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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Numbers</span> </div> </a> <ul id="toc-Numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Modern notation</span> </div> </a> <ul id="toc-Modern_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Typesetting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Typesetting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Typesetting</span> </div> </a> <ul id="toc-Typesetting-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-International_standard_mathematical_notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#International_standard_mathematical_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>International standard mathematical notation</span> </div> </a> <ul id="toc-International_standard_mathematical_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Latin-based_mathematical_notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-Latin-based_mathematical_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Non-Latin-based mathematical notation</span> </div> </a> <ul id="toc-Non-Latin-based_mathematical_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">9</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 id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mathematical notation</span></h1> <div id="p-lang-btn" 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Available in 30 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-30" 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">30 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%AA%D8%AF%D9%88%D9%8A%D9%86_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="تدوين رياضي – Arabic" lang="ar" hreflang="ar" data-title="تدوين رياضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</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%A4%E0%A7%80%E0%A6%95-%E0%A6%9A%E0%A6%BF%E0%A6%B9%E0%A7%8D%E0%A6%A8%E0%A6%BE%E0%A6%A6%E0%A6%BF" title="গাণিতিক প্রতীক-চিহ্নাদি – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক প্রতীক-চিহ্নাদি" 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%B8_%D0%BE%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%B8%D1%8F" title="Математически означения – Bulgarian" lang="bg" hreflang="bg" data-title="Математически означения" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Notaci%C3%B3_matem%C3%A0tica" title="Notació matemàtica – Catalan" lang="ca" hreflang="ca" data-title="Notació 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D1%80%D0%B8_%D0%BF%D0%B0%D0%BB%D0%BB%C4%83%D1%81%D0%B5%D0%BC" title="Математикăри паллăсем – Chuvash" lang="cv" hreflang="cv" data-title="Математикăри паллăсем" 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/Nodiant_mathemategol" title="Nodiant mathemategol – Welsh" lang="cy" hreflang="cy" data-title="Nodiant mathemategol" 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/Mathematische_Notation" title="Mathematische Notation – German" lang="de" hreflang="de" data-title="Mathematische Notation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Notaci%C3%B3n_matem%C3%A1tica" title="Notación matemática – Spanish" lang="es" hreflang="es" data-title="Notación 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/Matematika_notacio" title="Matematika notacio – Esperanto" lang="eo" hreflang="eo" data-title="Matematika notacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Notation_(math%C3%A9matiques)" title="Notation (mathématiques) – French" lang="fr" hreflang="fr" data-title="Notation (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Mathematical_notation" title="Mathematical notation – Hausa" lang="ha" hreflang="ha" data-title="Mathematical notation" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A5%87%E0%A4%A4%E0%A4%A8" title="गणितीय संकेतन – Hindi" lang="hi" hreflang="hi" data-title="गणितीय संकेतन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Notasi_matematika" title="Notasi matematika – Indonesian" lang="id" hreflang="id" data-title="Notasi 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-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Notasyon_(matematik)" title="Notasyon (matematik) – Haitian Creole" lang="ht" hreflang="ht" data-title="Notasyon (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Simboli_matematical" title="Simboli matematical – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Simboli matematical" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%BE%E0%A4%9A%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%B6%E0%A4%BE%E0%A4%96%E0%A4%BE" title="गणिताच्या शाखा – Marathi" lang="mr" hreflang="mr" data-title="गणिताच्या शाखा" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tatatanda_matematik" title="Tatatanda matematik – Malay" lang="ms" hreflang="ms" data-title="Tatatanda matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%80%E1%80%B1%E1%80%90%E1%80%A1%E1%80%9E%E1%80%AF%E1%80%B6%E1%80%B8%E1%80%A1%E1%80%94%E1%80%BE%E1%80%AF%E1%80%94%E1%80%BA%E1%80%B8%E1%80%99%E1%80%BB%E1%80%AC%E1%80%B8" title="သင်္ချာသင်္ကေတအသုံးအနှုန်းများ – Burmese" lang="my" hreflang="my" data-title="သင်္ချာသင်္ကေတအသုံးအနှုန်းများ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Nota%C3%A7%C3%A3o_matem%C3%A1tica" title="Notação matemática – Portuguese" lang="pt" hreflang="pt" data-title="Notação 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/Nota%C8%9Bie_matematic%C4%83" title="Notație matematică – Romanian" lang="ro" hreflang="ro" data-title="Notație 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D0%BE%D0%B1%D0%BE%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%B8%D1%8F" title="Математические обозначения – Russian" lang="ru" hreflang="ru" data-title="Математические обозначения" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_notation" title="Mathematical notation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical notation" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Matemati%C4%8Dka_notacija" title="Matematička notacija – Serbian" lang="sr" hreflang="sr" data-title="Matematička notacija" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_merkint%C3%A4" title="Matemaattinen merkintä – Finnish" lang="fi" hreflang="fi" data-title="Matemaattinen merkintä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Notasyong_matematikal" title="Notasyong matematikal – Tagalog" lang="tl" hreflang="tl" data-title="Notasyong matematikal" 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%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B1%E0%AE%BF%E0%AE%AF%E0%AF%80%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="கணிதக் குறியீடுகள் – Tamil" lang="ta" hreflang="ta" data-title="கணிதக் குறியீடுகள்" 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%AA%E0%B8%B1%E0%B8%8D%E0%B8%81%E0%B8%A3%E0%B8%93%E0%B9%8C%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="สัญกรณ์คณิตศาสตร์ – Thai" lang="th" hreflang="th" data-title="สัญกรณ์คณิตศาสตร์" 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/Matematiksel_g%C3%B6sterim" title="Matematiksel gösterim – Turkish" lang="tr" hreflang="tr" data-title="Matematiksel gösterim" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%BD%D0%BE%D1%82%D0%B0%D1%86%D1%96%D1%8F" title="Математична нотація – Ukrainian" lang="uk" hreflang="uk" data-title="Математична нотація" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E7%AC%A6%E8%99%9F" title="數學符號 – Chinese" lang="zh" hreflang="zh" data-title="數學符號" 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/Q1140046#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> 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class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Mathematical_notation" title="Special:EditPage/Mathematical notation">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Mathematical+notation%22">"Mathematical notation"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Mathematical+notation%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Mathematical+notation%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Mathematical+notation%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Mathematical+notation%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Mathematical+notation%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote 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 information on rendering mathematical formulae, see <a href="/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">Help:Displaying a formula</a> and <a href="/wiki/Wikipedia:Manual_of_Style/Mathematics" title="Wikipedia:Manual of Style/Mathematics">Wikipedia:Manual of Style/Mathematics</a>.</div> <p> <b>Mathematical notation</b> consists of using <a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> for representing <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a>, unspecified <a href="/wiki/Number" title="Number">numbers</a>, <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a>, and any other <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> and assembling them into <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> and <a href="/wiki/Formula" title="Formula">formulas</a>. Mathematical notation is widely used in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Science" title="Science">science</a>, and <a href="/wiki/Engineering" title="Engineering">engineering</a> for representing complex <a href="/wiki/Concept" title="Concept">concepts</a> and <a href="/wiki/Property_(philosophy)" title="Property (philosophy)">properties</a> in a concise, unambiguous, and accurate way. </p><p>For example, the physicist <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f73dbd37a0cac34406ee89057fa1b36a1e6a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}"></span> is the quantitative representation in mathematical notation of <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>.<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> </p><p>Mathematical notation was first introduced by <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> at the end of the 16th century and largely expanded during the 17th and 18th centuries by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, and overall <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Symbols">Symbols</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=1" title="Edit section: Symbols"><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/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></div> <p>The use of many symbols is the basis of mathematical notation. They play a similar role as words in <a href="/wiki/Natural_language" title="Natural language">natural languages</a>. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence. </p> <div class="mw-heading mw-heading3"><h3 id="Letters_as_symbols">Letters as symbols</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=2" title="Edit section: Letters as symbols"><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/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">List of letters used in mathematics, science, and engineering</a></div> <p>Letters are typically used for naming—in <a href="/wiki/List_of_mathematical_jargon" class="mw-redirect" title="List of mathematical jargon">mathematical jargon</a>, one says <i>representing</i>—<a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>. The <a href="/wiki/Latin_alphabet" title="Latin alphabet">Latin</a> and <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek</a> alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the <a href="/wiki/Hebrew_alphabet" title="Hebrew alphabet">Hebrew</a> <span class="nowrap">⁠<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 \aleph }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }"></span>⁠</span>, <a href="/wiki/Cyrillic_script" title="Cyrillic script">Cyrillic</a> <span class="texhtml">Ш</span>, and <a href="/wiki/Hiragana" title="Hiragana">Hiragana</a> <span class="texhtml">よ</span>. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, <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,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>R</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">r</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6628084b4eca01231c8b54ea09de51eb22591411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.149ex; height:2.509ex;" alt="{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}"></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 {\mathfrak {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}"></span> could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"></span>" of the <a href="/wiki/Sine_function" class="mw-redirect" title="Sine function">sine function</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> </p><p>In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, <a href="/wiki/Diacritic" title="Diacritic">diacritics</a>, <a href="/wiki/Subscript" class="mw-redirect" title="Subscript">subscripts</a> and <a href="/wiki/Superscript" class="mw-redirect" title="Superscript">superscripts</a> are often used. For example, <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 {\hat {f'_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f'_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca08b9d46d5a855ce0a075bc9372a4d13ac992cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.193ex; height:3.509ex;" alt="{\displaystyle {\hat {f'_{1}}}}"></span> may denote the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="/wiki/Derivative" title="Derivative">derivative</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15589063e81f3e5fa2699321756b83cab936bb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.509ex;" alt="{\displaystyle f_{1}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Other_symbols">Other symbols</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=3" title="Edit section: Other symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Symbols are not only used for naming mathematical objects. They can be used for <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+,-,/,\oplus ,\ldots ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>+</mo> <mo>,</mo> <mo>−</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>,</mo> <mo>⊕</mo> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (+,-,/,\oplus ,\ldots ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1838a740b7cd4f8586f00a1520699dcc093b5d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.902ex; height:2.843ex;" alt="{\displaystyle (+,-,/,\oplus ,\ldots ),}"></span> for <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>=</mo> <mo>,</mo> <mo><</mo> <mo>,</mo> <mo>≤</mo> <mo>,</mo> <mo>∼</mo> <mo>,</mo> <mo>≡</mo> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a985a49e8c74b68584afefe74722a8f87d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.39ex; height:2.843ex;" alt="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}"></span> for <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\implies ,\land ,\lor ,\ldots ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997ff31b3b9371a8becedb25fb044f7bc6120838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.478ex; height:2.843ex;" alt="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}"></span> for <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifiers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall ,\exists ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mo>,</mo> <mi mathvariant="normal">∃</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall ,\exists ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07b94cb636c83b580262f3b01af09b31eed3a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.075ex; height:2.843ex;" alt="{\displaystyle (\forall ,\exists ),}"></span> and for other purposes. </p><p>Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional <a href="/wiki/Typographic_symbol" class="mw-redirect" title="Typographic symbol">typographic symbols</a>, but many have been specially designed for mathematics. </p> <div class="mw-heading mw-heading2"><h2 id="Expressions">Expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=4" title="Edit section: Expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Mathematical_notation" title="Special:EditPage/Mathematical notation">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>An <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> is a finite combination of <a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> that is <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed</a> according to rules that depend on the context. In general, an expression denotes or names a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a>, and plays therefore in the <a href="/wiki/Language_of_mathematics" title="Language of mathematics">language of mathematics</a> the role of a <a href="/wiki/Noun_phrase" title="Noun phrase">noun phrase</a> in the natural language. </p><p>An expression contains often some <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operators</a>, and may therefore be <i>evaluated</i> by the action of the operators in it. For example, <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 3+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82be7d97da38d489ec230d0b3217453e8400e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 3+2}"></span> is an expression in which the operator <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> can be evaluated for giving 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 5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62cfda82378d02d6ff65a09e66873314c7013888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 5.}"></span> So, <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 3+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82be7d97da38d489ec230d0b3217453e8400e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 3+2}"></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 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"></span> are two different expressions that represent the same number. This is the meaning of the equality <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 3+2=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2=5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4011f9c8012f01a64b37e67426926efd6a9790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.073ex; height:2.343ex;" alt="{\displaystyle 3+2=5.}"></span> </p><p>A more complicated example is given by the expression<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="{\textstyle \int _{a}^{b}xdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>x</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{a}^{b}xdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbca8c9bb5a7c1b8b1fa995605b9e0852ffdde22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.714ex; height:3.676ex;" alt="{\textstyle \int _{a}^{b}xdx}"></span> that can be evaluated to <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="{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e31d9aed0d9638bb60a893b6253874b9b13389d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.398ex; height:4.009ex;" alt="{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}"></span> Although the resulting expression contains the operators of <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a> and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, it cannot be evaluated further because <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> denote unspecified numbers. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=5" title="Edit section: History"><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/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></div> <div class="mw-heading mw-heading3"><h3 id="Numbers">Numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=6" title="Edit section: Numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is believed that a notation to represent <a href="/wiki/Number" title="Number">numbers</a> was first developed at least 50,000 years ago.<sup id="cite_ref-Eves_1990_3-0" class="reference"><a href="#cite_note-Eves_1990-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Early mathematical ideas such as <a href="/wiki/Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a><sup id="cite_ref-Ifrah_2000_4-0" class="reference"><a href="#cite_note-Ifrah_2000-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The <a href="/wiki/Tally_stick" title="Tally stick">tally stick</a> is a way of counting dating back to the <a href="/wiki/Upper_Paleolithic" title="Upper Paleolithic">Upper Paleolithic</a>. Perhaps the oldest known mathematical texts are those of ancient <a href="/wiki/Sumer" title="Sumer">Sumer</a>. The <a href="/wiki/Census_quipu" class="mw-redirect" title="Census quipu">Census Quipu</a> of the Andes and the <a href="/wiki/Ishango_Bone" class="mw-redirect" title="Ishango Bone">Ishango Bone</a> from Africa both used the <a href="/wiki/Tally_mark" class="mw-redirect" title="Tally mark">tally mark</a> method of accounting for numerical concepts. </p><p>The concept of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the <a href="/wiki/Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonians</a> and <a href="/wiki/Greek_numerals" title="Greek numerals">Greek Egyptians</a>, and then as an <a href="/wiki/Integer" title="Integer">integer</a> by the <a href="/wiki/Maya_numerals" title="Maya numerals">Mayans</a>, <a href="/wiki/Indian_numerals" class="mw-redirect" title="Indian numerals">Indians</a> and <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabs</a> (see <a href="/wiki/History_of_zero" class="mw-redirect" title="History of zero">the history of zero</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Modern_notation">Modern notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=7" title="Edit section: Modern notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Until the 16th century, mathematics was essentially <a href="/wiki/Rhetorical_algebra" class="mw-redirect" title="Rhetorical algebra">rhetorical</a>, in the sense that everything but explicit numbers was expressed in words. However, some authors such as <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> used some symbols as abbreviations. </p><p>The first systematic use of formulas, and, in particular the use of symbols (<a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>) for unspecified numbers is generally attributed to <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (16th century). However, he used different symbols than those that are now standard. </p><p>Later, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (17th century) introduced the modern notation for variables and <a href="/wiki/Equation" title="Equation">equations</a>; in particular, the use of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> for <a href="/wiki/Unknown_(mathematics)" class="mw-redirect" title="Unknown (mathematics)">unknown</a> quantities 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 a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span> for known ones (<a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>). He introduced also the notation <span class="texhtml mvar" style="font-style:italic;">i</span> and the term "imaginary" for the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. </p><p>The 18th and 19th centuries saw the standardization of mathematical notation as used today. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> was responsible for many of the notations currently in use: the <a href="/wiki/Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"></span> <span class="texhtml"><i>e</i></span> for the base of the natural logarithm, <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="{\textstyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2b0b7618be940f4e8c0d27f05ab75fbc13e83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.454ex; height:2.843ex;" alt="{\textstyle \sum }"></span> for <a href="/wiki/Summation" title="Summation">summation</a>, etc.<sup id="cite_ref-Boyer-Merzbach_1991_5-0" class="reference"><a href="#cite_note-Boyer-Merzbach_1991-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> He also popularized the use of <span class="texhtml mvar" style="font-style:italic;">π</span> for the <a href="/wiki/Archimedes_constant" class="mw-redirect" title="Archimedes constant">Archimedes constant</a> (proposed by <a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a>, based on an earlier notation of <a href="/wiki/William_Oughtred" title="William Oughtred">William Oughtred</a>).<sup id="cite_ref-Arndt-Haenel_2006_6-0" class="reference"><a href="#cite_note-Arndt-Haenel_2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a>, <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>, <a href="/wiki/Einstein%27s_summation_convention" class="mw-redirect" title="Einstein's summation convention">Einstein's summation convention</a>, etc. </p> <div class="mw-heading mw-heading3"><h3 id="Typesetting">Typesetting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=8" title="Edit section: Typesetting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>General <a href="/wiki/Typesetting_system" class="mw-redirect" title="Typesetting system">typesetting systems</a> are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures, such as in: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f05c2431187190062e84cae7cc75ac50b68ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.418ex; height:10.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}"></span></dd></dl> <p><a href="/wiki/TeX" title="TeX">TeX</a> is a mathematically oriented typesetting system that was created in 1978 by <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. It is widely used in mathematics, through its extension called <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>, and is a <i>de facto</i> standard. (The above expression is written in LaTeX.) </p><p>More recently, another approach for mathematical typesetting is provided by <a href="/wiki/MathML" title="MathML">MathML</a>. However, it is not well supported in web browsers, which is its primary target. </p> <div class="mw-heading mw-heading2"><h2 id="International_standard_mathematical_notation">International standard mathematical notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=9" title="Edit section: International standard mathematical notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The international standard <a href="/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> (previously, <a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a>) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., <i>E</i>=<i>mc</i><sup>2</sup>) and roman (upright) fonts for mathematical constants (e.g., e or π). </p> <div class="mw-heading mw-heading2"><h2 id="Non-Latin-based_mathematical_notation">Non-Latin-based mathematical notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=10" title="Edit section: Non-Latin-based mathematical notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</a> is based mostly on the <a href="/wiki/Arabic_alphabet" title="Arabic alphabet">Arabic alphabet</a> and is used widely in the <a href="/wiki/Arab_world" title="Arab world">Arab world</a>, especially in pre-<a href="/wiki/Tertiary_education" title="Tertiary education">tertiary education</a>. (Western notation uses <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.) </p><p>In addition to Arabic notation, mathematics also makes use of <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek letters</a> to denote a wide variety of mathematical objects and variables. On some occasions, certain <a href="/wiki/Hebrew_alphabet" title="Hebrew alphabet">Hebrew letters</a> are also used (such as in the context of <a href="/wiki/Infinite_cardinal" class="mw-redirect" title="Infinite cardinal">infinite cardinals</a>). </p><p>Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are <a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a> and <a href="/wiki/Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter–Dynkin diagrams</a>. </p><p>Braille-based mathematical notations used by blind people include <a href="/wiki/Nemeth_Braille" title="Nemeth Braille">Nemeth Braille</a> and <a href="/wiki/GS8_Braille" class="mw-redirect" title="GS8 Braille">GS8 Braille</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=Mathematical_notation&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Abuse_of_notation" title="Abuse of notation">Abuse of notation</a></li> <li><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a> <ul><li><a href="/wiki/Bourbaki_dangerous_bend_symbol" title="Bourbaki dangerous bend symbol">Bourbaki dangerous bend symbol</a></li></ul></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></li> <li><a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a></li> <li><a href="/wiki/ISO/IEC_80000#Part_2:_Mathematics" title="ISO/IEC 80000">ISO 80000-2</a></li> <li><a href="/wiki/Knuth%27s_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li> <li><a href="/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li> <li><a href="/wiki/Mathematical_formula" class="mw-redirect" title="Mathematical formula">Mathematical formula</a></li> <li><a href="/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li> <li><a href="/wiki/Language_of_mathematics" title="Language of mathematics">Language of mathematics</a></li> <li><a href="/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li> <li><a href="/wiki/Semasiography" title="Semasiography">Semasiography</a></li> <li><a href="/wiki/Table_of_mathematical_symbols" class="mw-redirect" title="Table of mathematical symbols">Table of mathematical symbols</a></li> <li><a href="/wiki/Vector_notation" title="Vector notation">Vector notation</a></li> <li><a href="/wiki/Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</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=Mathematical_notation&action=edit&section=12" 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"><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 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.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 id="CITEREFEinstein1905" class="citation journal cs1 cs1-prop-foreign-lang-source">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314">"Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?"</a>. <i>Annalen der Physik</i> (in German). <b>323</b> (13): 639–641. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053231314">10.1002/andp.19053231314</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-3804">0003-3804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Ist+die+Tr%C3%A4gheit+eines+K%C3%B6rpers+von+seinem+Energieinhalt+abh%C3%A4ngig%3F&rft.volume=323&rft.issue=13&rft.pages=639-641&rft.date=1905&rft_id=info%3Adoi%2F10.1002%2Fandp.19053231314&rft.issn=0003-3804&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fandp.19053231314&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" 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">ISO 80000-2:2019</span> </li> <li id="cite_note-Eves_1990-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Eves_1990_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1990" class="citation book cs1"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard</a> (1990). <i>An Introduction to the History of Mathematics</i> (6 ed.). p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-029558-4" title="Special:BookSources/978-0-03-029558-4"><bdi>978-0-03-029558-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+History+of+Mathematics&rft.pages=9&rft.edition=6&rft.date=1990&rft.isbn=978-0-03-029558-4&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span> </li> <li id="cite_note-Ifrah_2000-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ifrah_2000_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfrah2000" class="citation book cs1"><a href="/wiki/Georges_Ifrah" title="Georges Ifrah">Ifrah, Georges</a> (2000). <i>The Universal History of Numbers: From prehistory to the invention of the computer</i>. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. <a href="/wiki/John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-39340-1" title="Special:BookSources/0-471-39340-1"><bdi>0-471-39340-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+History+of+Numbers%3A+From+prehistory+to+the+invention+of+the+computer.&rft.pages=48&rft.pub=John+Wiley+and+Sons&rft.date=2000&rft.isbn=0-471-39340-1&rft.aulast=Ifrah&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span> (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of <a href="/wiki/Boethius" title="Boethius">Boethius</a> (who lived 480–524 or 525) reckoning on his fingers.)</span> </li> <li id="cite_note-Boyer-Merzbach_1991-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer-Merzbach_1991_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyerMerzbach1991" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl Benjamin</a>; <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/442"><i>A History of Mathematics</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. 442–443. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=442-443&rft.pub=John+Wiley+%26+Sons&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+Benjamin&rft.au=Merzbach%2C+Uta+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye%2Fpage%2F442&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span> </li> <li id="cite_note-Arndt-Haenel_2006-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Arndt-Haenel_2006_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArndtHaenel2006" class="citation book cs1">Arndt, Jörg; Haenel, Christoph (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166"><i>Pi Unleashed</i></a>. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 166. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66572-4" title="Special:BookSources/978-3-540-66572-4"><bdi>978-3-540-66572-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+Unleashed&rft.pages=166&rft.pub=Springer-Verlag&rft.date=2006&rft.isbn=978-3-540-66572-4&rft.aulast=Arndt&rft.aufirst=J%C3%B6rg&rft.au=Haenel%2C+Christoph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQwwcmweJCDQC%26pg%3DPA166&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_notation&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Florian_Cajori" title="Florian Cajori">Florian Cajori</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7juWmvQSTvwC"><i>A History of Mathematical Notations</i></a> (1929), 2 volumes. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-67766-4" title="Special:BookSources/0-486-67766-4">0-486-67766-4</a></li> <li>Mazur, Joseph (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YZLzjwEACAAJ&q=enlightening+symbols"><i>Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers</i></a>. Princeton, New Jersey: Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-15463-3" title="Special:BookSources/978-0-691-15463-3">978-0-691-15463-3</a></li></ul> <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=Mathematical_notation&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_notation" class="extiw" title="commons:Category:Mathematical notation">Mathematical notation</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a></li> <li><a rel="nofollow" class="external text" href="http://www.apronus.com/math/mrwmath.htm">Mathematical ASCII Notation</a> how to type math notation in any text editor.</li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/language/index.shtml">Mathematics as a Language</a> at <a href="/wiki/Alexander_Bogomolny#Cut-the-Knot" title="Alexander Bogomolny">Cut-the-Knot</a></li> <li><a href="/wiki/Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a>: <a rel="nofollow" class="external text" href="http://www.stephenwolfram.com/publications/mathematical-notation-past-future/">Mathematical Notation: Past and Future</a>. October 2000. 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title="Unicode">Unicode</a> and <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> mathematical symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">List of mathematical symbols by subject</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li> <li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_symbols" style="font-size:114%;margin:0 4em">Lists of <a href="/wiki/Unicode" title="Unicode">Unicode</a> symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_Unicode_characters" title="List of Unicode characters">List of Unicode characters</a></li> <li><a href="/wiki/Unicode_block" title="Unicode block">Unicode block</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Alphanumeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li> <li><a href="/wiki/Blackboard_bold#Usage" title="Blackboard bold">Blackboard bold</a></li> <li><a href="/wiki/Letterlike_Symbols" title="Letterlike Symbols">Letterlike Symbols</a></li> <li><a href="/wiki/Symbols_for_zero" title="Symbols for zero">Symbols for zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a> and <a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a></li> <li><a href="/wiki/Miscellaneous_Symbols_and_Arrows" title="Miscellaneous Symbols and Arrows">Miscellaneous Symbols and Arrows</a></li> <li><a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes (Unicode block)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols</a></li> <li><a href="/wiki/Mathematical_Operators_(Unicode_block)" title="Mathematical Operators (Unicode block)">Mathematical Operators (Unicode block)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Math Operators</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Mathematical Operators</a></li> <li><a href="/wiki/Number_Forms" title="Number Forms">Number Forms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Miscellaneous_Mathematical_Symbols-A" title="Miscellaneous Mathematical Symbols-A">A</a></li> <li><a href="/wiki/Miscellaneous_Mathematical_Symbols-B" title="Miscellaneous Mathematical Symbols-B">B</a></li> <li><a href="/wiki/Miscellaneous_Technical" title="Miscellaneous Technical">Technical</a></li> <li><a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a> (Mathematical signs and symbols for use in physical sciences and technology)</li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Typographical_conventions_and_notations" style="font-size:114%;margin:0 4em">Typographical conventions and notations</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Language_of_mathematics" title="Language of mathematics">Language</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APL_syntax_and_symbols#Monadic_functions" title="APL syntax and symbols">APL syntax and symbols</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Letters</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Diacritic" title="Diacritic">Diacritic</a></li> <li><a href="/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">Letters in STEM</a> <ul><li><a href="/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering" title="Greek letters used in mathematics, science, and engineering">Greek letters in STEM</a></li> <li><a href="/wiki/Latin_letters_used_in_mathematics,_science,_and_engineering" title="Latin letters used in mathematics, science, and engineering">Latin letters in STEM</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Mathematical notation</a></li> <li><a href="/wiki/List_of_mathematical_abbreviations" title="List of mathematical abbreviations">Abbreviations</a></li> <li><a href="/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li> <li><a href="/wiki/List_of_common_physics_notations" title="List of common physics notations">List of common physics notations</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Meanings_of_symbols" style="font-size:114%;margin:0 4em">Meanings of symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li> <li><a href="/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li> <li><a href="/wiki/Physical_constant" title="Physical constant">Physical constants</a></li> <li><a href="/wiki/Table_of_mathematical_symbols_by_introduction_date" title="Table of mathematical symbols by introduction date">Table of mathematical symbols by introduction date</a></li> <li><a href="/wiki/List_of_typographical_symbols_and_punctuation_marks" title="List of typographical symbols and punctuation marks">List of typographical symbols and punctuation marks</a></li></ul> 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