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href="/wiki/Istimewa:Perubahan_terbaru" title="Daftar perubahan terbaru dalam wiki. [r]" accesskey="r"><span>Perubahan terbaru</span></a></li><li id="n-Artikel-pilihan" class="mw-list-item"><a href="/wiki/Wikipedia:Artikel_pilihan/Topik"><span>Artikel pilihan</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Peristiwa_terkini" title="Temukan informasi tentang peristiwa terkini"><span>Peristiwa terkini</span></a></li><li id="n-newpage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_baru"><span>Halaman baru</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_sembarang" title="Tampilkan sembarang halaman [x]" accesskey="x"><span>Halaman sembarang</span></a></li> </ul> </div> </div> <div id="p-Komunitas" class="vector-menu mw-portlet mw-portlet-Komunitas" > <div class="vector-menu-heading"> Komunitas </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Warung-Kopi" class="mw-list-item"><a href="/wiki/Wikipedia:Warung_Kopi"><span>Warung Kopi</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Komunitas" title="Tentang proyek, apa yang dapat Anda lakukan, di mana untuk mencari sesuatu"><span>Portal komunitas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Bantuan:Isi" title="Tempat mencari bantuan."><span>Bantuan</span></a></li> </ul> </div> </div> <div id="p-Wikipedia" class="vector-menu mw-portlet mw-portlet-Wikipedia" > <div class="vector-menu-heading"> Wikipedia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Perihal"><span>Tentang Wikipedia</span></a></li><li id="n-Pancapilar" class="mw-list-item"><a href="/wiki/Wikipedia:Pancapilar"><span>Pancapilar</span></a></li><li id="n-Kebijakan" class="mw-list-item"><a href="/wiki/Wikipedia:Kebijakan_dan_pedoman"><span>Kebijakan</span></a></li><li id="n-Hubungi-kami" class="mw-list-item"><a href="/wiki/Wikipedia:Hubungi_kami"><span>Hubungi kami</span></a></li><li id="n-Bak-pasir" class="mw-list-item"><a href="/wiki/Wikipedia:Bak_pasir"><span>Bak pasir</span></a></li> </ul> </div> </div> <div id="p-Bagikan" class="vector-menu mw-portlet mw-portlet-Bagikan emptyPortlet" > <div class="vector-menu-heading"> Bagikan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Halaman_Utama" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; 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meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Masuk_log&returnto=Bilangan+kompleks" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Opsi lainnya" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas pribadi" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Bilangan+kompleks" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Bilangan+kompleks" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Garis_besar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Garis_besar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Garis besar</span> </div> </a> <button aria-controls="toc-Garis_besar-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Garis besar</span> </button> <ul id="toc-Garis_besar-sublist" class="vector-toc-list"> <li id="toc-Notasi_dan_operasi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notasi_dan_operasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Notasi dan operasi</span> </div> </a> <ul id="toc-Notasi_dan_operasi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definisi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definisi</span> </div> </a> <ul id="toc-Definisi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Notasi</span> </div> </a> <button aria-controls="toc-Notasi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Notasi</span> </button> <ul id="toc-Notasi-sublist" class="vector-toc-list"> <li id="toc-Bentuk_Penjumlahan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bentuk_Penjumlahan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Bentuk Penjumlahan</span> </div> </a> <ul id="toc-Bentuk_Penjumlahan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bentuk_Polar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bentuk_Polar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Bentuk Polar</span> </div> </a> <ul id="toc-Bentuk_Polar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bentuk_Eksponen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bentuk_Eksponen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Bentuk Eksponen</span> </div> </a> <ul id="toc-Bentuk_Eksponen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bidang_kompleks" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bidang_kompleks"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bidang kompleks</span> </div> </a> <ul id="toc-Bidang_kompleks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konstruksi_formal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konstruksi_formal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Konstruksi formal</span> </div> </a> <button aria-controls="toc-Konstruksi_formal-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Konstruksi formal</span> </button> <ul id="toc-Konstruksi_formal-sublist" class="vector-toc-list"> <li id="toc-Konstruksi_sebagai_tatanan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konstruksi_sebagai_tatanan"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Konstruksi sebagai tatanan</span> </div> </a> <ul id="toc-Konstruksi_sebagai_tatanan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konstruksi_sebagai_medan_hasil_bagi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konstruksi_sebagai_medan_hasil_bagi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Konstruksi sebagai medan hasil bagi</span> </div> </a> <ul id="toc-Konstruksi_sebagai_medan_hasil_bagi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wakilan_matriks_dari_bilangan_kompleks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wakilan_matriks_dari_bilangan_kompleks"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Wakilan matriks dari bilangan kompleks</span> </div> </a> <ul id="toc-Wakilan_matriks_dari_bilangan_kompleks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometri" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometri"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Geometri</span> </div> </a> <button aria-controls="toc-Geometri-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Geometri</span> </button> <ul id="toc-Geometri-sublist" class="vector-toc-list"> <li id="toc-Bentuk" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bentuk"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Bentuk</span> </div> </a> <ul id="toc-Bentuk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometri_fraktal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometri_fraktal"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Geometri fraktal</span> </div> </a> <ul id="toc-Geometri_fraktal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Segitiga" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Segitiga"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Segitiga</span> </div> </a> <ul id="toc-Segitiga-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Catatan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Catatan"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Catatan</span> </div> </a> <ul id="toc-Catatan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Referensi</span> </div> </a> <button aria-controls="toc-Referensi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Referensi</span> </button> <ul id="toc-Referensi-sublist" class="vector-toc-list"> <li id="toc-Kutipan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kutipan"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Kutipan</span> </div> </a> <ul id="toc-Kutipan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bacaan_lebih_lanjut" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bacaan_lebih_lanjut"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bacaan lebih lanjut</span> </div> </a> <button aria-controls="toc-Bacaan_lebih_lanjut-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Bacaan lebih lanjut</span> </button> <ul id="toc-Bacaan_lebih_lanjut-sublist" class="vector-toc-list"> <li id="toc-Matematika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matematika"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Matematika</span> </div> </a> <ul id="toc-Matematika-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Sejarah</span> </div> </a> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> </ul> </li> </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="Daftar isi" 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="Gulingkan daftar isi" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Bilangan kompleks</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Pergi ke artikel dalam bahasa lain. Terdapat 132 bahasa" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-132" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">132 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Komplekse_getal" title="Komplekse getal – Afrikaans" lang="af" hreflang="af" data-title="Komplekse getal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – Jerman (Swiss)" lang="gsw" hreflang="gsw" data-title="Komplexe Zahl" data-language-autonym="Alemannisch" data-language-local-name="Jerman (Swiss)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%A0%E1%89%85%E1%8C%A3%E1%8C%AB_%E1%89%81%E1%8C%A5%E1%88%AD" title="የአቅጣጫ ቁጥር – Amharik" lang="am" hreflang="am" data-title="የአቅጣጫ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="Amharik" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_complexo" title="Numero complexo – Aragon" lang="an" hreflang="an" data-title="Numero complexo" data-language-autonym="Aragonés" data-language-local-name="Aragon" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – Angika" lang="anp" hreflang="anp" data-title="समिश्र संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%B1%D9%83%D8%A8" title="عدد مركب – Arab" lang="ar" hreflang="ar" data-title="عدد مركب" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – Assam" lang="as" hreflang="as" data-title="জটিল সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assam" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_complexu" title="Númberu complexu – Asturia" lang="ast" hreflang="ast" data-title="Númberu complexu" data-language-autonym="Asturianu" data-language-local-name="Asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kompleks_%C9%99d%C9%99dl%C9%99r" title="Kompleks ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Kompleks ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D9%85%D9%BE%D9%84%DA%A9%D8%B3_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="کومپلکس ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="کومپلکس ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D2%BB%D0%B0%D0%BD" title="Комплекслы һан – Bashkir" lang="ba" hreflang="ba" data-title="Комплекслы һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Kuompleks%C4%97nis_skaitlios" title="Kuompleksėnis skaitlios – Samogitian" lang="sgs" hreflang="sgs" data-title="Kuompleksėnis skaitlios" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – Belarusia" lang="be" hreflang="be" data-title="Камплексны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusia" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Камплексны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Камплексны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексно число – Bulgaria" lang="bg" hreflang="bg" data-title="Комплексно число" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="জটিল সংখ্যা – Bengali" lang="bn" hreflang="bn" data-title="জটিল সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – Bosnia" lang="bs" hreflang="bs" data-title="Kompleksan broj" data-language-autonym="Bosanski" data-language-local-name="Bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Комплекс тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_complex" title="Nombre complex – Katalan" lang="ca" hreflang="ca" data-title="Nombre complex" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%A6%D8%A7%D9%88%DB%8E%D8%AA%DB%95" title="ژمارەی ئاوێتە – Kurdi Sorani" lang="ckb" hreflang="ckb" data-title="ژمارەی ئاوێتە" data-language-autonym="کوردی" data-language-local-name="Kurdi Sorani" class="interlanguage-link-target"><span>کوردی</span></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 – Cheska" lang="cs" hreflang="cs" data-title="Komplexní číslo" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Комплекслă хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Комплекслă хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_cymhlyg" title="Rhif cymhlyg – Welsh" lang="cy" hreflang="cy" data-title="Rhif cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://da.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – Dansk" lang="da" hreflang="da" data-title="Komplekse tal" data-language-autonym="Dansk" data-language-local-name="Dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Komplexe_Zahl" title="Komplexe Zahl – Jerman" lang="de" hreflang="de" data-title="Komplexe Zahl" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_kompleks" title="Amaro kompleks – Zazaki" lang="diq" hreflang="diq" data-title="Amaro kompleks" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Μιγαδικός αριθμός – Yunani" lang="el" hreflang="el" data-title="Μιγαδικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_cumpl%C3%AAs" title="Nómmer cumplês – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer cumplês" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complex_number" title="Complex number – Inggris" lang="en" hreflang="en" data-title="Complex number" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_nombro" title="Kompleksa nombro – Esperanto" lang="eo" hreflang="eo" data-title="Kompleksa nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_complejo" title="Número complejo – Spanyol" lang="es" hreflang="es" data-title="Número complejo" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – Esti" lang="et" hreflang="et" data-title="Kompleksarv" data-language-autonym="Eesti" data-language-local-name="Esti" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_konplexu" title="Zenbaki konplexu – Basque" lang="eu" hreflang="eu" data-title="Zenbaki konplexu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%AE%D8%AA%D9%84%D8%B7" title="عدد مختلط – Persia" lang="fa" hreflang="fa" data-title="عدد مختلط" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kompleksiluku" title="Kompleksiluku – Suomi" lang="fi" hreflang="fi" data-title="Kompleksiluku" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kompleksarv" title="Kompleksarv – Võro" lang="vro" hreflang="vro" data-title="Kompleksarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Komplekst_tal" title="Komplekst tal – Faroe" lang="fo" hreflang="fo" data-title="Komplekst tal" data-language-autonym="Føroyskt" data-language-local-name="Faroe" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_complexe" title="Nombre complexe – Prancis" lang="fr" hreflang="fr" data-title="Nombre complexe" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kompleks_taal" title="Kompleks taal – Frisia Utara" lang="frr" hreflang="frr" data-title="Kompleks taal" data-language-autonym="Nordfriisk" data-language-local-name="Frisia Utara" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kompleks_getal" title="Kompleks getal – Frisia Barat" lang="fy" hreflang="fy" data-title="Kompleks getal" data-language-autonym="Frysk" data-language-local-name="Frisia Barat" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_choimpl%C3%A9ascach" title="Uimhir choimpléascach – Irlandia" lang="ga" hreflang="ga" data-title="Uimhir choimpléascach" data-language-autonym="Gaeilge" data-language-local-name="Irlandia" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Gan" lang="gan" hreflang="gan" data-title="複數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_kompleks" title="Nonm kompleks – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm kompleks" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – Galisia" lang="gl" hreflang="gl" data-title="Número complexo" data-language-autonym="Galego" data-language-local-name="Galisia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapy_rypy%27%C5%A9" title="Papapy rypy'ũ – Guarani" lang="gn" hreflang="gn" data-title="Papapy rypy'ũ" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%95%E0%AA%B0_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE%E0%AA%93" title="સંકર સંખ્યાઓ – Gujarat" lang="gu" hreflang="gu" data-title="સંકર સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarat" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A8%D7%95%D7%9B%D7%91" title="מספר מרוכב – Ibrani" lang="he" hreflang="he" data-title="מספר מרוכב" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="समिश्र संख्या – Hindi" lang="hi" hreflang="hi" data-title="समिश्र संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Jatil_ginti" title="Jatil ginti – Hindi Fiji" lang="hif" hreflang="hif" data-title="Jatil ginti" data-language-autonym="Fiji Hindi" data-language-local-name="Hindi Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kompleksni_broj" title="Kompleksni broj – Kroasia" lang="hr" hreflang="hr" data-title="Kompleksni broj" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Komplex_sz%C3%A1mok" title="Komplex számok – Hungaria" lang="hu" hreflang="hu" data-title="Komplex számok" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A9%D5%AB%D5%BE" title="Կոմպլեքս թիվ – Armenia" lang="hy" hreflang="hy" data-title="Կոմպլեքս թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_complexe" title="Numero complexe – Interlingua" lang="ia" hreflang="ia" data-title="Numero complexe" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_kompleks" title="Lumur kompleks – Iban" lang="iba" hreflang="iba" data-title="Lumur kompleks" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Komplexa_nombro" title="Komplexa nombro – Ido" lang="io" hreflang="io" data-title="Komplexa nombro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tvinnt%C3%B6lur" title="Tvinntölur – Islandia" lang="is" hreflang="is" data-title="Tvinntölur" data-language-autonym="Íslenska" data-language-local-name="Islandia" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_complesso" title="Numero complesso – Italia" lang="it" hreflang="it" data-title="Numero complesso" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E6%95%B0" title="複素数 – Jepang" lang="ja" hreflang="ja" data-title="複素数" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Komplex_nomba" title="Komplex nomba – 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"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/relcimdyna%27u" title="relcimdyna'u – Lojban" lang="jbo" hreflang="jbo" data-title="relcimdyna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A1%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="კომპლექსური რიცხვი – Georgia" lang="ka" hreflang="ka" data-title="კომპლექსური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Am%E1%B8%8Dan_asemlal" title="Amḍan asemlal – Kabyle" lang="kab" hreflang="kab" data-title="Amḍan asemlal" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Nd%C9%A9_nd%C9%A9_%C3%B1%CA%8A%C5%8B" title="Ndɩ ndɩ ñʊŋ – Kabiye" lang="kbp" hreflang="kbp" data-title="Ndɩ ndɩ ñʊŋ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%88%D0%B5%D0%BD_%D1%81%D0%B0%D0%BD" title="Кешен сан – Kazakh" lang="kk" hreflang="kk" data-title="Кешен сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%80%E1%9E%BB%E1%9F%86%E1%9E%95%E1%9F%92%E1%9E%9B%E1%9E%B7%E1%9E%85" title="ចំនួនកុំផ្លិច – Khmer" lang="km" hreflang="km" data-title="ចំនួនកុំផ្លិច" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수 – Korea" lang="ko" hreflang="ko" data-title="복소수" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Niver_kompleth" title="Niver kompleth – Kornish" lang="kw" hreflang="kw" data-title="Niver kompleth" data-language-autonym="Kernowek" data-language-local-name="Kornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D1%82%D2%AF%D2%AF_%D1%81%D0%B0%D0%BD" title="Комплекстүү сан – Kirgiz" lang="ky" hreflang="ky" data-title="Комплекстүү сан" data-language-autonym="Кыргызча" data-language-local-name="Kirgiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_complexus" title="Numerus complexus – Latin" lang="la" hreflang="la" data-title="Numerus complexus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Complex_getal" title="Complex getal – Limburgia" lang="li" hreflang="li" data-title="Complex getal" data-language-autonym="Limburgs" data-language-local-name="Limburgia" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_compless" title="Numer compless – Lombard" lang="lmo" hreflang="lmo" data-title="Numer compless" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AA%E0%BA%BB%E0%BA%99" title="ຈຳນວນສົນ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນສົນ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kompleksinis_skai%C4%8Dius" title="Kompleksinis skaičius – Lituavi" lang="lt" hreflang="lt" data-title="Kompleksinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lituavi" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komplekss_skaitlis" title="Komplekss skaitlis – Latvi" lang="lv" hreflang="lv" data-title="Komplekss skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvi" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_haro" title="Isa haro – Malagasi" lang="mg" hreflang="mg" data-title="Isa haro" data-language-autonym="Malagasy" data-language-local-name="Malagasi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексен број – Makedonia" lang="mk" hreflang="mk" data-title="Комплексен број" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B0%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="മിശ്രസംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="മിശ്രസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%82%D0%BE%D0%BE" title="Комплекс тоо – Mongolia" lang="mn" hreflang="mn" data-title="Комплекс тоо" data-language-autonym="Монгол" data-language-local-name="Mongolia" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="संमिश्र संख्या – Marathi" lang="mr" hreflang="mr" data-title="संमिश्र संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_kompleks" title="Nombor kompleks – Melayu" lang="ms" hreflang="ms" data-title="Nombor kompleks" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%94%E1%80%BA%E1%80%95%E1%80%9C%E1%80%80%E1%80%BA%E1%80%85%E1%80%BA%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="ကွန်ပလက်စ်ကိန်း – Burma" lang="my" hreflang="my" data-title="ကွန်ပလက်စ်ကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Komplexe_Tall" title="Komplexe Tall – Jerman Rendah" lang="nds" hreflang="nds" data-title="Komplexe Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Jerman Rendah" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Complex_getal" title="Complex getal – Belanda" lang="nl" hreflang="nl" data-title="Complex getal" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Komplekse_tal" title="Komplekse tal – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Komplekse tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Komplekst_tall" title="Komplekst tall – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Komplekst tall" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_compl%C3%A8xe" title="Nombre complèxe – Ositania" lang="oc" hreflang="oc" data-title="Nombre complèxe" data-language-autonym="Occitan" data-language-local-name="Ositania" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_Xaxxamaa" title="Lakkoofsa Xaxxamaa – Oromo" lang="om" hreflang="om" data-title="Lakkoofsa Xaxxamaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BE%D0%BD_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Комплексон нымæц – Ossetia" lang="os" hreflang="os" data-title="Комплексон нымæц" data-language-autonym="Ирон" data-language-local-name="Ossetia" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AA%E0%A8%B2%E0%A9%88%E0%A8%95%E0%A8%B8_%E0%A8%A8%E0%A9%B0%E0%A8%AC%E0%A8%B0" title="ਕੰਪਲੈਕਸ ਨੰਬਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੰਪਲੈਕਸ ਨੰਬਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_zespolone" title="Liczby zespolone – Polski" lang="pl" hreflang="pl" data-title="Liczby zespolone" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_compless" title="Nùmer compless – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer compless" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%85%D9%BE%D9%84%DB%8C%DA%A9%D8%B3_%D9%86%D9%85%D8%A8%D8%B1" title="کمپلیکس نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کمپلیکس نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_complexo" title="Número complexo – Portugis" lang="pt" hreflang="pt" data-title="Número complexo" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_complex" title="Număr complex – Rumania" lang="ro" hreflang="ro" data-title="Număr complex" data-language-autonym="Română" data-language-local-name="Rumania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="artikel pilihan"><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 – Rusia" lang="ru" hreflang="ru" data-title="Комплексное число" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D1%96%D1%81%D0%BB%D0%BE" title="Комплексне чісло – Rusyn" lang="rue" hreflang="rue" data-title="Комплексне чісло" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D0%B0%D1%85%D1%81%D0%B0%D0%B0%D0%BD" title="Комплекс ахсаан – Sakha" lang="sah" hreflang="sah" data-title="Комплекс ахсаан" data-language-autonym="Саха тыла" data-language-local-name="Sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_cumplessu" title="Nùmmuru cumplessu – Sisilia" lang="scn" hreflang="scn" data-title="Nùmmuru cumplessu" data-language-autonym="Sicilianu" data-language-local-name="Sisilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Complex_nummer" title="Complex nummer – Skotlandia" lang="sco" hreflang="sco" data-title="Complex nummer" data-language-autonym="Scots" data-language-local-name="Skotlandia" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kompleksan_broj" title="Kompleksan broj – Serbo-Kroasia" lang="sh" hreflang="sh" data-title="Kompleksan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9A%E0%B7%93%E0%B6%BB%E0%B7%8A%E0%B6%AB_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="සංකීර්ණ සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="සංකීර්ණ සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Complex_number" title="Complex number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Complex number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Komplexn%C3%A9_%C4%8D%C3%ADslo" title="Komplexné číslo – Slovak" lang="sk" hreflang="sk" data-title="Komplexné číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kompleksno_%C5%A1tevilo" title="Kompleksno število – Sloven" lang="sl" hreflang="sl" data-title="Kompleksno število" data-language-autonym="Slovenščina" data-language-local-name="Sloven" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kompleksloho" title="Kompleksloho – Inari Sami" lang="smn" hreflang="smn" data-title="Kompleksloho" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_kakan" title="Thiin kakan – Somalia" lang="so" hreflang="so" data-title="Thiin kakan" data-language-autonym="Soomaaliga" data-language-local-name="Somalia" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_kompleks%C3%AB" title="Numrat kompleksë – Albania" lang="sq" hreflang="sq" data-title="Numrat kompleksë" data-language-autonym="Shqip" data-language-local-name="Albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Комплексан број – Serbia" lang="sr" hreflang="sr" data-title="Комплексан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komplexa_tal" title="Komplexa tal – Swedia" lang="sv" hreflang="sv" data-title="Komplexa tal" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_changamano" title="Namba changamano – Swahili" lang="sw" hreflang="sw" data-title="Namba changamano" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%86%E0%AE%A3%E0%AF%8D" title="சிக்கலெண் – Tamil" lang="ta" hreflang="ta" data-title="சிக்கலெண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%82%E0%B0%95%E0%B1%80%E0%B0%B0%E0%B1%8D%E0%B0%A3_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="సంకీర్ణ సంఖ్యలు – Telugu" lang="te" hreflang="te" data-title="సంకీర్ణ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D3%A3" title="Адади комплексӣ – Tajik" lang="tg" hreflang="tg" data-title="Адади комплексӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99" title="จำนวนเชิงซ้อน – Thai" lang="th" hreflang="th" data-title="จำนวนเชิงซ้อน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Komplikadong_bilang" title="Komplikadong bilang – Tagalog" lang="tl" hreflang="tl" data-title="Komplikadong bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karma%C5%9F%C4%B1k_say%C4%B1" title="Karmaşık sayı – Turki" lang="tr" hreflang="tr" data-title="Karmaşık sayı" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D1%81%D0%B0%D0%BD" title="Комплекс сан – Tatar" lang="tt" hreflang="tt" data-title="Комплекс сан" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Комплексне число – Ukraina" lang="uk" hreflang="uk" data-title="Комплексне число" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%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"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kompleks_sonlar" title="Kompleks sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Kompleks sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Numaro_conpleso" title="Numaro conpleso – Venesia" lang="vec" hreflang="vec" data-title="Numaro conpleso" data-language-autonym="Vèneto" data-language-local-name="Venesia" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_ph%E1%BB%A9c" title="Số phức – Vietnam" lang="vi" hreflang="vi" data-title="Số phức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Complexe_getalln" title="Complexe getalln – 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"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Komplikado_nga_ihap" title="Komplikado nga ihap – Warai" lang="war" hreflang="war" data-title="Komplikado nga ihap" data-language-autonym="Winaray" data-language-local-name="Warai" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="复数（数学） – Wu Tionghoa" lang="wuu" hreflang="wuu" data-title="复数（数学）" data-language-autonym="吴语" data-language-local-name="Wu Tionghoa" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B8%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Комплексин тойг – Kalmuk" lang="xal" hreflang="xal" data-title="Комплексин тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmuk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%9E%D7%A4%D7%9C%D7%A2%D7%A7%D7%A1%D7%A2_%D7%A6%D7%90%D7%9C" title="קאמפלעקסע צאל – Yiddish" lang="yi" hreflang="yi" data-title="קאמפלעקסע צאל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_t%C3%B3%E1%B9%A3%C3%B2ro" title="Nọ́mbà tóṣòro – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà tóṣòro" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学) – Tionghoa" lang="zh" hreflang="zh" data-title="复数 (数学)" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="複數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ho%CC%8Dk-cha%CC%8Dp-s%C3%B2%CD%98" title="Ho̍k-cha̍p-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Ho̍k-cha̍p-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A4%87%E6%95%B8" title="複數 – Kanton" lang="yue" hreflang="yue" data-title="複數" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11567#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bilangan_kompleks" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Bilangan_kompleks" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bilangan_kompleks"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Perkakas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Bilangan_kompleks"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Bilangan_kompleks" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Bilangan_kompleks" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa" title="Daftar semua halaman istimewa [q]" accesskey="q"><span>Halaman istimewa</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&oldid=24101384" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Bilangan_kompleks&id=24101384&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBilangan_kompleks"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBilangan_kompleks"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Bilangan+kompleks"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Bilangan_kompleks&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bilangan_kompleks&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Complex_numbers" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11567" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Complex_number_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/220px-Complex_number_illustration.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/330px-Complex_number_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/440px-Complex_number_illustration.svg.png 2x" data-file-width="180" data-file-height="180" /></a><figcaption>Bilangan kompleks secara visual dapat direpresentasikan sebagai sepasang angka <span class="texhtml" style="white-space: nowrap;">(<i>a</i>, <i>b</i>)</span> membentuk vektor pada diagram yang disebut <a href="/w/index.php?title=Diagram_Argand&action=edit&redlink=1" class="new" title="Diagram Argand (halaman belum tersedia)">diagram Argand</a>, mewakili <a href="/wiki/Bidang_kompleks" title="Bidang kompleks">bidang kompleks</a>. "Re"adalah sumbu nyata,"Im"adalah sumbu imajiner, dan <span class="texhtml" style="white-space: nowrap;"><i>i</i></span> memuaskan <span class="texhtml" style="white-space: nowrap;"><i>i</i><sup>2</sup> = −1</span>.</figcaption></figure> <p><b>Bilangan kompleks</b> dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, adalah bilangan yang dinotasikan oleh <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e8eb1f43c87d1631c9cf33779f92c83431626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.257ex; height:2.343ex;" alt="{\displaystyle a+bi\,}"></span>, di mana <i>a</i> dan <i>b</i> adalah <a href="/wiki/Bilangan_riil" title="Bilangan riil">bilangan riil</a>, dan <i>i</i> adalah suatu <a href="/wiki/Bilangan_imajiner" title="Bilangan imajiner">bilangan imajiner</a> di mana <i>i</i> <sup>2</sup> = −1. Bilangan riil <i>a</i> disebut juga <i><a href="/wiki/Bagian_riil" class="mw-redirect" title="Bagian riil">bagian riil</a></i> dari bilangan kompleks, dan bilangan real <i>b</i> disebut <i><a href="/wiki/Bagian_imajiner" class="mw-redirect" title="Bagian imajiner">bagian imajiner</a></i>. Jika pada suatu bilangan kompleks, nilai <i>b</i> adalah 0, maka bilangan kompleks tersebut menjadi sama dengan bilangan real <i>a</i>. </p><p>Sebagai contoh, 3 + 2<i>i</i> adalah <i>bilangan kompleks</i> dengan bagian riil 3 dan bagian imajiner 2<i>i</i>. </p><p>Bilangan kompleks dapat ditambah, dikurang, dikali, dan dibagi seperti bilangan riil; namun bilangan kompleks juga mempunyai sifat-sifat tambahan yang menarik. Misalnya, setiap persamaan aljabar <a href="/wiki/Polinomial" title="Polinomial">polinomial</a> mempunyai solusi bilangan kompleks, tidak seperti bilangan riil yang hanya memiliki sebagian. </p><p>Dalam bidang-bidang tertentu (seperti <a href="/wiki/Teknik_elektro" class="mw-redirect" title="Teknik elektro">teknik elektro</a>, di mana <i>i</i> digunakan sebagai simbol untuk <a href="/wiki/Arus_listrik" title="Arus listrik">arus listrik</a>), bilangan kompleks ditulis <i>a</i> + <i>bj</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Garis_besar">Garis besar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=1" title="Sunting bagian: Garis besar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=1" title="Sunting kode sumber bagian: Garis besar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notasi_dan_operasi">Notasi dan operasi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=2" title="Sunting bagian: Notasi dan operasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=2" title="Sunting kode sumber bagian: Notasi dan operasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Himpunan" class="mw-redirect" title="Himpunan">Himpunan</a> bilangan kompleks umumnya dinotasikan dengan <b>C</b>, atau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. Bilangan real, <b>R</b>, dapat dinyatakan sebagai bagian dari himpunan <b>C</b> dengan menyatakan setiap bilangan real sebagai bilangan kompleks: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a+0i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a+0i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406b84a58489330d31d2d07d902196baf79caa41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.363ex; height:2.343ex;" alt="{\displaystyle a=a+0i}"></span>. </p><p>Bilangan kompleks ditambah, dikurang, dan dikali dengan menggunakan sifat-sifat aljabar seperti <a href="/wiki/Asosiatif" class="mw-redirect" title="Asosiatif">asosiatif</a>, <a href="/wiki/Komutatif" class="mw-redirect" title="Komutatif">komutatif</a>, dan <a href="/wiki/Distributif" class="mw-redirect" title="Distributif">distributif</a>, dan dengan persamaan <i>i</i> <sup>2</sup> = −1: </p> <dl><dd>(<i>a</i> + <i>bi</i>) + (<i>c</i> + <i>di</i>) = (<i>a</i>+<i>c</i>) + (<i>b</i>+<i>d</i>)<i>i</i></dd> <dd>(<i>a</i> + <i>bi</i>) − (<i>c</i> + <i>di</i>) = (<i>a</i>−<i>c</i>) + (<i>b</i>−<i>d</i>)<i>i</i></dd> <dd>(<i>a</i> + <i>bi</i>)(<i>c</i> + <i>di</i>) = <i>ac</i> + <i>bci</i> + <i>adi</i> + <i>bd i</i> <sup>2</sup> = (<i>ac</i>−<i>bd</i>) + (<i>bc</i>+<i>ad</i>)<i>i</i></dd></dl> <p>Pembagian bilangan kompleks juga dapat didefinisikan (lihat di bawah). Jadi, himpunan bilangan kompleks membentuk bidang matematika yang, berbeda dengan bilangan real, berupa aljabar tertutup. </p><p>Dalam matematika, adjektif "kompleks" berarti bilangan kompleks digunakan sebagai dasar teori angka yang digunakan. Sebagai contoh, <a href="/wiki/Analisis_kompleks" title="Analisis kompleks">analisis kompleks</a>, <a href="/w/index.php?title=Matriks_kompleks&action=edit&redlink=1" class="new" title="Matriks kompleks (halaman belum tersedia)">matriks kompleks</a>, <a href="/w/index.php?title=Polinomial_kompleks&action=edit&redlink=1" class="new" title="Polinomial kompleks (halaman belum tersedia)">polinomial kompleks</a>, dan <a href="/w/index.php?title=Aljabar_Lie_kompleks&action=edit&redlink=1" class="new" title="Aljabar Lie kompleks (halaman belum tersedia)">aljabar Lie kompleks</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Definisi">Definisi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=3" title="Sunting bagian: Definisi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=3" title="Sunting kode sumber bagian: Definisi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas: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/230px-Illustration_of_a_complex_number.svg.png" decoding="async" width="230" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Illustration_of_a_complex_number.svg/345px-Illustration_of_a_complex_number.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Illustration_of_a_complex_number.svg/460px-Illustration_of_a_complex_number.svg.png 2x" data-file-width="832" data-file-height="754" /></a><figcaption>Ilustrasi dari bilangan kompleks <span class="texhtml" style="white-space: nowrap;"><i>z</i> = <i>x</i> + <i>iy</i></span> dalam <a href="/w/index.php?title=Medan_kompleks&action=edit&redlink=1" class="new" title="Medan kompleks (halaman belum tersedia)">medan kompleks</a>. Bagian yang sebenarnya adalah <span class="texhtml mvar" style="font-style:italic;">x</span>, dan bagian imajinernya adalah <span class="texhtml mvar" style="font-style:italic;">y</span>.</figcaption></figure> <p>Definisi formal bilangan kompleks adalah sepasang bilangan real (<i>a</i>, <i>b</i>) dengan operasi sebagai berikut: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4831924a9066bf2a688d1ac5fa210269f8bbda01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.437ex; height:2.843ex;" alt="{\displaystyle (a,b)+(c,d)=(a+c,b+d)\,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\cdot (c,d)=(ac-bd,bc+ad).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\cdot (c,d)=(ac-bd,bc+ad).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ab38955157364dfe79efeb4015ef01c8a3c429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.372ex; height:2.843ex;" alt="{\displaystyle (a,b)\cdot (c,d)=(ac-bd,bc+ad).\,}"></span></li></ul> <p>Dengan definisi di atas, bilangan-bilangan kompleks yang ada membentuk suatu himpunan bilangan kompleks yang dinotasikan dengan <b>C</b>. </p><p>Karena bilangan kompleks <i>a</i> + <i>bi</i> merupakan spesifikasi unik yang berdasarkan sepasang bilangan riil (<i>a</i>, <i>b</i>), bilangan kompleks mempunyai hubungan korespondensi satu-satu dengan titik-titik pada satu bidang yang dinamakan <a href="/wiki/Bidang_kompleks" title="Bidang kompleks">bidang kompleks</a>. </p><p>Bilangan riil <i>a</i> dapat disebut juga dengan bilangan kompleks (<i>a</i>, 0), dan dengan cara ini, himpunan bilangan riil <b>R</b> menjadi bagian dari himpunan bilangan kompleks <b>C</b>. </p><p>Dalam <b>C</b>, berlaku sebagai berikut: </p> <ul><li>identitas penjumlahan ("nol"): (0, 0)</li> <li>identitas perkalian ("satu"): (1, 0)</li> <li>invers penjumlahan (<i>a</i>,<i>b</i>): (−<i>a</i>, −<i>b</i>)</li> <li><a href="/wiki/Invers_perkalian" title="Invers perkalian">invers perkalian</a> (reciprocal) bukan nol (<i>a</i>, <i>b</i>): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({a \over a^{2}+b^{2}},{-b \over a^{2}+b^{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <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> <mrow> <mo>−</mo> <mi>b</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> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({a \over a^{2}+b^{2}},{-b \over a^{2}+b^{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cd2bf24565d8bd66d86eb3e8c085d8d6962c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.514ex; height:6.176ex;" alt="{\displaystyle \left({a \over a^{2}+b^{2}},{-b \over a^{2}+b^{2}}\right).}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notasi">Notasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=4" title="Sunting bagian: Notasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=4" title="Sunting kode sumber bagian: Notasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bentuk_Penjumlahan">Bentuk Penjumlahan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=5" title="Sunting bagian: Bentuk Penjumlahan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=5" title="Sunting kode sumber bagian: Bentuk Penjumlahan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bilangan kompleks pada umumnya dinyatakan sebagai penjumlahan dua suku, dengan suku pertama adalah bilangan riil, dan suku kedua adalah bilangan imajiner. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bentuk_Polar">Bentuk Polar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=6" title="Sunting bagian: Bentuk Polar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=6" title="Sunting kode sumber bagian: Bentuk Polar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dengan menganggap bahwa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\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"> <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={\sqrt {a^{2}+b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06427751d7f71ba70ddfae47fb47e6386324ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.647ex; height:3.509ex;" alt="{\displaystyle r={\sqrt {a^{2}+b^{2}}}}"></span></dd></dl> <p>dan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan \left({\frac {b}{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan \left({\frac {b}{a}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c148038e274492ee5aab2c570dd3edbbb56ce192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.142ex; height:6.176ex;" alt="{\displaystyle \theta =\arctan \left({\frac {b}{a}}\right)}"></span></dd></dl> <p>maka </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi=r(\cos \theta +i\sin \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi=r(\cos \theta +i\sin \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124491bd98ebc28952396bb014ff75ed46e89738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.778ex; height:2.843ex;" alt="{\displaystyle a+bi=r(\cos \theta +i\sin \theta )}"></span></dd></dl> <p>Untuk mempersingkat penulisan, bentuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\cos \theta +i\sin \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\cos \theta +i\sin \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840b9a8077588228c19631d22c2472304dd91199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.81ex; height:2.843ex;" alt="{\displaystyle r(\cos \theta +i\sin \theta )}"></span> juga sering ditulis sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\,cis\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\,cis\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac2e424a7563e89ca1ad934683eabcd7a9e2ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle r\,cis\theta }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Bentuk_Eksponen">Bentuk Eksponen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=7" title="Sunting bagian: Bentuk Eksponen" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=7" title="Sunting kode sumber bagian: Bentuk Eksponen"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bentuk lain adalah bentuk eksponen, yaitu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle re^{i\theta }=r(\cos \theta +i\sin \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle re^{i\theta }=r(\cos \theta +i\sin \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83e436496380e3e5e397b10c7bda60174da61a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.611ex; height:3.176ex;" alt="{\displaystyle re^{i\theta }=r(\cos \theta +i\sin \theta )}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Bidang_kompleks">Bidang kompleks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=8" title="Sunting bagian: Bidang kompleks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=8" title="Sunting kode sumber bagian: Bidang kompleks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="float:right; margin-left:3px; margin-right:3px" title="Graphic Representation"> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Berkas:Complex.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c2/Complex.png" decoding="async" width="200" height="300" class="mw-file-element" data-file-width="200" data-file-height="300" /></a></span> </p> </div> <p>Bilangan kompleks dapat divisualisasikan sebagai titik atau vektor posisi pada sistem koordinat dua dimensi yang dinamakan <b><a href="/wiki/Bidang_kompleks" title="Bidang kompleks">bidang kompleks</a></b> atau <b>Diagram Argand</b>. </p><p><a href="/wiki/Sistem_koordinat_Kartesius" class="mw-redirect" title="Sistem koordinat Kartesius">Koordinat Kartesius</a> bilangan kompleks adalah bagian riil <i>x</i> dan bagian imajiner <i>y</i>, sedangkan koordinat sirkulernya adalah <i>r</i> = |<i>z</i>|, yang disebut <a href="/wiki/Modulus" class="mw-redirect" title="Modulus">modulus</a>, dan φ = arg(<i>z</i>), yang disebut juga <i>argumen kompleks</i> dari <i>z</i> (Format ini disebut format mod-arg). Dikombinasikan dengan <a href="/wiki/Rumus_Euler" title="Rumus Euler">Rumus Euler</a>, dapat diperoleh: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy=r(\cos \phi +i\sin \phi )=re^{i\phi }.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy=r(\cos \phi +i\sin \phi )=re^{i\phi }.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d28bec1e80153b61509f81426b9e06bcc5b9807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.857ex; height:3.176ex;" alt="{\displaystyle z=x+iy=r(\cos \phi +i\sin \phi )=re^{i\phi }.\,}"></span></dd></dl> <p>Kadang-kadang, notasi <i>r</i> cis φ dapat juga ditemui. </p><p>Perlu diperhatikan bahwa argumen kompleks adalah unik <a href="/wiki/Modulo" class="mw-redirect" title="Modulo">modulo</a> 2π, jadi, jika terdapat dua nilai argumen kompleks yang berbeda sebanyak kelipatan <a href="/wiki/Bilangan_bulat" title="Bilangan bulat">bilangan bulat</a> dari 2π, kedua argumen kompleks tersebut adalah sama (ekivalen). </p><p>Dengan menggunakan <a href="/wiki/Identitas_trigonometri" class="mw-redirect" title="Identitas trigonometri">identitas trigonometri</a> dasar, dapat diperoleh: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}e^{i\phi _{1}}\cdot r_{2}e^{i\phi _{2}}=r_{1}r_{2}e^{i(\phi _{1}+\phi _{2})}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}e^{i\phi _{1}}\cdot r_{2}e^{i\phi _{2}}=r_{1}r_{2}e^{i(\phi _{1}+\phi _{2})}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f27178b79d8ea373bad05ebef26bba9365d8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.028ex; height:3.176ex;" alt="{\displaystyle r_{1}e^{i\phi _{1}}\cdot r_{2}e^{i\phi _{2}}=r_{1}r_{2}e^{i(\phi _{1}+\phi _{2})}\,}"></span></dd></dl> <p>dan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{1}e^{i\phi _{1}}}{r_{2}e^{i\phi _{2}}}}={\frac {r_{1}}{r_{2}}}e^{i(\phi _{1}-\phi _{2})}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <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> <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> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r_{1}e^{i\phi _{1}}}{r_{2}e^{i\phi _{2}}}}={\frac {r_{1}}{r_{2}}}e^{i(\phi _{1}-\phi _{2})}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11cce3bd3a265787764fa65574d2e463f4fdf06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.768ex; height:6.343ex;" alt="{\displaystyle {\frac {r_{1}e^{i\phi _{1}}}{r_{2}e^{i\phi _{2}}}}={\frac {r_{1}}{r_{2}}}e^{i(\phi _{1}-\phi _{2})}.\,}"></span></dd></dl> <p>Penjumlahan dua bilangan kompleks sama seperti <a href="/wiki/Penjumlahan_vektor" class="mw-redirect" title="Penjumlahan vektor">penjumlahan vektor</a> dari dua vektor, dan perkalian dengan bilangan kompleks dapat divisualisasikan sebagai rotasi dan pemanjangan secara bersamaan. </p><p>Perkalian dengan i adalah rotasi 90 derajat berlawanan dengan arah jarum jam (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}"></span> <a href="/wiki/Radian" title="Radian">radian</a>). Secara geometris, persamaan <i>i</i><sup>2</sup> = −1 adalah dua kali rotasi 90 derajat yang sama dengan rotasi 180 derajat (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> radian). </p> <div class="mw-heading mw-heading2"><h2 id="Konstruksi_formal">Konstruksi formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=9" title="Sunting bagian: Konstruksi formal" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=9" title="Sunting kode sumber bagian: Konstruksi formal"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Konstruksi_sebagai_tatanan">Konstruksi sebagai tatanan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=10" title="Sunting bagian: Konstruksi sebagai tatanan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=10" title="Sunting kode sumber bagian: Konstruksi sebagai tatanan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> memperkenalkan pendekatan untuk mendefinisikan himpunan <span class="texhtml" style="white-space: nowrap;"><b>C</b></span> dari bilangan kompleks<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> sebagai himpunan <span class="texhtml" style="white-space: nowrap;"><b>R</b><sup>2</sup></span> <span class="nowrap"><a href="/w/index.php?title=Pasangan_order&action=edit&redlink=1" class="new" title="Pasangan order (halaman belum tersedia)">pasangan order</a> <span class="texhtml" style="white-space: nowrap;">(<i>a</i>, <i>b</i>)</span></span> dari bilangan real, di mana aturan penjumlahan dan perkalian berikut diterapkan:<sup id="cite_ref-Apostol_1981_2-0" class="reference"><a href="#cite_note-Apostol_1981-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24db4fe63b72661c549790a3b13936cd3c8e665b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.898ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}}}"></span></dd></dl> <p>Kemudian hanya masalah notasi untuk diungkapkan <span class="texhtml" style="white-space: nowrap;">(<i>a</i>, <i>b</i>)</span> sebagai <span class="texhtml" style="white-space: nowrap;"><i>a</i> + <i>bi</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Konstruksi_sebagai_medan_hasil_bagi">Konstruksi sebagai medan hasil bagi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=11" title="Sunting bagian: Konstruksi sebagai medan hasil bagi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=11" title="Sunting kode sumber bagian: Konstruksi sebagai medan hasil bagi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meskipun konstruksi tingkat rendah ini secara akurat mendeskripsikan struktur bilangan kompleks, definisi ekuivalen berikut mengungkapkan sifat aljabar <span class="texhtml" style="white-space: nowrap;"><b>C</b></span> lebih segera. Karakterisasi ini bergantung pada pengertian bidang dan polinomial. Bidang adalah himpunan yang diberkahi dengan operasi penjumlahan, pengurangan, perkalian dan pembagian yang berperilaku seperti yang biasa dari, katakanlah, bilangan rasional. Contohnya, <a href="/wiki/Hukum_distributif" class="mw-redirect" title="Hukum distributif">hukum distributif</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)z=xz+yz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mi>z</mi> <mo>+</mo> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)z=xz+yz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2403554b5d138ef68108772a7bd573477c6d905f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.823ex; height:2.843ex;" alt="{\displaystyle (x+y)z=xz+yz}"></span></dd></dl> <p>harus memegang untuk tiga elemen <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> dan <span class="texhtml mvar" style="font-style:italic;">z</span> dari sebuah lapangan. Himpunan <span class="texhtml" style="white-space: nowrap;"><b>R</b></span> bilangan real memang membentuk bidang. Polinomial <span class="texhtml" style="white-space: nowrap;"><i>p</i>(<i>X</i>)</span> dengan <a href="/wiki/Koefisien" title="Koefisien">koefisien</a> nyata adalah ekspresi dari bentuk </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}"> <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> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3558a6da3a54245fc82ca46cbb60e6f2787fe4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.103ex; height:2.676ex;" alt="{\displaystyle a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}"></span></dd></dl> <p>Dimana <span class="texhtml" style="white-space: nowrap;"><i>a</i><sub>0</sub>, ..., <i>a</i><sub><i>n</i></sub></span> adalah bilangan real. Penambahan dan perkalian polinomial biasa memberikan himpunan <span class="texhtml" style="white-space: nowrap;"><b>R</b>[<i>X</i>]</span> dari semua polinomial dengan struktur <a href="/wiki/Gelanggang_(matematika)" title="Gelanggang (matematika)">gelanggang</a>. Gelanggang ini disebut <a href="/w/index.php?title=Gelanggang_polinomial&action=edit&redlink=1" class="new" title="Gelanggang polinomial (halaman belum tersedia)">gelanggang polinomial</a> di atas bilangan riil. </p><p>Kumpulan bilangan kompleks ditentukan sebagai <a href="/wiki/Gelanggang_hasil_bagi" title="Gelanggang hasil bagi">gelanggang hasil bagi</a> <span class="texhtml" style="white-space: nowrap;"><b>R</b>[<i>X</i>]/(<i>X</i> <sup>2</sup> + 1)</span>.<sup id="cite_ref-Bourbaki_3-0" class="reference"><a href="#cite_note-Bourbaki-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Bidang ekstensi ini berisi dua akar kuadrat dari <span class="texhtml" style="white-space: nowrap;">−1</span>, yaitu (<a href="/wiki/Coset" class="mw-redirect" title="Coset">coset</a> dari) <span class="texhtml" style="white-space: nowrap;"><i>X</i></span> dan <span class="texhtml" style="white-space: nowrap;">−<i>X</i></span>, masing-masing. (Koset dari) <span class="texhtml" style="white-space: nowrap;">1</span> dan <span class="texhtml" style="white-space: nowrap;"><i>X</i></span> membentuk dasar dari <span class="texhtml" style="white-space: nowrap;"><b>R</b>[<i>X</i>]/(<i>X</i> <sup>2</sup> + 1)</span> sebagai <a href="/wiki/Ruang_vektor" title="Ruang vektor">ruang vektor</a> nyata, yang berarti bahwa setiap elemen bidang ekstensi dapat ditulis secara unik sebagai <a href="/w/index.php?title=Kombinasi_linier&action=edit&redlink=1" class="new" title="Kombinasi linier (halaman belum tersedia)">kombinasi linier</a> di kedua elemen ini. Dengan kata lain, elemen bidang ekstensi dapat ditulis sebagai pasangan berurutan <span class="texhtml" style="white-space: nowrap;">(<i>a</i>, <i>b</i>)</span> dari bilangan real. Cincin hasil bagi adalah bidang, karena <span class="texhtml" style="white-space: nowrap;"><i>X</i><sup>2</sup> + 1</span> adalah <a href="/w/index.php?title=Polinomial_tak_tersederhanakan&action=edit&redlink=1" class="new" title="Polinomial tak tersederhanakan (halaman belum tersedia)">tak tersederhanakan</a> berakhir <span class="texhtml" style="white-space: nowrap;"><b>R</b></span>, sehingga ideal yang dihasilkan adalah <a href="/w/index.php?title=Ideal_maksimal&action=edit&redlink=1" class="new" title="Ideal maksimal (halaman belum tersedia)">maksimal</a>. </p><p>Rumus penjumlahan dan perkalian di ring <span class="texhtml" style="white-space: nowrap;"><b>R</b>[<i>X</i>]</span>, modulo the relation <span class="texhtml" style="white-space: nowrap;"><i>X</i><sup>2</sup> = −1</span>, sesuai dengan rumus untuk penjumlahan dan perkalian bilangan kompleks yang didefinisikan sebagai pasangan berurutan. </p> <div class="mw-heading mw-heading3"><h3 id="Wakilan_matriks_dari_bilangan_kompleks">Wakilan matriks dari bilangan kompleks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=12" title="Sunting bagian: Wakilan matriks dari bilangan kompleks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=12" title="Sunting kode sumber bagian: Wakilan matriks dari bilangan kompleks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bilangan kompleks <span class="texhtml" style="white-space: nowrap;"><i>a</i> + <i>bi</i></span> bisa juga diwakili oleh <span class="texhtml" style="white-space: nowrap;">2 × 2</span> <a href="/wiki/Matriks_(matematika)" title="Matriks (matematika)">matriks</a> yang memiliki bentuk sebagai berikut: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874762a513e504f9a2b5854c093217705dc57b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.531ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}"></span></dd></dl> <p>Di sini entri <span class="texhtml mvar" style="font-style:italic;">a</span> dan <span class="texhtml mvar" style="font-style:italic;">b</span> adalah bilangan riil. Jumlah dan produk dari dua matriks tersebut lagi-lagi dalam bentuk ini, dan jumlah dan hasil kali bilangan kompleks sesuai dengan jumlah dan <a href="/wiki/Perkalian_matriks" title="Perkalian matriks">perkalian</a> dari matriks-matriks tersebut, hasil perkaliannya adalah: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}{\begin{pmatrix}c&-d\\d&\;\;c\end{pmatrix}}={\begin{pmatrix}ac-bd&-ad-bc\\bc+ad&\;\;-bd+ac\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mo>−</mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> </mtd> <mtd> <mo>−</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}{\begin{pmatrix}c&-d\\d&\;\;c\end{pmatrix}}={\begin{pmatrix}ac-bd&-ad-bc\\bc+ad&\;\;-bd+ac\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc99ea6a6a9c79091f7891879ae9f97e608daac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.539ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}{\begin{pmatrix}c&-d\\d&\;\;c\end{pmatrix}}={\begin{pmatrix}ac-bd&-ad-bc\\bc+ad&\;\;-bd+ac\end{pmatrix}}}"></span></dd></dl> <p>Deskripsi geometris dari perkalian bilangan kompleks juga dapat diekspresikan dalam <a href="/wiki/Matriks_rotasi" title="Matriks rotasi">matriks rotasi</a> dengan menggunakan korespondensi antara bilangan kompleks dan sejenisnya. Selain itu, kuadrat dari nilai absolut dari bilangan kompleks yang dinyatakan sebagai matriks sama dengan <a href="/wiki/Determinan" title="Determinan">determinan</a> matriks tersebut: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=a^{2}+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=a^{2}+b^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903a73456b7ca6e884a22dbaa963302e2c603040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.859ex; height:6.176ex;" alt="{\displaystyle |z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=a^{2}+b^{2}.}"></span></dd></dl> <p>Konjugasi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}}"></span> sesuai dengan <a href="/wiki/Transpos" title="Transpos">transpos</a> dari matriks. </p><p>Meskipun representasi bilangan kompleks dengan matriks ini adalah yang paling umum, banyak representasi lain yang muncul dari matriks <i> selain </i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7081c52d076c6fb434e4a99c194ce05fe0da899e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.004ex; margin-bottom: -0.334ex; width:6.577ex; height:3.509ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr )}}"></span> kuadrat itu ke negatif dari <a href="/wiki/Matriks_identitas" title="Matriks identitas">matriks identitas</a>. Lihat artikel tentang <a href="/w/index.php?title=2_%C3%97_2_matriks_riil&action=edit&redlink=1" class="new" title="2 × 2 matriks riil (halaman belum tersedia)">2 × 2 matriks riil</a> untuk representasi lain dari bilangan kompleks. </p> <div class="mw-heading mw-heading2"><h2 id="Geometri">Geometri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=13" title="Sunting bagian: Geometri" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=13" title="Sunting kode sumber bagian: Geometri"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bentuk">Bentuk</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=14" title="Sunting bagian: Bentuk" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=14" title="Sunting kode sumber bagian: Bentuk"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tiga poin <a href="/w/index.php?title=Collinearity&action=edit&redlink=1" class="new" title="Collinearity (halaman belum tersedia)">non-collinear</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cabca98f60f9ee828adb0d73276eb90eb2ee56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.189ex; height:2.009ex;" alt="{\displaystyle u,v,w}"></span> di pesawat tentukan <i> 'bentuk' </i> segitiga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v,w\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v,w\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b64a901f197da5658f531c5b4cbf0ec9c425265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.514ex; height:2.843ex;" alt="{\displaystyle \{u,v,w\}}"></span>. Menemukan titik-titik dalam bidang kompleks, bentuk segitiga ini dapat diekspresikan dengan aritmatika kompleks sebagai </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1275fc01560cb752cb3f02f3da8a2087a30cd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.913ex; height:5.176ex;" alt="{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}"></span></dd></dl> <p>Bentuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> sebuah segitiga akan tetap sama, ketika bidang kompleks diubah oleh translasi atau dilasi (dengan <a href="/w/index.php?title=Transformasi_affin&action=edit&redlink=1" class="new" title="Transformasi affin (halaman belum tersedia)">transformasi affin</a>), sesuai dengan pengertian intuitif tentang bentuk, dan mendeskripsikan <a href="/w/index.php?title=Kesamaan_(geometri)&action=edit&redlink=1" class="new" title="Kesamaan (geometri) (halaman belum tersedia)">kesamaan</a>. Demikianlah setiap segitiga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v,w\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v,w\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b64a901f197da5658f531c5b4cbf0ec9c425265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.514ex; height:2.843ex;" alt="{\displaystyle \{u,v,w\}}"></span> berada dalam <a href="/wiki/Bentuk#kelas_kesamaan" title="Bentuk">kelas kesamaan</a> segitiga dengan bentuk yang sama.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometri_fraktal">Geometri fraktal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=15" title="Sunting bagian: Geometri fraktal" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=15" title="Sunting kode sumber bagian: Geometri fraktal"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Mandelset_hires.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/250px-Mandelset_hires.png" decoding="async" width="250" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/375px-Mandelset_hires.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/500px-Mandelset_hires.png 2x" data-file-width="3121" data-file-height="2288" /></a><figcaption>Set Mandelbrot dengan sumbu nyata dan imajiner berlabel.</figcaption></figure> <p><a href="/wiki/Himpunan_Mandelbrot" title="Himpunan Mandelbrot">Himpunan Mandelbrot</a> adalah contoh populer dari fraktal yang terbentuk pada bidang kompleks. Ini didefinisikan dengan memplot setiap lokasi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> tempat melakukan iterasi urutan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}(z)=z^{2}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{c}(z)=z^{2}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191627a3eebdd6608c9b226786defc468b747502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:3.176ex;" alt="{\displaystyle f_{c}(z)=z^{2}+c}"></span> tidak <a href="/w/index.php?title=Menyimpang_(teori_stabilitas)&action=edit&redlink=1" class="new" title="Menyimpang (teori stabilitas) (halaman belum tersedia)">menyimpang</a> ketika <a href="/wiki/Iterasi" title="Iterasi">iterasi</a> tanpa batas. Demikian pula, <a href="/wiki/Himpunan_Julia" title="Himpunan Julia">himpunan Julia</a> memiliki aturan yang sama, kecuali di mana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> tetap konstan. </p> <div class="mw-heading mw-heading3"><h3 id="Segitiga">Segitiga</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=16" title="Sunting bagian: Segitiga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=16" title="Sunting kode sumber bagian: Segitiga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Setiap segitiga memiliki <a href="/w/index.php?title=Steiner_inellipse&action=edit&redlink=1" class="new" title="Steiner inellipse (halaman belum tersedia)">Steiner inellipse</a> unik sebuah <a href="/wiki/Elips" title="Elips">elips</a> di dalam segitiga dan bersinggungan dengan titik tengah ketiga sisi segitiga. <a href="/w/index.php?title=Fokus_(geometri)&action=edit&redlink=1" class="new" title="Fokus (geometri) (halaman belum tersedia)">fokus</a> dari segitiga inellipse Steiner dapat ditemukan sebagai berikut, menurut <a href="/w/index.php?title=Teorema_Marden&action=edit&redlink=1" class="new" title="Teorema Marden (halaman belum tersedia)">teorema Marden</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Nyatakan simpul segitiga pada bidang kompleks sebagai <span class="texhtml" style="white-space: nowrap;"><i>a</i> = <i>x</i><sub><i>A</i></sub> + <i>y</i><sub><i>A</i></sub><i>i</i></span>, <span class="texhtml" style="white-space: nowrap;"><i>b</i> = <i>x</i><sub><i>B</i></sub> + <i>y</i><sub><i>B</i></sub><i>i</i></span>, and <span class="texhtml" style="white-space: nowrap;"><i>c</i> = <i>x</i><sub><i>C</i></sub> + <i>y</i><sub><i>C</i></sub><i>i</i></span>. Tulis <a href="/wiki/Persamaan_kubik" title="Persamaan kubik">persamaan kubik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (x-a)(x-b)(x-c)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (x-a)(x-b)(x-c)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80235fc812f7ac55a184d03a0e117abad5a8656d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.882ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (x-a)(x-b)(x-c)=0}"></span>, ambil turunannya, dan samakan turunan (kuadratik) menjadi nol. <a href="/w/index.php?title=Teorema_Marden&action=edit&redlink=1" class="new" title="Teorema Marden (halaman belum tersedia)">Teorema Marden</a> mengatakan bahwa solusi dari persamaan ini adalah bilangan kompleks yang menunjukkan lokasi dari dua fokus. </p> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=17" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=17" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23035139">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style 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tright"> <ul> <li><span><span class="noviewer" typeof="mw:File"><a href="/wiki/Berkas:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span><a href="/wiki/Portal:Matematika" title="Portal:Matematika">Portal matematika </a></span></li></ul></div> <ul><li><a href="/wiki/Bilangan_asli" title="Bilangan asli">Bilangan asli</a></li> <li><a href="/wiki/Bilangan_bulat" title="Bilangan bulat">Bilangan bulat</a></li> <li><a href="/wiki/Bilangan_cacah" title="Bilangan cacah">Bilangan cacah</a></li> <li><a href="/wiki/Bilangan_imajiner" title="Bilangan imajiner">Bilangan imajiner</a></li> <li><a href="/wiki/Bilangan_riil" title="Bilangan riil">Bilangan riil</a></li> <li><a href="/wiki/Bilangan_rasional" title="Bilangan rasional">Bilangan rasional</a></li> <li><a href="/wiki/Bilangan_irasional" title="Bilangan irasional">Bilangan irasional</a></li> <li><a href="/wiki/Bilangan_prima" title="Bilangan prima">Bilangan prima</a></li> <li><a href="/wiki/Bilangan_komposit" title="Bilangan komposit">Bilangan komposit</a></li> <li><a href="/wiki/Pecahan" title="Pecahan">Pecahan</a></li> <li><a href="/w/index.php?title=Permukaan_aljabar&action=edit&redlink=1" class="new" title="Permukaan aljabar (halaman belum tersedia)">Permukaan aljabar</a></li> <li><a href="/w/index.php?title=Gerakan_melingkar&action=edit&redlink=1" class="new" title="Gerakan melingkar (halaman belum tersedia)">Gerakan melingkar menggunakan bilangan kompleks</a></li> <li><a href="/w/index.php?title=Sistem_basis_kompleks&action=edit&redlink=1" class="new" title="Sistem basis kompleks (halaman belum tersedia)">Sistem basis kompleks</a></li> <li><a href="/wiki/Geometri_kompleks" title="Geometri kompleks">Geometri kompleks</a></li> <li><a href="/w/index.php?title=Bilangan_kompleks_ganda&action=edit&redlink=1" class="new" title="Bilangan kompleks ganda (halaman belum tersedia)">Bilangan kompleks ganda</a></li> <li><a href="/w/index.php?title=Bilangan_bulat_Eisenstein&action=edit&redlink=1" class="new" title="Bilangan bulat Eisenstein (halaman belum tersedia)">Bilangan bulat Eisenstein</a></li> <li><a href="/wiki/Identitas_Euler" title="Identitas Euler">Identitas Euler</a></li> <li><a href="/w/index.php?title=Aljabar_geometri&action=edit&redlink=1" class="new" title="Aljabar geometri (halaman belum tersedia)">Aljabar geometri</a> (yang menyertakan bidang kompleks sebagai subruang 2-dimensi <a href="/w/index.php?title=Spinor&action=edit&redlink=1" class="new" title="Spinor (halaman belum tersedia)">spinor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{2}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{2}^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6023850da07089febe34ebd02728b8c7a3e05cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.906ex; height:3.009ex;" alt="{\displaystyle {\mathcal {G}}_{2}^{+}}"></span>)</li> <li><a href="/wiki/Akar_persatuan" class="mw-redirect" title="Akar persatuan">Akar persatuan</a></li> <li><a href="/w/index.php?title=Bilangan_kompleks_satuan&action=edit&redlink=1" class="new" title="Bilangan kompleks satuan (halaman belum tersedia)">Bilangan kompleks satuan</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Catatan">Catatan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=18" title="Sunting bagian: Catatan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=18" title="Sunting kode sumber bagian: Catatan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18833634">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=19" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=19" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18833634"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation book">Corry, Leo (2015). <a rel="nofollow" class="external text" href="https://archive.org/details/briefhistoryofnu0000corr"><i>A Brief History of Numbers</i></a>. Oxford University Press. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/briefhistoryofnu0000corr/page/215">215</a>–16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brief+History+of+Numbers&rft.pages=215-16&rft.pub=Oxford+University+Press&rft.date=2015&rft.aulast=Corry&rft.aufirst=Leo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbriefhistoryofnu0000corr&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Apostol_1981-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Apostol_1981_2-0">^</a></b></span> <span class="error mw-ext-cite-error" lang="id" dir="ltr">Kesalahan pengutipan: Tag <code><ref></code> tidak sah; tidak ditemukan teks untuk ref bernama <code>Apostol 1981</code></span></li> <li id="cite_note-Bourbaki-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bourbaki_3-0">^</a></b></span> <span class="error mw-ext-cite-error" lang="id" dir="ltr">Kesalahan pengutipan: Tag <code><ref></code> tidak sah; tidak ditemukan teks untuk ref bernama <code>Bourbaki</code></span></li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFLester1994" class="citation">Lester, J.A. (1994), "Triangles I: Shapes", <i><a href="/w/index.php?title=Aequationes_Mathematicae&action=edit&redlink=1" class="new" title="Aequationes Mathematicae (halaman belum tersedia)">Aequationes Mathematicae</a></i>, <b>52</b>: 30–54, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01818325">10.1007/BF01818325</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Aequationes+Mathematicae&rft.atitle=Triangles+I%3A+Shapes&rft.volume=52&rft.pages=30-54&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2FBF01818325&rft.aulast=Lester&rft.aufirst=J.A.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|s2cid=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFKalman2008a" class="citation">Kalman, Dan (2008a), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1">"An Elementary Proof of Marden's Theorem"</a>, <i><a href="/w/index.php?title=American_Mathematical_Monthly&action=edit&redlink=1" class="new" title="American Mathematical Monthly (halaman belum tersedia)">American Mathematical Monthly</a></i>, <b>115</b> (4): 330–38, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2008.11920532">10.1080/00029890.2008.11920532</a>, <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0002-9890">0002-9890</a>, diarsipkan dari <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1">versi asli</a> tanggal 8 March 2012<span class="reference-accessdate">, diakses tanggal <span class="nowrap">1 January</span> 2012</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=An+Elementary+Proof+of+Marden%27s+Theorem&rft.volume=115&rft.issue=4&rft.pages=330-38&rft.date=2008&rft_id=info%3Adoi%2F10.1080%2F00029890.2008.11920532&rft.issn=0002-9890&rft.aulast=Kalman&rft.aufirst=Dan&rft_id=http%3A%2F%2Fmathdl.maa.org%2FmathDL%2F22%2F%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D3338%26pf%3D1&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|s2cid=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>); </span><span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFKalman2008b" class="citation">Kalman, Dan (2008b), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663">"The Most Marvelous Theorem in Mathematics"</a>, <i><a href="/w/index.php?title=Journal_of_Online_Mathematics_and_its_Applications&action=edit&redlink=1" class="new" title="Journal of Online Mathematics and its Applications (halaman belum tersedia)">Journal of Online Mathematics and its Applications</a></i>, diarsipkan dari <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663">versi asli</a> tanggal 8 February 2012<span class="reference-accessdate">, diakses tanggal <span class="nowrap">1 January</span> 2012</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Online+Mathematics+and+its+Applications&rft.atitle=The+Most+Marvelous+Theorem+in+Mathematics&rft.date=2008&rft.aulast=Kalman&rft.aufirst=Dan&rft_id=http%3A%2F%2Fmathdl.maa.org%2FmathDL%2F4%2F%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D1663&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Kutipan">Kutipan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=20" title="Sunting bagian: Kutipan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=20" title="Sunting kode sumber bagian: Kutipan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFAhlfors1979" class="citation"><a href="/w/index.php?title=Lars_Ahlfors&action=edit&redlink=1" class="new" title="Lars Ahlfors (halaman belum tersedia)">Ahlfors, Lars</a> (1979), <i>Complex analysis</i> (edisi ke-3rd), McGraw-Hill, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-07-000657-7" title="Istimewa:Sumber buku/978-0-07-000657-7">978-0-07-000657-7</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1979&rft.isbn=978-0-07-000657-7&rft.aulast=Ahlfors&rft.aufirst=Lars&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFApostol1981" class="citation book"><a href="/w/index.php?title=Tom_Apostol&action=edit&redlink=1" class="new" title="Tom Apostol (halaman belum tersedia)">Apostol, Tom</a> (1981). <i>Mathematical analysis</i>. Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+analysis&rft.pub=Addison-Wesley&rft.date=1981&rft.aulast=Apostol&rft.aufirst=Tom&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFSolomentsev2001" class="citation">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=c/c024140">"Complex number"</a>, dalam <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-1-55608-010-4" title="Istimewa:Sumber buku/978-1-55608-010-4">978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Complex+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Springer+Science%2BBusiness+Media+B.V.+%2F+Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1-55608-010-4&rft.aulast=Solomentsev&rft.aufirst=E.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dc%2Fc024140&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bacaan_lebih_lanjut">Bacaan lebih lanjut</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=21" title="Sunting bagian: Bacaan lebih lanjut" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=21" title="Sunting kode sumber bagian: Bacaan lebih lanjut"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23035139"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782729"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/40px-Wikiversity-logo.svg.png" decoding="async" width="40" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/60px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/80px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></span></span></div> <div class="side-box-text plainlist">Wikiversity memiliki bahan belajar tentang <i><b><a href="https://id.wikiversity.org/wiki/Bilangan_Kompleks" class="extiw" title="v:Bilangan Kompleks">Bilangan Kompleks</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23035139"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782729"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikibooks" class="mw-redirect" title="Wikibooks">Wikibooks</a> memiliki buku di: <div style="margin-left:10px;"><i><a href="https://id.wikibooks.org/wiki/id:Kalkulus_/_Bilangan_Kompleks" class="extiw" title="b:id:Kalkulus / Bilangan Kompleks">Kalkulus / Bilangan Kompleks</a></i></div></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23035139"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782729"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" class="mw-redirect" title="Wikisource">Wikisource</a> memiliki teks artikel the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 <i>Encyclopædia Britannica</i></a> tentang <i><b><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Number/Complex_Numbers" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Number/Complex Numbers">Number/Complex Numbers</a></b></i>.</div></div> </div> <ul><li><cite id="CITEREFPenrose2005" class="citation"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, Roger</a> (2005), <a rel="nofollow" class="external text" href="https://archive.org/details/roadtorealitycom00penr_0"><i>The Road to Reality: A Complete Guide to the Laws of the Universe</i></a>, Alfred A. Knopf, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-679-45443-4" title="Istimewa:Sumber buku/978-0-679-45443-4">978-0-679-45443-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality%3A+A+Complete+Guide+to+the+Laws+of+the+Universe&rft.pub=Alfred+A.+Knopf&rft.date=2005&rft.isbn=978-0-679-45443-4&rft.aulast=Penrose&rft.aufirst=Roger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Froadtorealitycom00penr_0&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFDerbyshire2006" class="citation"><a href="/wiki/John_Derbyshire" title="John Derbyshire">Derbyshire, John</a> (2006), <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780309096577"><i>Unknown Quantity: A Real and Imaginary History of Algebra</i></a>, Joseph Henry Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-309-09657-7" title="Istimewa:Sumber buku/978-0-309-09657-7">978-0-309-09657-7</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unknown+Quantity%3A+A+Real+and+Imaginary+History+of+Algebra&rft.pub=Joseph+Henry+Press&rft.date=2006&rft.isbn=978-0-309-09657-7&rft.aulast=Derbyshire&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780309096577&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFNeedham1997" class="citation">Needham, Tristan (1997), <i>Visual Complex Analysis</i>, Clarendon Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-19-853447-1" title="Istimewa:Sumber buku/978-0-19-853447-1">978-0-19-853447-1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+Complex+Analysis&rft.pub=Clarendon+Press&rft.date=1997&rft.isbn=978-0-19-853447-1&rft.aulast=Needham&rft.aufirst=Tristan&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Matematika">Matematika</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=22" title="Sunting bagian: Matematika" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=22" title="Sunting kode sumber bagian: Matematika"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFAhlfors1979" class="citation"><a href="/w/index.php?title=Lars_Ahlfors&action=edit&redlink=1" class="new" title="Lars Ahlfors (halaman belum tersedia)">Ahlfors, Lars</a> (1979), <i>Complex analysis</i> (edisi ke-3rd), McGraw-Hill, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-07-000657-7" title="Istimewa:Sumber buku/978-0-07-000657-7">978-0-07-000657-7</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1979&rft.isbn=978-0-07-000657-7&rft.aulast=Ahlfors&rft.aufirst=Lars&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFConway1986" class="citation">Conway, John B. (1986), <i>Functions of One Complex Variable I</i>, Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-90328-6" title="Istimewa:Sumber buku/978-0-387-90328-6">978-0-387-90328-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+One+Complex+Variable+I&rft.pub=Springer&rft.date=1986&rft.isbn=978-0-387-90328-6&rft.aulast=Conway&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFJoshi1989" class="citation">Joshi, Kapil D. (1989), <i>Foundations of Discrete Mathematics</i>, New York: <a href="/wiki/John_Wiley_%26_Sons" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-470-21152-6" title="Istimewa:Sumber buku/978-0-470-21152-6">978-0-470-21152-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Discrete+Mathematics&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=978-0-470-21152-6&rft.aulast=Joshi&rft.aufirst=Kapil+D.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFPedoe1988" class="citation"><a href="/w/index.php?title=Daniel_Pedoe&action=edit&redlink=1" class="new" title="Daniel Pedoe (halaman belum tersedia)">Pedoe, Dan</a> (1988), <i>Geometry: A comprehensive course</i>, Dover, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-486-65812-4" title="Istimewa:Sumber buku/978-0-486-65812-4">978-0-486-65812-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+A+comprehensive+course&rft.pub=Dover&rft.date=1988&rft.isbn=978-0-486-65812-4&rft.aulast=Pedoe&rft.aufirst=Dan&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=225">"Section 5.5 Complex Arithmetic"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (edisi ke-3rd), New York: Cambridge University Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-521-88068-8" title="Istimewa:Sumber buku/978-0-521-88068-8">978-0-521-88068-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+5.5+Complex+Arithmetic&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=WH&rft.au=Teukolsky%2C+SA&rft.au=Vetterling%2C+WT&rft.au=Flannery%2C+BP&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%3Fpg%3D225&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFSolomentsev2001" class="citation">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=c/c024140">"Complex number"</a>, dalam <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-1-55608-010-4" title="Istimewa:Sumber buku/978-1-55608-010-4">978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Complex+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Springer+Science%2BBusiness+Media+B.V.+%2F+Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1-55608-010-4&rft.aulast=Solomentsev&rft.aufirst=E.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dc%2Fc024140&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Sejarah">Sejarah</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bilangan_kompleks&veaction=edit&section=23" title="Sunting bagian: Sejarah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bilangan_kompleks&action=edit&section=23" title="Sunting kode sumber bagian: Sejarah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFBourbaki1998" class="citation"><a href="/w/index.php?title=Nicolas_Bourbaki&action=edit&redlink=1" class="new" title="Nicolas Bourbaki (halaman belum tersedia)">Bourbaki, Nicolas</a> (1998), "Foundations of mathematics § logic: set theory", <i>Elements of the history of mathematics</i>, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foundations+of+mathematics+%C2%A7+logic%3A+set+theory&rft.btitle=Elements+of+the+history+of+mathematics&rft.pub=Springer&rft.date=1998&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFBurton1995" class="citation">Burton, David M. (1995), <i>The History of Mathematics</i> (edisi ke-3rd), New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-07-009465-9" title="Istimewa:Sumber buku/978-0-07-009465-9">978-0-07-009465-9</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematics&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1995&rft.isbn=978-0-07-009465-9&rft.aulast=Burton&rft.aufirst=David+M.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFKatz2004" class="citation">Katz, Victor J. (2004), <i>A History of Mathematics, Brief Version</i>, <a href="/w/index.php?title=Addison-Wesley&action=edit&redlink=1" class="new" title="Addison-Wesley (halaman belum tersedia)">Addison-Wesley</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-321-16193-2" title="Istimewa:Sumber buku/978-0-321-16193-2">978-0-321-16193-2</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%2C+Brief+Version&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-321-16193-2&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFNahin1998" class="citation">Nahin, Paul J. (1998), <i>An Imaginary Tale: The Story of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba09297ec8ad80d38116c988c033ae42e0d03ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.469ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\sqrt {-1}}}"></span></i>, Princeton University Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-691-02795-1" title="Istimewa:Sumber buku/978-0-691-02795-1">978-0-691-02795-1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Imaginary+Tale%3A+The+Story+of+MATH+RENDER+ERROR&rft.pub=Princeton+University+Press&rft.date=1998&rft.isbn=978-0-691-02795-1&rft.aulast=Nahin&rft.aufirst=Paul+J.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span> <dl><dd>A gentle introduction to the history of complex numbers and the beginnings of complex analysis.</dd></dl></li> <li><cite id="CITEREFEbbinghausHermesHirzebruchKoecher1991" class="citation">Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991), <i>Numbers</i> (edisi ke-hardcover), Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-97497-2" title="Istimewa:Sumber buku/978-0-387-97497-2">978-0-387-97497-2</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers&rft.edition=hardcover&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-97497-2&rft.aulast=Ebbinghaus&rft.aufirst=H.+D.&rft.au=Hermes%2C+H.&rft.au=Hirzebruch%2C+F.&rft.au=Koecher%2C+M.&rft.au=Mainzer%2C+K.&rft.au=Neukirch%2C+J.&rft.au=Prestel%2C+A.&rft.au=Remmert%2C+R.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABilangan+kompleks" class="Z3988"><span style="display:none;"> </span></span> <dl><dd>An advanced perspective on the historical development of the concept of number.</dd></dl></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23782733">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output 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.navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Sistem_Bilangan" title="Templat:Sistem Bilangan"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Sistem_Bilangan" title="Pembicaraan Templat:Sistem Bilangan"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Sistem_Bilangan&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div><div id="Sistem_bilangan" style="font-size:114%;margin:0 4em">Sistem <a href="/wiki/Bilangan" title="Bilangan">bilangan</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;"><a href="/wiki/Himpunan_terhitung" title="Himpunan terhitung">Himpunan terhitung</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bilangan_asli" title="Bilangan asli">Bilangan asli</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/wiki/Bilangan_bulat" title="Bilangan bulat">Bilangan bulat</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/wiki/Bilangan_rasional" title="Bilangan rasional">Bilangan rasional</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/wiki/Bilangan_aljabar" title="Bilangan aljabar">Bilangan aljabar</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }}}"></span>)</li> <li><a href="/w/index.php?title=Perioda&action=edit&redlink=1" class="new" title="Perioda (halaman belum tersedia)">Perioda</a></li> <li><a href="/w/index.php?title=Bilangan_terkomputasi&action=edit&redlink=1" class="new" title="Bilangan terkomputasi (halaman belum tersedia)">Bilangan terkomputasi</a></li> <li><a href="/w/index.php?title=Bilangan_aritmetis&action=edit&redlink=1" class="new" title="Bilangan aritmetis (halaman belum tersedia)">Bilangan aritmetis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Bilangan riil dan<br />cabangan</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bilangan_riil" title="Bilangan riil">Bilangan riil</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a class="mw-selflink selflink">Bilangan kompleks</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/wiki/Kuaternion" title="Kuaternion">Kuaternion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/wiki/Oktonion" title="Oktonion">Oktonion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/w/index.php?title=Sedenion&action=edit&redlink=1" class="new" title="Sedenion (halaman belum tersedia)">Sedenion</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }"></span>)</li> <li><a href="/w/index.php?title=Aljabar_Cayley%E2%80%93Dickson&action=edit&redlink=1" class="new" title="Aljabar Cayley–Dickson (halaman belum tersedia)">Aljabar Cayley–Dickson</a></li> <li><a href="/w/index.php?title=Bilangan_rangkap&action=edit&redlink=1" class="new" title="Bilangan rangkap (halaman belum tersedia)">Bilangan rangkap</a></li> <li><a href="/w/index.php?title=Bilangan_kompleks_hiperbolik&action=edit&redlink=1" class="new" title="Bilangan kompleks hiperbolik (halaman belum tersedia)">Bilangan kompleks hiperbolik</a></li> <li><a href="/wiki/Bilangan_hiperkompleks" title="Bilangan hiperkompleks">Bilangan hiperkompleks</a></li> <li><a href="/w/index.php?title=Bilangan_superreal&action=edit&redlink=1" class="new" title="Bilangan superreal (halaman belum tersedia)">Bilangan superreal</a></li> <li><a href="/wiki/Bilangan_irasional" title="Bilangan irasional">Bilangan irasional</a></li> <li><a href="/wiki/Bilangan_transenden" title="Bilangan transenden">Bilangan transenden</a></li> <li><a href="/w/index.php?title=Bilangan_hiperreal&action=edit&redlink=1" class="new" title="Bilangan hiperreal (halaman belum tersedia)">Bilangan hiperreal</a></li> <li><a href="/w/index.php?title=Bilangan_surreal&action=edit&redlink=1" class="new" title="Bilangan surreal (halaman belum tersedia)">Bilangan surreal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;">Sistem lain</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bilangan_kardinal" title="Bilangan kardinal">Bilangan kardinal</a></li> <li><a href="/wiki/Bilangan_ordinal" title="Bilangan ordinal">Bilangan ordinal</a></li> <li><a href="/w/index.php?title=Bilangan_p-adik&action=edit&redlink=1" class="new" title="Bilangan p-adik (halaman belum tersedia)">Bilangan p-adik</a></li> <li><a href="/w/index.php?title=Bilangan_supernatural&action=edit&redlink=1" class="new" title="Bilangan supernatural (halaman belum tersedia)">Bilangan supernatural</a></li></ul> </div></td></tr></tbody></table></div> <p><a href="/w/index.php?title=Templat:Bilangan_kompleks&action=edit&redlink=1" class="new" title="Templat:Bilangan kompleks (halaman belum tersedia)">Templat:Bilangan kompleks</a> </p> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25847331"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Pengawasan_otoritas_frameless_&#124;text-top_&#124;10px_&#124;alt=Sunting_ini_di_Wikidata_&#124;link=https&#58;//www.wikidata.org/wiki/Q11567#identifiers&#124;Sunting_ini_di_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pengawasan_otoritas_frameless_&#124;text-top_&#124;10px_&#124;alt=Sunting_ini_di_Wikidata_&#124;link=https&#58;//www.wikidata.org/wiki/Q11567#identifiers&#124;Sunting_ini_di_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Bantuan:Pengawasan_otoritas" title="Bantuan:Pengawasan otoritas">Pengawasan otoritas</a> <span class="mw-valign-text-top" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11567#identifiers" title="Sunting ini di Wikidata"><img alt="Sunting ini di Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Umum</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4128698-4">Integrated Authority File (Jerman)</a></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Perpustakaan nasional</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11981946j">Prancis</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11981946j">(data)</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85093211">Amerika Serikat</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000082623&P_CON_LNG=ENG">Latvia</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00563643">Jepang</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph121761&CON_LNG=ENG">Republik Ceko</a></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lain-lain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://academic.microsoft.com/v2/detail/45862120">Microsoft Academic</a></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Diperoleh dari "<a dir="ltr" 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