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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main 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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quotient</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 44 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-44" 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">44 languages</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/Kwosi%C3%ABnt" title="Kwosiënt – Afrikaans" lang="af" hreflang="af" data-title="Kwosiënt" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%A7%D8%AA%D8%AC_%D9%82%D8%B3%D9%85%D8%A9" title="ناتج قسمة – Arabic" lang="ar" hreflang="ar" data-title="ناتج قسمة" 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/Qism%C9%99t_(riyaziyyat)" title="Qismət (riyaziyyat) – Azerbaijani" lang="az" hreflang="az" data-title="Qismət (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%AD%E0%A6%BE%E0%A6%97%E0%A6%AB%E0%A6%B2" title="ভাগফল – Bangla" lang="bn" hreflang="bn" data-title="ভাগফল" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B7%D0%B5%D0%BB%D1%8C" title="Дзель – Belarusian" lang="be" hreflang="be" data-title="Дзель" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%94%D0%B7%D0%B5%D0%BB%D1%8C" title="Дзель – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дзель" 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 badge-Q70893996 mw-list-item" title=""><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%BD%D0%BE" title="Частно – Bulgarian" lang="bg" hreflang="bg" data-title="Частно" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B9%D0%BB%D0%B0%D0%BD%C4%83%D1%88" title="Пайланăш – Chuvash" lang="cv" hreflang="cv" data-title="Пайланăш" 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/Cyniferydd" title="Cyniferydd – Welsh" lang="cy" hreflang="cy" data-title="Cyniferydd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quotient" title="Quotient – German" lang="de" hreflang="de" data-title="Quotient" 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/Jagatis" title="Jagatis – Estonian" lang="et" hreflang="et" data-title="Jagatis" 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%CE%B7%CE%BB%CE%AF%CE%BA%CE%BF" title="Πηλίκο – Greek" lang="el" hreflang="el" data-title="Πηλίκο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvociento" title="Kvociento – Esperanto" lang="eo" hreflang="eo" data-title="Kvociento" 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/Zatidura_(matematika)" title="Zatidura (matematika) – Basque" lang="eu" hreflang="eu" data-title="Zatidura (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%AE%D8%A7%D8%B1%D8%AC_%D9%82%D8%B3%D9%85%D8%AA" title="خارج قسمت – Persian" lang="fa" hreflang="fa" data-title="خارج قسمت" 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/Quotient" title="Quotient – French" lang="fr" hreflang="fr" data-title="Quotient" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%AB" title="몫 – Korean" lang="ko" hreflang="ko" data-title="몫" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%97%E0%A4%AB%E0%A4%B2" title="भागफल – Hindi" lang="hi" hreflang="hi" data-title="भागफल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Quociento" title="Quociento – Ido" lang="io" hreflang="io" data-title="Quociento" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hasil_bagi" title="Hasil bagi – Indonesian" lang="id" hreflang="id" data-title="Hasil bagi" 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-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Quotient" title="I-Quotient – Xhosa" lang="xh" hreflang="xh" data-title="I-Quotient" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kv%C3%B3ti_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Kvóti (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Kvóti (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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/Quoziente" title="Quoziente – Italian" lang="it" hreflang="it" data-title="Quoziente" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Hisa_(hisabati)" title="Hisa (hisabati) – Swahili" lang="sw" hreflang="sw" data-title="Hisa (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dalmuo" title="Dalmuo – Lithuanian" lang="lt" hreflang="lt" data-title="Dalmuo" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B8%D0%BA" title="Количник – Macedonian" lang="mk" hreflang="mk" data-title="Количник" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Quoti%C3%ABnt" title="Quotiënt – Dutch" lang="nl" hreflang="nl" data-title="Quotiënt" 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/%E5%95%86_(%E6%95%B0%E5%AD%A6)" title="商 (数学) – Japanese" lang="ja" hreflang="ja" data-title="商 (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/Kvotient" title="Kvotient – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvotient" 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/Kvotient" title="Kvotient – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvotient" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%8B%E0%A8%B8%E0%A8%BC%E0%A9%B0%E0%A8%9F" title="ਕੋਸ਼ੰਟ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੋਸ਼ੰਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/C%C3%A2t" title="Cât – Romanian" lang="ro" hreflang="ro" data-title="Cât" 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%A7%D0%B0%D1%81%D1%82%D0%BD%D0%BE%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Частное (математика) – Russian" lang="ru" hreflang="ru" data-title="Частное (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/Her%C3%ABsi" title="Herësi – Albanian" lang="sq" hreflang="sq" data-title="Herësi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quotient" title="Quotient – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quotient" 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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%95%D8%B1%DA%A9%DB%95%D9%88%D8%AA" title="بەرکەوت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بەرکەوت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B8%D0%BA" title="Количник – Serbian" lang="sr" hreflang="sr" data-title="Количник" 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/Koli%C4%8Dnik" title="Količnik – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Količnik" 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/Kvot" title="Kvot – Swedish" lang="sv" hreflang="sv" data-title="Kvot" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B5%E0%AF%81" title="ஈவு – Tamil" lang="ta" hreflang="ta" data-title="ஈவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/B%C3%B6l%C3%BCm" title="Bölüm – Turkish" lang="tr" hreflang="tr" data-title="Bölüm" 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%A7%D0%B0%D1%81%D1%82%D0%BA%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Частка (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Частка (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%95%86%E6%95%B8" title="商數 – Cantonese" lang="yue" hreflang="yue" data-title="商數" 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/%E5%95%86%E6%95%B8" title="商數 – Chinese" lang="zh" hreflang="zh" data-title="商數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q41118#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Quotient_(disambiguation)" class="mw-disambig" title="Quotient (disambiguation)">Quotient (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divide12by3.svg" class="mw-file-description"><img alt="12 apples divided into 4 groups of 3 each." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Divide12by3.svg/220px-Divide12by3.svg.png" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Divide12by3.svg/330px-Divide12by3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Divide12by3.svg/440px-Divide12by3.svg.png 2x" data-file-width="750" data-file-height="625" /></a><figcaption>The quotient of 12 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.navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-above" style="background:#efefef;"> <span style="font-size:130%;"><a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">Arithmetic operations</a></span><div class="navbar plainlinks hlist navbar-mini" style="float:right"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr><tr><td class="sidebar-content" style="font-size:130%;"> <table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="/wiki/Addition" title="Addition">Addition</a> (+)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>augend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea99a27b5a763ef48889c450ac8157083ea97118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.217ex; height:9.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 {\text{sum}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sum</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{sum}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8609baca9fdbc4c529f5894884a08122d695dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\text{sum}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>minuend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>subtrahend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2780b756445a5f8f95b16c33e3b924f976958ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 {\text{difference}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>difference</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{difference}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac22c4e24eef2036cff5bfea924cc0dbb30c5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.857ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{difference}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplier</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplicand</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f7b476e32221c7b05d356289c8085aef54059b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.176ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a href="/wiki/Product_(mathematics)" title="Product (mathematics)"><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 {\text{product}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>product</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{product}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c8b7509b8be1043622cb7b1b9a36ca8bfc2616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.578ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\text{product}}}"></span></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.83em 0.4em" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>dividend</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>divisor</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>numerator</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>denominator</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5d22ff59234f0d437be740306e8dd905991e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.15ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a class="mw-selflink selflink"><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 \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>fraction</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>quotient</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ratio</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2359c3ca6e50e7ae8065baa710440b3c79895023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.197ex; height:7.176ex;" alt="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"></span></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> (^)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>exponent</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb107371002b62a60fcbd13e742f4d81f872b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.618ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 {\text{power}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{power}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d0a9fbffb659c0055d5ee6fde3f7f28e96f45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.297ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {\text{power}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Nth_root" title="Nth root"><i>n</i>th root</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>radicand</mtext> </mrow> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <mtext>degree</mtext> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5582d567e7e7fbcdb728291770905e09beb0ea18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.422ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 {\text{root}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>root</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{root}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a015c1122190da3f1f1732d88b8bb03a8d7eb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{root}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Logarithm" title="Logarithm">Logarithm</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>anti-logarithm</mtext> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2435266fcae4aa91d3d70a74bb91b5b35ef52edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.454ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 {\text{logarithm}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logarithm</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{logarithm}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5d50baa86b950ff6d15760b7a38df1f8d8c868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{logarithm}}}"></span></td></tr></tbody></table></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, a <b>quotient</b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la">quotiens</i> 'how many times', pronounced <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'k' in 'kind'">k</span><span title="'w' in 'wind'">w</span><span title="/oʊ/: 'o' in 'code'">oʊ</span><span title="/ʃ/: 'sh' in 'shy'">ʃ</span><span title="/ən/: 'on' in 'button'">ən</span><span title="'t' in 'tie'">t</span></span>/</a></span></span>) is a quantity produced by the <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> of two numbers.<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> The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>)<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> or a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> or <a href="/wiki/Ratio" title="Ratio">ratio</a> (in the case of a general <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>). For example, when dividing 20 (the <i>dividend</i>) by 3 (the <i>divisor</i>), the <i>quotient</i> is 6 (with a remainder of 2) in the first sense 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 6{\tfrac {2}{3}}=6.66...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>6.66...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6{\tfrac {2}{3}}=6.66...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d4205b8965368b276f923211d42514a5aaa05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.994ex; height:3.676ex;" alt="{\displaystyle 6{\tfrac {2}{3}}=6.66...}"></span> (a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a>) in the second sense. </p><p><span class="anchor" id="Metrology"></span>In <a href="/wiki/Metrology" title="Metrology">metrology</a> (<a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a> and the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a>), "quotient" refers to the general case with respect to the <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> of <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a>.<sup id="cite_ref-ISO_80000-1_3-0" class="reference"><a href="#cite_note-ISO_80000-1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> <sup id="cite_ref-International_Electrotechnical_Vocabulary_e707_5-0" class="reference"><a href="#cite_note-International_Electrotechnical_Vocabulary_e707-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <i><a href="/wiki/Ratio" title="Ratio">Ratios</a></i> is the special case for <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> quotients of two quantities of the same <a href="/wiki/Kind_of_quantity" class="mw-redirect" title="Kind of quantity">kind</a>.<sup id="cite_ref-ISO_80000-1_3-1" class="reference"><a href="#cite_note-ISO_80000-1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-International_Electrotechnical_Vocabulary_g891_6-0" class="reference"><a href="#cite_note-International_Electrotechnical_Vocabulary_g891-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Quotients with a non-trivial <a href="/wiki/Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimension</a> and <a href="/wiki/Compound_unit" class="mw-redirect" title="Compound unit">compound units</a>, especially when the divisor is a duration (e.g., "<a href="/wiki/Per_second" class="mw-redirect" title="Per second">per second</a>"), are known as <a href="/wiki/Rate_(mathematics)" title="Rate (mathematics)"><i>rates</i></a>.<sup id="cite_ref-International_Electrotechnical_Vocabulary_x558_7-0" class="reference"><a href="#cite_note-International_Electrotechnical_Vocabulary_x558-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Density" title="Density">density</a> (mass divided by volume, in units of <a href="/wiki/Kilogram_per_cubic_metre" title="Kilogram per cubic metre">kg/m<sup>3</sup></a>) is said to be a "quotient", whereas <a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">mass fraction</a> (mass divided by mass, in kg/kg or in percent) is a "ratio".<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> <i><a href="/wiki/Specific_quantity" title="Specific quantity">Specific quantities</a></i> are <a href="/wiki/Intensive_quantity" class="mw-redirect" title="Intensive quantity">intensive quantities</a> resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".<sup id="cite_ref-ISO_80000-1_3-2" class="reference"><a href="#cite_note-ISO_80000-1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quotient&action=edit&section=1" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Division_(mathematics)#Notation" title="Division (mathematics)">Division (mathematics) § Notation</a></div> <p>The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="1em" /> <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 /> <mtd> <mi></mi> <mo stretchy="false">←</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>dividend or numerator</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">←</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>divisor or denominator</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">}</mo> </mrow> </mrow> <mo stretchy="false">←</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>quotient</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08896d32d4e0157d7030f0b483be72f0b75b12a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.634ex; height:7.509ex;" alt="{\displaystyle {\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Integer_part_definition">Integer part definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quotient&action=edit&section=2" title="Edit section: Integer part definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quotient is also less commonly defined as the greatest <a href="/wiki/Natural_number" title="Natural number">whole number</a> of times a divisor may be subtracted from a dividend—before making the <a href="/wiki/Remainder" title="Remainder">remainder</a> negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative: </p> <dl><dd>20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,</dd></dl> <p>while </p> <dl><dd>20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.</dd></dl> <p>In this sense, a quotient is the <a href="/wiki/Integer_part" class="mw-redirect" title="Integer part">integer part</a> of the ratio of two numbers.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Quotient_of_two_integers">Quotient of two integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quotient&action=edit&section=3" title="Edit section: Quotient of two integers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rational_number" title="Rational number">Rational number</a></div> <p>A <a href="/wiki/Rational_number" title="Rational number">rational number</a> can be defined as the quotient of two <a href="/wiki/Integer" title="Integer">integers</a> (as long as the denominator is non-zero). </p><p>A more detailed definition goes as follows:<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> <dl><dd>A real number <i>r</i> is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.</dd></dl> <p>Or more formally: </p> <dl><dd>Given a real number <i>r</i>, <i>r</i> is rational if and only if there exists integers <i>a</i> and <i>b</i> such that <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={\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\tfrac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d0da0f704c247f15b6ca1ba70b8472eecf300d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.853ex; height:3.343ex;" alt="{\displaystyle r={\tfrac {a}{b}}}"></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 b\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad073253b4c817f2ec7e3dd7517b7f89a8e581dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 0}"></span>.</dd></dl> <p>The existence of <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.<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> </p> <div class="mw-heading mw-heading2"><h2 id="More_general_quotients">More general quotients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quotient&action=edit&section=4" title="Edit section: More general quotients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> with an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> defined on it, a "<a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a>" may be created which contains those equivalence classes as elements. A <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> may be formed by breaking a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> into a number of similar <a href="/wiki/Cosets" class="mw-redirect" title="Cosets">cosets</a>, while a <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> may be formed in a similar process by breaking a <a href="/wiki/Vector_space" title="Vector space">vector space</a> into a number of similar <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspaces</a>. </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=Quotient&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Product_(mathematics)" title="Product (mathematics)">Product (mathematics)</a></li> <li><a href="/wiki/Quotient_category" title="Quotient category">Quotient category</a></li> <li><a href="/wiki/Quotient_graph" title="Quotient graph">Quotient graph</a></li> <li><a href="/wiki/Division_(mathematics)#Of_integers" title="Division (mathematics)">Integer division</a></li> <li><a href="/wiki/Quotient_module" title="Quotient module">Quotient module</a></li> <li><a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">Quotient object</a></li> <li><a href="/wiki/Quotient_of_a_formal_language" title="Quotient of a formal language">Quotient of a formal language</a>, also left and right quotient</li> <li><a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a></li> <li><a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">Quotient set</a></li> <li><a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">Quotient space (topology)</a></li> <li><a href="/wiki/Quotient_type" title="Quotient type">Quotient type</a></li> <li><a href="/wiki/Quotition_and_partition" title="Quotition and partition">Quotition and partition</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=Quotient&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.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 class="mw-references-wrap mw-references-columns"><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"><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="http://dictionary.reference.com/browse/quotient">"Quotient"</a>. <i>Dictionary.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dictionary.com&rft.atitle=Quotient&rft_id=http%3A%2F%2Fdictionary.reference.com%2Fbrowse%2Fquotient&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" 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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/IntegerDivision.html#:~:text=Integer%20division%20is%20division%20in,and%20is%20the%20floor%20function.">"Integer Division"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Integer+Division&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FIntegerDivision.html%23%3A~%3Atext%3DInteger%2520division%2520is%2520division%2520in%2Cand%2520is%2520the%2520floor%2520function.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-ISO_80000-1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-ISO_80000-1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ISO_80000-1_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ISO_80000-1_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFiso.org" class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en">"ISO 80000-1:2022(en) Quantities and units — Part 1: General"</a>. <i>iso.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-07-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=iso.org&rft.atitle=ISO+80000-1%3A2022%28en%29+Quantities+and+units+%E2%80%94+Part+1%3A+General&rft_id=https%3A%2F%2Fwww.iso.org%2Fobp%2Fui%2F%23iso%3Astd%3Aiso%3A80000%3A-1%3Aed-2%3Av1%3Aen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames1992" class="citation book cs1">James, R. C. (1992-07-31). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UyIfgBIwLMQC&dq=dictionary+ratio&pg=PA349"><i>Mathematics Dictionary</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-99041-0" title="Special:BookSources/978-0-412-99041-0"><bdi>978-0-412-99041-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Dictionary&rft.pub=Springer+Science+%26+Business+Media&rft.date=1992-07-31&rft.isbn=978-0-412-99041-0&rft.aulast=James&rft.aufirst=R.+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUyIfgBIwLMQC%26dq%3Ddictionary%2Bratio%26pg%3DPA349&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-International_Electrotechnical_Vocabulary_e707-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-International_Electrotechnical_Vocabulary_e707_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-22">"IEC 60050 - Details for IEV number 102-01-22: "quotient"<span class="cs1-kern-right"></span>"</a>. <i>International Electrotechnical Vocabulary</i> (in Japanese)<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=International+Electrotechnical+Vocabulary&rft.atitle=IEC+60050+-+Details+for+IEV+number+102-01-22%3A+%22quotient%22&rft_id=https%3A%2F%2Fwww.electropedia.org%2Fiev%2Fiev.nsf%2Fdisplay%3Fopenform%26ievref%3D102-01-22&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-International_Electrotechnical_Vocabulary_g891-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-International_Electrotechnical_Vocabulary_g891_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-23">"IEC 60050 - Details for IEV number 102-01-23: "ratio"<span class="cs1-kern-right"></span>"</a>. <i>International Electrotechnical Vocabulary</i> (in Japanese)<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=International+Electrotechnical+Vocabulary&rft.atitle=IEC+60050+-+Details+for+IEV+number+102-01-23%3A+%22ratio%22&rft_id=https%3A%2F%2Fwww.electropedia.org%2Fiev%2Fiev.nsf%2Fdisplay%3Fopenform%26ievref%3D102-01-23&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-International_Electrotechnical_Vocabulary_x558-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-International_Electrotechnical_Vocabulary_x558_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=112-03-18">"IEC 60050 - Details for IEV number 112-03-18: "rate"<span class="cs1-kern-right"></span>"</a>. <i>International Electrotechnical Vocabulary</i> (in Japanese)<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=International+Electrotechnical+Vocabulary&rft.atitle=IEC+60050+-+Details+for+IEV+number+112-03-18%3A+%22rate%22&rft_id=https%3A%2F%2Fwww.electropedia.org%2Fiev%2Fiev.nsf%2Fdisplay%3Fopenform%26ievref%3D112-03-18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompsonTaylor2020" class="citation book cs1">Thompson, A.; Taylor, B. N. (March 4, 2020). <a rel="nofollow" class="external text" href="https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values">"NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities"</a>. <i>Special Publication 811 | The NIST Guide for the use of the International System of Units</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a><span class="reference-accessdate">. Retrieved <span class="nowrap">October 25,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=NIST+Guide+to+the+SI%2C+Chapter+7%3A+Rules+and+Style+Conventions+for+Expressing+Values+of+Quantities&rft.btitle=Special+Publication+811+%7C+The+NIST+Guide+for+the+use+of+the+International+System+of+Units&rft.pub=National+Institute+of+Standards+and+Technology&rft.date=2020-03-04&rft.aulast=Thompson&rft.aufirst=A.&rft.au=Taylor%2C+B.+N.&rft_id=https%3A%2F%2Fwww.nist.gov%2Fpml%2Fspecial-publication-811%2Fnist-guide-si-chapter-7-rules-and-style-conventions-expressing-values&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Quotient"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Quotient.html">"Quotient"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Quotient&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FQuotient.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEpp2011" class="citation book cs1">Epp, Susanna S. (2011-01-01). <i>Discrete mathematics with applications</i>. Brooks/Cole. p. 163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780495391326" title="Special:BookSources/9780495391326"><bdi>9780495391326</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/970542319">970542319</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+mathematics+with+applications&rft.pages=163&rft.pub=Brooks%2FCole&rft.date=2011-01-01&rft_id=info%3Aoclcnum%2F970542319&rft.isbn=9780495391326&rft.aulast=Epp&rft.aufirst=Susanna+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.math.utah.edu/~pa/math/q1.html">"Irrationality of the square root of 2"</a>. <i>www.math.utah.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-27</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.math.utah.edu&rft.atitle=Irrationality+of+the+square+root+of+2.&rft_id=https%3A%2F%2Fwww.math.utah.edu%2F~pa%2Fmath%2Fq1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuotient" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quotient&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Quotients" class="extiw" title="commons:Category:Quotients">Quotients</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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.num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style></div><div role="navigation" class="navbox" aria-labelledby="Fractions_and_ratios" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="4"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Fractions_and_ratios" title="Template:Fractions and ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fractions_and_ratios" title="Template talk:Fractions and ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fractions_and_ratios" title="Special:EditPage/Template:Fractions and ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fractions_and_ratios" style="font-size:114%;margin:0 4em"><a href="/wiki/Fraction" title="Fraction">Fractions</a> and <a href="/wiki/Ratio" title="Ratio">ratios</a></div></th></tr><tr><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 2px 0 0;min-width: 60px"><div><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Unicode_0x0025.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/50px-Unicode_0x0025.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/75px-Unicode_0x0025.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/100px-Unicode_0x0025.svg.png 2x" data-file-width="16" data-file-height="16" /></a></span></div></td><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Division and ratio</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/Division_(mathematics)" title="Division (mathematics)">Dividend</a> ÷ <a href="/wiki/Divisor" title="Divisor">Divisor</a> = <a class="mw-selflink selflink">Quotient</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File"><span><img alt="The ratio of width to height of standard-definition television." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/66px-Aspect-ratio-4x3.svg.png" decoding="async" width="66" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/99px-Aspect-ratio-4x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/131px-Aspect-ratio-4x3.svg.png 2x" data-file-width="139" data-file-height="106" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Fraction</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><span style="font-size:120%"><span class="sfrac">⁠<span class="tion"><span class="num">Numerator</span><span class="sr-only">/</span><span class="den">Denominator</span></span>⁠</span></span> = Quotient</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" 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