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class="mw-page-title-main">Operation (mathematics)</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="Go to an article in another language. Available in 54 languages" > <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-54" 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">54 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="عملية (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="عملية (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fm%C9%99liyyat_(riyaziyyat)" title="Əməliyyat (riyaziyyat) – Azerbaijani" lang="az" hreflang="az" data-title="Əməliyyat (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE" title="গাণিতিক প্রক্রিয়া – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক প্রক্রিয়া" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Математическа операция – Bulgarian" lang="bg" hreflang="bg" data-title="Математическа операция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Operacija_(matematika)" title="Operacija (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Operacija (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Operaci%C3%B3_matem%C3%A0tica" title="Operació matemàtica – Catalan" lang="ca" hreflang="ca" data-title="Operació matemàtica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Операци (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Операци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Operace_(matematika)" title="Operace (matematika) – Czech" lang="cs" hreflang="cs" data-title="Operace (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Verkn%C3%BCpfung_(Mathematik)" title="Verknüpfung (Mathematik) – German" lang="de" hreflang="de" data-title="Verknüpfung (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tehe" title="Tehe – Estonian" lang="et" hreflang="et" data-title="Tehe" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%AC%CE%BE%CE%B7_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Πράξη (μαθηματικά) – Greek" lang="el" hreflang="el" data-title="Πράξη (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Operaci%C3%B3n_(matem%C3%A1tica)" title="Operación (matemática) – Spanish" lang="es" hreflang="es" data-title="Operación (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Operacio_(matematiko)" title="Operacio (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Operacio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eragiketa_(matematika)" title="Eragiketa (matematika) – Basque" lang="eu" hreflang="eu" data-title="Eragiketa (matematika)" 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%D9%85%D9%84_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="عمل (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="عمل (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Op%C3%A9ration_(math%C3%A9matiques)" title="Opération (mathématiques) – French" lang="fr" hreflang="fr" data-title="Opération (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Operaci%C3%B3n_(matem%C3%A1ticas)" title="Operación (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Operación (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B0%EC%82%B0_(%EC%88%98%ED%95%99)" title="연산 (수학) – Korean" lang="ko" hreflang="ko" data-title="연산 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%BE%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A3%D5%B8%D6%80%D5%AE%D5%B8%D5%B2%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80" title="Հանրահաշվական գործողություններ – Armenian" lang="hy" hreflang="hy" data-title="Հանրահաշվական գործողություններ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Operasi_(matematika)" title="Operasi (matematika) – Indonesian" lang="id" hreflang="id" data-title="Operasi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operazione_aritmetica" title="Operazione aritmetica – Italian" lang="it" hreflang="it" data-title="Operazione aritmetica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="פעולה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="פעולה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B2%B0%E0%B3%8D%E0%B2%AE" title="ಪರಿಕರ್ಮ – Kannada" lang="kn" hreflang="kn" data-title="ಪರಿಕರ್ಮ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Tendo_(hisabati)" title="Tendo (hisabati) – Swahili" lang="sw" hreflang="sw" data-title="Tendo (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C5%B1velet" title="Művelet – Hungarian" lang="hu" hreflang="hu" data-title="Művelet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Операција (математика) – Macedonian" lang="mk" hreflang="mk" data-title="Операција (математика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Veisele_(fika)" title="Veisele (fika) – Fijian" lang="fj" hreflang="fj" data-title="Veisele (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Operatie_(wiskunde)" title="Operatie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Operatie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%BC%94%E7%AE%97_(%E6%95%B0%E5%AD%A6)" title="演算 (数学) – Japanese" lang="ja" hreflang="ja" data-title="演算 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Operasjon_(matematikk)" title="Operasjon (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Operasjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_operasjon" title="Matematisk operasjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk operasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Qooyyaba" title="Qooyyaba – Oromo" lang="om" hreflang="om" data-title="Qooyyaba" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Operacja_n-arna" title="Operacja n-arna – Polish" lang="pl" hreflang="pl" data-title="Operacja n-arna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Opera%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Operação (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Operação (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Opera%C8%9Bie_(matematic%C4%83)" title="Operație (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Operație (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Операция (математика) – Russian" lang="ru" hreflang="ru" data-title="Операция (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Veprimi_(matematik%C3%AB)" title="Veprimi (matematikë) – Albanian" lang="sq" hreflang="sq" data-title="Veprimi (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Upirazzioni_(matim%C3%A0tica)" title="Upirazzioni (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Upirazzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Operation_(mathematics)" title="Operation (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Operation (mathematics)" 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/Matematick%C3%A1_oper%C3%A1cia" title="Matematická operácia – Slovak" lang="sk" hreflang="sk" data-title="Matematická operácia" 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/Matemati%C4%8Dna_operacija" title="Matematična operacija – Slovenian" lang="sl" hreflang="sl" data-title="Matematična operacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Операција (математика) – Serbian" lang="sr" hreflang="sr" data-title="Операција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Operacija_(matematika)" title="Operacija (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Operacija (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Operation_(matematik)" title="Operation (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Operation (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Operasyon_(matematika)" title="Operasyon (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Operasyon (matematika)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%86%E0%AE%AF%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="செயல் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="செயல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%94%E0%B8%B3%E0%B9%80%E0%B8%99%E0%B8%B4%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="การดำเนินการ (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="การดำเนินการ (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0%C5%9Flem" title="İşlem – Turkish" lang="tr" hreflang="tr" data-title="İşlem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Операція (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Операція (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_to%C3%A1n" title="Phép toán – Vietnamese" lang="vi" hreflang="vi" data-title="Phép toán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A4%D7%A2%D7%A8%D7%90%D7%A6%D7%99%D7%A2_(%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7)" title="אפעראציע (מאטעמאטיק) – Yiddish" lang="yi" hreflang="yi" data-title="אפעראציע (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E9%81%8B%E7%AE%97" title="數學運算 – Cantonese" lang="yue" hreflang="yue" data-title="數學運算" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BF%90%E7%AE%97" title="运算 – Chinese" lang="zh" hreflang="zh" data-title="运算" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%8E%E2%B5%80%E2%B5%8D%E2%B5%9C_(%E2%B5%9C%E2%B5%93%E2%B5%99%E2%B5%8F%E2%B4%B0%E2%B4%BD%E2%B5%9C)" title="ⵜⴰⵎⵀⵍⵜ (ⵜⵓⵙⵏⴰⴽⵜ) – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵎⵀⵍⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</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/Q3884033#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> 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role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">Operator (mathematics)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Addition, multiplication, division, ...</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arithmetic_operations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/220px-Arithmetic_operations.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/330px-Arithmetic_operations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Arithmetic_operations.svg/440px-Arithmetic_operations.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption><a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">Elementary arithmetic</a> operations:<style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="margin-left:1.6em;"><ul><li>+, plus (addition)</li><li>−, minus (subtraction)</li><li>÷, obelus (division)</li><li>×, times (multiplication)</li></ul></div></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>operation</b> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> to itself. For example, an operation on <a href="/wiki/Real_number" title="Real number">real numbers</a> will take in real numbers and return a real number. An operation can take zero or more input values (also called "<i><a href="/wiki/Operand" title="Operand">operands</a></i>" or "arguments") to a well-defined output value. The number of operands is the <a href="/wiki/Arity" title="Arity">arity</a> of the operation. </p><p>The most commonly studied operations are <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> (i.e., operations of arity 2), such as <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Unary_operation" title="Unary operation">unary operations</a> (i.e., operations of arity 1), such as <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> and <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>. An operation of <a href="/wiki/Arity" title="Arity">arity</a> zero, or <a href="/wiki/Nullary_operation" class="mw-redirect" title="Nullary operation">nullary operation</a>, is a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Mixed_product" class="mw-redirect" title="Mixed product">mixed product</a> is an example of an operation of arity 3, also called <a href="/wiki/Ternary_operation" title="Ternary operation">ternary operation</a>. </p><p>Generally, the arity is taken to be finite. However, <a href="/wiki/Infinitary_operation" class="mw-redirect" title="Infinitary operation">infinitary operations</a> are sometimes considered,<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in which case the "usual" operations of finite arity are called <b>finitary operations</b>. </p><p>A <b>partial operation</b> is defined similarly to an operation, but with a <a href="/wiki/Partial_function" title="Partial function">partial function</a> in place of a function. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types_of_operation">Types of operation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operation_(mathematics)&action=edit&section=1" title="Edit section: Types of operation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_operations_as_black_box.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/220px-Binary_operations_as_black_box.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/330px-Binary_operations_as_black_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/440px-Binary_operations_as_black_box.svg.png 2x" data-file-width="142" data-file-height="142" /></a><figcaption>A binary operation takes two arguments <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, and returns the result <span class="mwe-math-element"><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\circ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ba86902ee98c41deb1275ddb8693977f27e1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.68ex; height:2.009ex;" alt="{\displaystyle x\circ y}"></span>.</figcaption></figure> <p>There are two common types of operations: <a href="/wiki/Unary_operation" title="Unary operation">unary</a> and <a href="/wiki/Binary_operation" title="Binary operation">binary</a>. Unary operations involve only one value, such as <a href="/wiki/Negation" title="Negation">negation</a> and <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Binary operations, on the other hand, take two values, and include <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>.<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><p>Operations can involve mathematical objects other than numbers. The <a href="/wiki/Truth_value" title="Truth value">logical values</a> <i>true</i> and <i>false</i> can be combined using <a href="/wiki/Logic_operation" class="mw-redirect" title="Logic operation">logic operations</a>, such as <i>and</i>, <i>or,</i> and <i>not</i>. <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">Vectors</a> can be added and subtracted.<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> <a href="/wiki/Rotation" title="Rotation">Rotations</a> can be combined using the <a href="/wiki/Function_composition" title="Function composition">function composition</a> operation, performing the first rotation and then the second. Operations on <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> include the binary operations <i><a href="/wiki/Union_(mathematics)" class="mw-redirect" title="Union (mathematics)">union</a></i> and <i><a href="/wiki/Intersection_(mathematics)" class="mw-redirect" title="Intersection (mathematics)">intersection</a></i> and the unary operation of <i><a href="/wiki/Complementation_(mathematics)" class="mw-redirect" title="Complementation (mathematics)">complementation</a></i>.<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><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Operations on <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> include <a href="/wiki/Function_composition" title="Function composition">composition</a> and <a href="/wiki/Convolution" title="Convolution">convolution</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Operations may not be defined for every possible value of its <i><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></i>. For example, in the real numbers one cannot divide by zero<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> or take square roots of negative numbers. The values for which an operation is defined form a set called its <i>domain of definition</i> or <i>active domain</i>. The set which contains the values produced is called the <i><a href="/wiki/Codomain" title="Codomain">codomain</a></i>, but the set of actual values attained by the operation is its codomain of definition, active codomain, <i><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></i> or <i><a href="/wiki/Range_of_a_function" title="Range of a function">range</a></i>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers. </p><p>Operations can involve dissimilar objects: a vector can be multiplied by a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> to form another vector (an operation known as <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>),<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> operation on two vectors produces a quantity that is scalar.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> An operation may or may not have certain properties, for example it may be <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, <a href="/wiki/Anticommutative" class="mw-redirect" title="Anticommutative">anticommutative</a>, <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>, and so on. </p><p>The values combined are called <i>operands</i>, <i>arguments</i>, or <i>inputs</i>, and the value produced is called the <i>value</i>, <i>result</i>, or <i>output</i>. Operations can have fewer or more than two inputs (including the case of zero input and infinitely many inputs<sup id="cite_ref-:1_1-2" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>). </p><p>An <b>operator</b> is similar to an operation in that it refers to the symbol or the process used to denote the operation, hence their point of view is different. For instance, one often speaks of "the operation of addition" or "the addition operation", when focusing on the operands and result, but one switches to "addition operator" (rarely "operator of addition"), when focusing on the process, or from the more symbolic viewpoint, the function <span class="nowrap">+: <i>X</i> × <i>X</i> → <i>X</i></span> (where X is a set such as the set of real numbers). </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operation_(mathematics)&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b><i>n</i>-ary operation</b> <i>ω</i> on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>X</i> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="nowrap"><i>ω</i>: <i>X</i><sup>n</sup> → <i>X</i></span>. The set <span class="nowrap"><i>X</i><sup>n</sup></span> is called the <i>domain</i> of the operation, the output set is called the <i><a href="/wiki/Codomain" title="Codomain">codomain</a></i> of the operation, and the fixed non-negative integer <i>n</i> (the number of operands) is called the <i><a href="/wiki/Arity" title="Arity">arity</a></i> of the operation. Thus a <a href="/wiki/Unary_operation" title="Unary operation">unary operation</a> has arity one, and a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> has arity two. An operation of arity zero, called a <i>nullary</i> operation, is simply an element of the codomain <i>Y</i>. An <i>n</i>-ary operation can also be viewed as an <span class="nowrap">(<i>n</i> + 1)</span>-ary <a href="/wiki/Finitary_relation" title="Finitary relation">relation</a> that is <a href="/wiki/Finitary_relation#Definitions" title="Finitary relation">total</a> on its <i>n</i> input domains and <a href="/wiki/Finitary_relation#Definitions" title="Finitary relation">unique</a> on its output domain. </p><p>An <b><i>n</i>-ary partial operation</b> <i>ω</i> from <span class="nowrap"><i>X</i><sup>n</sup> to <i>X</i></span> is a <a href="/wiki/Partial_function" title="Partial function">partial function</a> <span class="nowrap"><i>ω</i>: <i>X</i><sup>n</sup> → <i>X</i></span>. An <i>n</i>-ary partial operation can also be viewed as an <span class="nowrap">(<i>n</i> + 1)</span>-ary relation that is unique on its output domain. </p><p>The above describes what is usually called a <b>finitary operation</b>, referring to the finite number of operands (the value <i>n</i>). There are obvious extensions where the arity is taken to be an infinite <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> or <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal</a>,<sup id="cite_ref-:1_1-3" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or even an arbitrary set indexing the operands. </p><p>Often, the use of the term <i>operation</i> implies that the domain of the function includes a power of the codomain (i.e. the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of one or more copies of the codomain),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> although this is by no means universal, as in the case of <a href="/wiki/Dot_product" title="Dot product">dot product</a>, where vectors are multiplied and result in a scalar. An <i>n</i>-ary operation <span class="nowrap"><i>ω</i>: <i>X</i><sup><i>n</i></sup> → <i>X</i></span> is called an <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="internal_operation"></span><span class="vanchor-text">internal operation</span></span></b>. An <i>n</i>-ary operation <span class="nowrap"><i>ω</i>: <i>X</i><sup><i>i</i></sup> × <i>S</i> × <i>X</i><sup><i>n</i> − <i>i</i> − 1</sup> → <i>X</i></span> where <span class="nowrap">0 ≤ <i>i</i> < <i>n</i></span> is called an <b>external operation</b> by the <i>scalar set</i> or <i>operator set</i> <i>S</i>. In particular for a binary operation, <span class="nowrap"><i>ω</i>: <i>S</i> × <i>X</i> → <i>X</i></span> is called a <b>left-external operation</b> by <i>S</i>, and <span class="nowrap"><i>ω</i>: <i>X</i> × <i>S</i> → <i>X</i></span> is called a <b>right-external operation</b> by <i>S</i>. An example of an internal operation is <a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a>, where two vectors are added and result in a vector. An example of an external operation is <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>, where a vector is multiplied by a scalar and result in a vector. </p><p>An <b><i>n</i>-ary multifunction</b> or <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="multioperation"></span><span class="vanchor-text">multioperation</span></span></b> <i>ω</i> is a mapping from a Cartesian power of a set into the set of subsets of that set, formally <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega :X^{n}\rightarrow {\mathcal {P}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega :X^{n}\rightarrow {\mathcal {P}}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e05768579f0e50c4bf6fa313b6c4afb61c8f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.705ex; height:2.843ex;" alt="{\displaystyle \omega :X^{n}\rightarrow {\mathcal {P}}(X)}"></span>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operation_(mathematics)&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Finitary_relation" title="Finitary relation">Finitary relation</a></li> <li><a href="/wiki/Hyperoperation" title="Hyperoperation">Hyperoperation</a></li> <li><a href="/wiki/Infix_notation" title="Infix notation">Infix notation</a></li> <li><a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">Operator (mathematics)</a></li> <li><a href="/wiki/Order_of_operations" title="Order of operations">Order of operations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operation_(mathematics)&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Algebraic_operation">"Algebraic operation - Encyclopedia of Mathematics"</a>. <i>www.encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.encyclopediaofmath.org&rft.atitle=Algebraic+operation+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FAlgebraic_operation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOperation+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeMeo2010" class="citation web cs1">DeMeo, William (August 26, 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210519000135/http://www.math.hawaii.edu/~williamdemeo/latticetheory/Glossary.pdf">"Universal Algebra Notes"</a> <span class="cs1-format">(PDF)</span>. <i>math.hawaii.edu</i>. 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