date=2014&rft.genre=unknown&rft.pub=Funda%C3%A7%C3%A3o+CECIERJ&rft_id=https%3A%2F%2Fcanal.cecierj.edu.br%2F012016%2F453f95169841f5f28c400aed94d56d18.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> <cite class="citation web"><a rel="nofollow" class="external text" href="https://www.ime.unicamp.br/~ftorres/ENSINO/MONOGRAFIAS/NC1.pdf">«Números complexos»</a> (PDF). Universidade Estadual de Campinas. 20 de junho de 2014. Consultado em 30 de novembro de 2019</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.btitle=N%C3%BAmeros+complexos&rft.date=2014-06-20&rft.genre=unknown&rft.pub=Universidade+Estadual+de+Campinas&rft_id=https%3A%2F%2Fwww.ime.unicamp.br%2F~ftorres%2FENSINO%2FMONOGRAFIAS%2FNC1.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-USP-7"><a href="#cite_ref-USP_7-0">↑</a> <cite class="citation web">Cerri, Cristina; Monteiro, Martha S. (setembro de 2001). <a rel="nofollow" class="external text" href="http://www.ime.usp.br/~martha/caem/complexos.pdf">«História dos Números Complexos»</a> (PDF). Instituto de Matemática e Estatística da Universidade de São Paulo. Consultado em 17 de janeiro de 2012</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.au=Cerri%2C+Cristina%3B+Monteiro%2C+Martha+S.&rft.btitle=Hist%C3%B3ria+dos+N%C3%BAmeros+Complexos&rft.date=2001-09&rft.genre=unknown&rft.pub=Instituto+de+Matem%C3%A1tica+e+Estat%C3stica+da+Universidade+de+S%C3%A3o+Paulo&rft_id=http%3A%2F%2Fwww.ime.usp.br%2F~martha%2Fcaem%2Fcomplexos.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  !CS1 manut: Nomes múltiplos: lista de autores (<a href="/wiki/Categoria:!CS1_manut:_Nomes_m%C3%BAltiplos:_lista_de_autores" title="Categoria:!CS1 manut: Nomes múltiplos: lista de autores">link</a>) </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> <cite class="citation web"><a rel="nofollow" class="external text" href="https://www.scribd.com/doc/52289490/Apostol-Tom-M-Mathematical-Analysis">«Tom M. Apostol - Mathematical Analysis (5ed 1981).pdf»</a>. Scribd. Consultado em 6 de novembro de 2020</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.atitle=Tom+M.+Apostol+-+Mathematical+Analysis+%285ed+1981%29.pdf&rft.genre=unknown&rft.jtitle=Scribd&rft_id=https%3A%2F%2Fwww.scribd.com%2Fdoc%2F52289490%2FApostol-Tom-M-Mathematical-Analysis&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988">  </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> <cite class="citation web">GARCIA, Vera. <a rel="nofollow" class="external text" href="http://www.mat.ufrgs.br/~vclotilde/disciplinas/html/complexos_trigonometria-web/complexos_trigonometria_leitura_complexos_trigonometria_tarefa_1.htm">«Números complexos na forma trigonométrica»</a>. Universidade Federal do Rio Grande do Sul. Consultado em 30 de novembro de 2019</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.aufirst=Vera&rft.aulast=GARCIA&rft.btitle=N%C3%BAmeros+complexos+na+forma+trigonom%C3%A9trica&rft.genre=unknown&rft.pub=Universidade+Federal+do+Rio+Grande+do+Sul&rft_id=http%3A%2F%2Fwww.mat.ufrgs.br%2F~vclotilde%2Fdisciplinas%2Fhtml%2Fcomplexos_trigonometria-web%2Fcomplexos_trigonometria_leitura_complexos_trigonometria_tarefa_1.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas">Ligações externas</h2>[<a href="/w/index.php?title=N%C3%BAmero_complexo&veaction=edit&section=27" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor">editar</a> | <a href="/w/index.php?title=N%C3%BAmero_complexo&action=edit&section=27" title="Editar código-fonte da secção: Ligações externas">editar código-fonte</a>]</div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="http://sorzal-df.fc.unesp.br/~edvaldo/dominiocores.htm">«Domínio de Cores, Funções Complexas»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.btitle=Dom%C3nio+de+Cores%2C+Fun%C3%A7%C3%B5es+Complexas&rft.genre=unknown&rft_id=http%3A%2F%2Fsorzal-df.fc.unesp.br%2F~edvaldo%2Fdominiocores.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><a rel="nofollow" class="external text" href="http://www.interaula.com/matweb/medio/213/ncomplex.htm">Projeto MatWeb - Números Complexos</a></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.educ.fc.ul.pt/docentes/opombo/seminario/euler/numeroscomplexos.htm">«Números Complexos, uma abordagem científica»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.btitle=N%C3%BAmeros+Complexos%2C+uma+abordagem+cient%C3fica&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.educ.fc.ul.pt%2Fdocentes%2Fopombo%2Fseminario%2Feuler%2Fnumeroscomplexos.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://wwwp.fc.unesp.br/~edvaldo/">«Funções de uma Variável Complexa: Visualização e Interpretação Gráfica»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AN%C3%BAmero+complexo&rft.btitle=Fun%C3%A7%C3%B5es+de+uma+Vari%C3%A1vel+Complexa%3A+Visualiza%C3%A7%C3%A3o+e+Interpreta%C3%A7%C3%A3o+Gr%C3%A1fica&rft.genre=unknown&rft_id=http%3A%2F%2Fwwwp.fc.unesp.br%2F~edvaldo%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li></ul> <div role="navigation" class="navbox" aria-labelledby="Números" style="padding:3px"><table class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:N%C3%BAmeros" title="Predefinição:Números"><abbr title="Ver esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">v</abbr></a></li><li class="nv-discutir"><a href="/w/index.php?title=Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:N%C3%BAmeros&action=edit&redlink=1" class="new" title="Predefinição Discussão:Números (página não existe)"><abbr title="Discutir esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:N%C3%BAmeros&action=edit"><abbr title="Editar esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">e</abbr></a></li></ul></div><div id="Números" style="font-size:114%;margin:0 4em"><a href="/wiki/N%C3%BAmero" title="Número">Números</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;"><a href="/wiki/Conjunto_cont%C3%A1vel" title="Conjunto contável">Conjuntos contáveis</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/N%C3%BAmero_natural" title="Número natural">Números naturais</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40eac26c488d3257e3fbe63619729673145d228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {N} }">)</li> <li><a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro">Números inteiros</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c672518c0350ca035befd41c26633a2d399431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.096ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {Z} }">)</li> <li><a href="/wiki/N%C3%BAmero_racional" title="Número racional">Números racionais</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaa5ab94a056a5a25944ddf0c52c92a404715ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {Q} }">)</li> <li><a href="/wiki/N%C3%BAmero_alg%C3%A9brico" title="Número algébrico">Números algébricos</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93d69135c22d5b1b10f65fa49c7ece56ae561fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {\overline {\mathbb {Q} }}}">)</li> <li><a href="/wiki/N%C3%BAmero_comput%C3%A1vel" title="Número computável">Números computáveis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Números reais e suas extensões</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/N%C3%BAmero_real" title="Número real">Números reais</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7df6838b44979c6531f6a0306206fbdb0477ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {R} }">)</li> <li><a class="mw-selflink selflink">Números complexos</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe3a54bb4e56c039e18c3af24ba70ab377f7a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {C} }">)</li> <li><a href="/wiki/Quaterni%C3%B5es" class="mw-redirect" title="Quaterniões">Quaterniões</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d00daea5df233d805f1ec5d5ae84845bac2ad06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {H} }">)</li> <li><a href="/wiki/Octoni%C3%B5es" class="mw-redirect" title="Octoniões">Octoniões</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5cf3960cf7ba384648447c15581d5d4589a6d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {O} }">)</li> <li><a href="/wiki/Sedeni%C3%B5es" class="mw-redirect" title="Sedeniões">Sedeniões</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef48a593f4503abeab608e8781ba478b7d1b304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.914ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbb {S} }">)</li> <li><a href="/wiki/N%C3%BAmero_complexo_hiperb%C3%B3lico" title="Número complexo hiperbólico">Números complexos hiperbólicos</a></li> <li><a href="/wiki/N%C3%BAmero_hipercomplexo" title="Número hipercomplexo">Números hipercomplexos</a></li> <li><a href="/w/index.php?title=N%C3%BAmero_superreal&action=edit&redlink=1" class="new" title="Número superreal (página não existe)">Números superreais</a></li> <li><a href="/wiki/N%C3%BAmero_irracional" title="Número irracional">Números irracionais</a></li> <li><a href="/wiki/N%C3%BAmero_transcendente" title="Número transcendente">Números transcendentais</a></li> <li><a href="/wiki/N%C3%BAmero_transfinito" title="Número transfinito">Números transfinitos</a></li> <li><a href="/wiki/N%C3%BAmero_hiper-real" title="Número hiper-real">Números hiper-reais</a></li> <li><a href="/wiki/N%C3%BAmero_surreal" title="Número surreal">Números surreais</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Outros sistemas</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/N%C3%BAmero_cardinal" title="Número cardinal">Números cardinais</a></li> <li><a href="/wiki/N%C3%BAmero_ordinal" title="Número ordinal">Números ordinais</a></li> <li><a href="/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico">Números p-ádicos</a></li> <li><a href="/w/index.php?title=N%C3%BAmero_supernatural&action=edit&redlink=1" class="new" title="Número supernatural (página não existe)">Números supernaturais</a></li></ul> </div></td></tr></tbody></table></div> <ul class="noprint navigation-box" style="border-top: solid silver 1px; 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