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id="toc-Symbolic_stage_and_early_arithmetic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Variables_and_evaluation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variables_and_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Variables and evaluation</span> </div> </a> <button aria-controls="toc-Variables_and_evaluation-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 Variables and evaluation subsection</span> </button> <ul id="toc-Variables_and_evaluation-sublist" class="vector-toc-list"> <li id="toc-Equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Equivalence</span> </div> </a> <ul id="toc-Equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Polynomial evaluation</span> </div> </a> <ul id="toc-Polynomial_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Computation</span> </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> <li id="toc-Rewriting" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rewriting"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Rewriting</span> </div> </a> <ul id="toc-Rewriting-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Well-defined_expressions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Well-defined_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Well-defined expressions</span> </div> </a> <button aria-controls="toc-Well-defined_expressions-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 Well-defined expressions subsection</span> </button> <ul id="toc-Well-defined_expressions-sublist" class="vector-toc-list"> <li id="toc-Well-formed" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Well-formed"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Well-formed</span> </div> </a> <ul id="toc-Well-formed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Well-defined" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Well-defined"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Well-defined</span> </div> </a> <ul id="toc-Well-defined-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Formal definition</span> </div> </a> <button aria-controls="toc-Formal_definition-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 Formal definition subsection</span> </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Lambda_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lambda_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Lambda calculus</span> </div> </a> <ul id="toc-Lambda_calculus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_expressions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Types of expressions</span> </div> </a> <button aria-controls="toc-Types_of_expressions-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 Types of expressions subsection</span> </button> <ul id="toc-Types_of_expressions-sublist" class="vector-toc-list"> <li id="toc-Algebraic_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Algebraic expression</span> </div> </a> <ul id="toc-Algebraic_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Polynomial expression</span> </div> </a> <ul id="toc-Polynomial_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Computational expression</span> </div> </a> <ul id="toc-Computational_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logical_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Logical expression</span> </div> </a> <ul id="toc-Logical_expression-sublist" class="vector-toc-list"> </ul> </li> </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-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_Cited" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Works_Cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Works Cited</span> </div> </a> <ul id="toc-Works_Cited-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" 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Available in 39 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-39" 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">39 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%A8%D8%A7%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="عبارة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="عبارة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B7%D1%80%D0%B0%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Израз (математика) – Bulgarian" lang="bg" hreflang="bg" data-title="Израз (математика)" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BA%D0%B0%D0%BB%D0%B0%D0%BD%C4%83%D0%BB%C4%83%D1%85" title="Математикăлла каланăлăх – Chuvash" lang="cv" hreflang="cv" data-title="Математикăлла каланăлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%BD_v%C3%BDraz" title="Matematický výraz – Czech" lang="cs" hreflang="cs" data-title="Matematický výraz" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mynegiad_(mathemateg)" title="Mynegiad (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Mynegiad (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Ausdruck_(Mathematik)" title="Ausdruck (Mathematik) – German" lang="de" hreflang="de" data-title="Ausdruck (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Avaldis" title="Avaldis – Estonian" lang="et" hreflang="et" data-title="Avaldis" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es badge-Q70893996 mw-list-item" title=""><a href="https://es.wikipedia.org/wiki/Expresi%C3%B3n_matem%C3%A1tica" title="Expresión matemática – Spanish" lang="es" hreflang="es" data-title="Expresión 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/Esprimo_(matematiko)" title="Esprimo (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Esprimo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%A8%D8%A7%D8%B1%D8%AA_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="عبارت (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="عبارت (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Expression_(math%C3%A9matiques)" title="Expression (mathématiques) – French" lang="fr" hreflang="fr" data-title="Expression (mathématiques)" 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/%EC%88%98%EC%8B%9D" title="수식 – Korean" lang="ko" hreflang="ko" data-title="수식" 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%B5%E0%A5%8D%E0%A4%AF%E0%A4%82%E0%A4%9C%E0%A4%95" title="व्यंजक – Hindi" lang="hi" hreflang="hi" data-title="व्यंजक" 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/Ekspresi_(matematika)" title="Ekspresi (matematika) – Indonesian" lang="id" hreflang="id" data-title="Ekspresi (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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6%C3%B0a_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Stæða (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Stæða (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/Espressione_matematica" title="Espressione matematica – Italian" lang="it" hreflang="it" data-title="Espressione matematica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%99%D7%98%D7%95%D7%99_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="ביטוי (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="ביטוי (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%B7%D1%80%D0%B0%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Израз (математика) – Macedonian" lang="mk" hreflang="mk" data-title="Израз (математика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%9E%E0%B5%8D%E0%B4%9C%E0%B4%95%E0%B4%82" title="വ്യഞ്ജകം – Malayalam" lang="ml" hreflang="ml" data-title="വ്യഞ്ജകം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A4%A6%E0%A4%BE%E0%A4%B5%E0%A4%B2%E0%A5%80" title="पदावली – Marathi" lang="mr" hreflang="mr" data-title="पदावली" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Uitdrukking_(wiskunde)" title="Uitdrukking (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Uitdrukking (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%BC%8F" title="数式 – Japanese" lang="ja" hreflang="ja" data-title="数式" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%87%E0%A8%AC%E0%A8%BE%E0%A8%B0%E0%A8%A4_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਇਬਾਰਤ (ਗਣਿਤ) – Punjabi" lang="pa" hreflang="pa" data-title="ਇਬਾਰਤ (ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Express%C3%A3o_matem%C3%A1tica" title="Expressão matemática – Portuguese" lang="pt" hreflang="pt" data-title="Expressão 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/Expresie_matematic%C4%83" title="Expresie matematică – Romanian" lang="ro" hreflang="ro" data-title="Expresie 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%92%D1%8B%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Выражение (математика) – Russian" lang="ru" hreflang="ru" data-title="Выражение (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Expression_(mathematics)" title="Expression (mathematics) – Scots" lang="sco" hreflang="sco" data-title="Expression (mathematics)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematick%C3%BD_v%C3%BDraz" title="Matematický výraz – Slovak" lang="sk" hreflang="sk" data-title="Matematický výraz" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matemati%C4%8Dni_izraz" title="Matematični izraz – Slovenian" lang="sl" hreflang="sl" data-title="Matematični izraz" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%DB%95%D8%B1%D8%A8%DA%95%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="دەربڕە (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دەربڕە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lauseke_(matematiikka)" 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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/Expression_(disambiguation)" class="mw-redirect mw-disambig" title="Expression (disambiguation)">Expression (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Equation_vs_Expression.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Equation_vs_Expression.png/239px-Equation_vs_Expression.png" decoding="async" width="239" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Equation_vs_Expression.png/359px-Equation_vs_Expression.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e8/Equation_vs_Expression.png 2x" data-file-width="390" data-file-height="207" /></a><figcaption>In the <a href="/wiki/Equation" title="Equation">equation</a> 7x − 5 = 2, the <a href="/wiki/Sides_of_an_equation" title="Sides of an equation">sides of the equation</a> are expressions.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>expression</b> is a written arrangement of <a href="/wiki/Symbol_(mathematics)" class="mw-redirect" title="Symbol (mathematics)">symbols</a> following the context-dependent, <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntactic</a> conventions of <a href="/wiki/Mathematical_notation" title="Mathematical notation">mathematical notation</a>. Symbols can denote <a href="/wiki/Numbers" class="mw-redirect" title="Numbers">numbers</a>, <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a>, and <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</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> Other symbols include <a href="/wiki/Punctuation" title="Punctuation">punctuation</a> marks and <a href="/wiki/Bracket_(mathematics)" title="Bracket (mathematics)">brackets</a>, used for <a href="/wiki/Symbols_of_grouping" title="Symbols of grouping">grouping</a> where there is not a well-defined <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a>. </p><p>Expressions are commonly distinguished from <i><a href="/wiki/Mathematical_formula" class="mw-redirect" title="Mathematical formula">formulas</a></i>: expressions are a kind of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a>, whereas formulas are statements <i>about</i> mathematical objects.<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> This is analogous to <a href="/wiki/Natural_language" title="Natural language">natural language</a>, where a <a href="/wiki/Noun_phrase" title="Noun phrase">noun phrase</a> refers to an object, and a whole <a href="/wiki/Sentence_(linguistics)" title="Sentence (linguistics)">sentence</a> refers to a <a href="/wiki/Fact" title="Fact">fact</a>. 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 8x-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8x-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8612a66aaa50a47781f9c936cdc985dbd979e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.495ex; height:2.343ex;" alt="{\displaystyle 8x-5}"></span> is an expression, while the <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</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 8x-5\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8x-5\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c798bbe752dacb43f38bc6c5d12da206ce867b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.756ex; height:2.343ex;" alt="{\displaystyle 8x-5\geq 3}"></span> is a formula. </p><p>To <i>evaluate</i> an expression means to find a numerical <a href="/wiki/Value_(mathematics)" title="Value (mathematics)">value</a> equivalent to the expression.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> Expressions can be <i>evaluated</i> or <i>simplified</i> by replacing <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> that appear in them with their result. For example, 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="{\displaystyle 8\times 2-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>×</mo> <mn>2</mn> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\times 2-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68fc09bbe5a30bbfee1a138ffd33103ca09f0e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.168ex; height:2.343ex;" alt="{\displaystyle 8\times 2-5}"></span> simplifies 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="{\displaystyle 16-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7303a3de69a07585ead57108a953a6207c91c849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 16-5}"></span>, and evaluates 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="{\displaystyle 11.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>11.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9b5e5fac088741f78d27719ffc4774c9815c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 11.}"></span> </p><p>An expression is often used to define a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, by taking the variables to be <a href="/wiki/Argument_of_a_function" title="Argument of a function">arguments</a>, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression.<sup id="cite_ref-Codd1970_5-0" class="reference"><a href="#cite_note-Codd1970-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 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 x\mapsto x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8840cd9ef722e99165d3d2456a8f26b62dc98ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle x\mapsto x^{2}+1}"></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 f(x)=x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060789c792de89c492dc74c0f3e74edd0cbf87c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.903ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+1}"></span> define the function that associates to each number its <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square</a> plus one. An expression with no variables would define a <a href="/wiki/Constant_function" title="Constant function">constant function</a>. Usually, two expressions are considered <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> or <i>equivalent</i> if they define the same function. Such an equality is called a "<a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">semantic</a> equality", that is, both expressions "mean the same thing." </p><p>A <b>formal expression</b> is a kind of <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a> of <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbols</a>, created by the same <a class="mw-selflink-fragment" href="#Formal_definition">production rules</a> as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two <i>formal expressions</i> are considered equal only if they are <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntactically</a> equal, that is, if they are the exact same expression.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For instance, the formal expressions "2" and "1+1" are not equal. </p> <meta property="mw:PageProp/toc" /> <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=Expression_(mathematics)&action=edit&section=1" 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">For broader coverage of this topic, see <a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a> and <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="Early_written_mathematics">Early written mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=2" title="Edit section: Early written mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:222px;max-width:222px"><div class="trow"><div class="tsingle" style="width:88px;max-width:88px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Ishango_bone_(cropped).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/86px-Ishango_bone_%28cropped%29.jpg" decoding="async" width="86" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/129px-Ishango_bone_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/172px-Ishango_bone_%28cropped%29.jpg 2x" data-file-width="891" data-file-height="1230" /></a></span></div></div><div class="tsingle" style="width:130px;max-width:130px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Ybc7289-bw.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/128px-Ybc7289-bw.jpg" decoding="async" width="128" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/192px-Ybc7289-bw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/256px-Ybc7289-bw.jpg 2x" data-file-width="338" data-file-height="315" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:220px;max-width:220px"><div class="thumbimage" style="height:104px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Moskou-papyrus.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Moskou-papyrus.jpg/218px-Moskou-papyrus.jpg" decoding="async" width="218" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Moskou-papyrus.jpg/327px-Moskou-papyrus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/f/fd/Moskou-papyrus.jpg 2x" data-file-width="375" data-file-height="180" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a> at the <a href="/wiki/Royal_Belgian_Institute_of_Natural_Sciences" class="mw-redirect" title="Royal Belgian Institute of Natural Sciences">RBINS</a>. A <a href="/wiki/Babylonian_tablet" class="mw-redirect" title="Babylonian tablet">Babylonian tablet</a> approximating the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>. Problem 14 from the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>.</div></div></div></div> <p>The earliest written mathematics likely began with <a href="/wiki/Tally_marks" title="Tally marks">tally marks</a>, where each mark represented one unit, carved into wood or stone. An example of early <a href="/wiki/Counting" title="Counting">counting</a> is the <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>, found near the <a href="/wiki/Nile" title="Nile">Nile</a> and dating back over <a href="/wiki/Upper_Paleolithic" title="Upper Paleolithic">20,000 years ago</a>, which is thought to show a six-month <a href="/wiki/Lunar_calendar" title="Lunar calendar">lunar calendar</a>.<sup id="cite_ref-Marshack2_8-0" class="reference"><a href="#cite_note-Marshack2-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egypt</a> developed a symbolic system using <a href="/wiki/Hieroglyphics" class="mw-redirect" title="Hieroglyphics">hieroglyphics</a>, assigning symbols for powers of ten and using addition and subtraction symbols resembling legs in motion.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This system, recorded in texts like the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> (c. 2000–1800 BC), influenced other <a href="/wiki/History_of_the_Mediterranean_region" title="History of the Mediterranean region">Mediterranean cultures</a>. In <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a>, a similar system evolved, with numbers written in a base-60 (<a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a>) format on <a href="/wiki/Clay_tablets" class="mw-redirect" title="Clay tablets">clay tablets</a> written in <a href="/wiki/Cuneiform" title="Cuneiform">Cuneiform</a>, a technique originating with the <a href="/wiki/Sumerians" class="mw-redirect" title="Sumerians">Sumerians</a> around 3000 BC. This base-60 system persists today in measuring time and <a href="/wiki/Angle" title="Angle">angles</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Syncopated_stage">Syncopated stage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=3" title="Edit section: Syncopated stage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "syncopated" stage of mathematics introduced symbolic abbreviations for commonly used operations and quantities, marking a shift from purely <a href="/wiki/Geometry" title="Geometry">geometric</a> reasoning. <a href="/wiki/Ancient_Greek_mathematics" class="mw-redirect" title="Ancient Greek mathematics">Ancient Greek mathematics</a>, largely geometric in nature, drew on <a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian numerical systems</a> (especially <a href="/wiki/Attic_numerals" title="Attic numerals">Attic numerals</a>),<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> with little interest in algebraic symbols, until the arrival of <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> of <a href="/wiki/History_of_Alexandria" title="History of Alexandria">Alexandria</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> who pioneered a form of <a href="/wiki/Syncopated_algebra" class="mw-redirect" title="Syncopated algebra">syncopated algebra</a> in his <i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a>,</i> which introduced symbolic manipulation of expressions.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> His notation represented unknowns and powers symbolically, but without modern symbols for <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a> (such as <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a> or <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a>) or <a href="/wiki/Exponentiation" title="Exponentiation">exponents</a>.<sup id="cite_ref-Boyer_14-0" class="reference"><a href="#cite_note-Boyer-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> An unknown number was 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 \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The square 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 \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span> was <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 \Delta ^{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4c7e133f61d18dff15b36ff814bbb2b92648c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.965ex; height:2.343ex;" alt="{\displaystyle \Delta ^{v}}"></span>; the cube was <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 K^{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce4d49f974be73816365cf9b1fc7900c90c12b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.124ex; height:2.343ex;" alt="{\displaystyle K^{v}}"></span>; the fourth power was <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 \Delta ^{v}\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{v}\Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4e3907f0ccaa0acce4d02357b28c7ab5d28162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.901ex; height:2.343ex;" alt="{\displaystyle \Delta ^{v}\Delta }"></span>; and the fifth power was <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 \Delta K^{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta K^{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10ff4e71e861767c5caa61ee46f3865081e3266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.059ex; height:2.343ex;" alt="{\displaystyle \Delta K^{v}}"></span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> So for example, what would be written in modern notation as:<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 x^{3}-2x^{2}+10x-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-2x^{2}+10x-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30af3cb1ddc55e79976bed83d850a2840f71e74d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.915ex; height:3.009ex;" alt="{\displaystyle x^{3}-2x^{2}+10x-1,}"></span>Would be written in Diophantus's syncopated notation as: </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 \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thickmathspace" /> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ι</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mo>⋔</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6f34b992f64ad82a6963fea1a3f486c0d7a67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.039ex; height:3.343ex;" alt="{\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}"></span></dd></dl> <p>In the 7th century, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> used different colours to represent the unknowns in algebraic equations in the <i><a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a></i>. Greek and other ancient mathematical advances, were often trapped in cycles of bursts of creativity, followed by long periods of stagnation, but this began to change as knowledge spread in the <a href="/wiki/Early_modern_period" title="Early modern period">early modern period</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Symbolic_stage_and_early_arithmetic">Symbolic stage and early arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=4" title="Edit section: Symbolic stage and early arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Johannes_Widmann-Mercantile_Arithmetic_1489.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Johannes_Widmann-Mercantile_Arithmetic_1489.jpg/160px-Johannes_Widmann-Mercantile_Arithmetic_1489.jpg" decoding="async" width="160" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Johannes_Widmann-Mercantile_Arithmetic_1489.jpg/240px-Johannes_Widmann-Mercantile_Arithmetic_1489.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Johannes_Widmann-Mercantile_Arithmetic_1489.jpg/320px-Johannes_Widmann-Mercantile_Arithmetic_1489.jpg 2x" data-file-width="440" data-file-height="756" /></a><figcaption>The 1489 use of the <a href="/wiki/Plus_and_minus_signs" title="Plus and minus signs">plus and minus signs</a> in print.</figcaption></figure> <p>The transition to fully symbolic algebra began with <a href="/wiki/Ibn_al-Banna%27_al-Marrakushi" title="Ibn al-Banna' al-Marrakushi">Ibn al-Banna' al-Marrakushi</a> (1256–1321) and <a href="/wiki/Ab%C5%AB_al-%E1%B8%A4asan_ibn_%CA%BFAl%C4%AB_al-Qala%E1%B9%A3%C4%81d%C4%AB" class="mw-redirect" title="Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī">Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī</a>, (1412–1482) who introduced symbols for operations using <a href="/wiki/Arabic_script" title="Arabic script">Arabic characters</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gullberg2_18-0" class="reference"><a href="#cite_note-Gullberg2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Qalasadi2_19-0" class="reference"><a href="#cite_note-Qalasadi2-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Plus_sign" class="mw-redirect" title="Plus sign">plus sign</a> (+) appeared around 1351 with <a href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> likely derived from the Latin <i>et</i> (meaning "and"), while the minus sign (−) was first used in 1489 by <a href="/wiki/Johannes_Widmann" title="Johannes Widmann">Johannes Widmann</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a> included these symbols in his works, though much was based on earlier contributions by <a href="/wiki/Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a>. The <a href="/wiki/Radical_symbol" title="Radical symbol">radical symbol</a> (√) for <a href="/wiki/Square_root" title="Square root">square root</a> was introduced by <a href="/wiki/Christoph_Rudolff" title="Christoph Rudolff">Christoph Rudolff</a> in the 1500s, and <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</a> for <a href="/wiki/Precedence_(mathematics)" class="mw-redirect" title="Precedence (mathematics)">precedence</a> by <a href="/wiki/Niccol%C3%B2_Tartaglia" class="mw-redirect" title="Niccolò Tartaglia">Niccolò Tartaglia</a> in 1556. <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a>’s <i>New Algebra</i> (1591) formalized modern symbolic manipulation. The <a href="/wiki/Multiplication_sign" title="Multiplication sign">multiplication sign</a> (×) was first used by <a href="/wiki/William_Oughtred" title="William Oughtred">William Oughtred</a> and the <a href="/wiki/Division_sign" title="Division sign">division sign</a> (÷) by <a href="/wiki/Johann_Rahn" title="Johann Rahn">Johann Rahn</a>. </p><p><a href="/wiki/Paris_Descartes_University" title="Paris Descartes University">René Descartes</a> further advanced algebraic symbolism in <i><a href="/wiki/La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></i> (1637), where he introduced the use of letters at at the end of the alphabet (x, y, z) for <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, along with the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, which bridged algebra and geometry.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> independently developed <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus</a> in the late 17th century, with <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a> becoming the standard. </p> <div class="mw-heading mw-heading2"><h2 id="Variables_and_evaluation">Variables and evaluation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=5" title="Edit section: Variables and evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="evaluation"></span> In <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>, a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)"><i>variable</i></a> in an expression is a <a href="/wiki/Letter_(alphabet)" title="Letter (alphabet)">letter</a> that represents a number whose value may change. To <i>evaluate an expression</i> with a variable means to find the value of the expression when the variable is <a href="/wiki/Assignment_(computer_science)" title="Assignment (computer science)">assigned</a> a given number. Expressions can be <i>evaluated</i> or <i>simplified</i> by replacing <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> that appear in them with their result, or by combining <a href="/wiki/Like_terms" title="Like terms">like-terms</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>For example, take 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="{\displaystyle 4x^{2}+8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}+8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a8be3e2c0c922d299475b506c4727f3ecb1537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 4x^{2}+8}"></span>; it can be evaluated at <span class="texhtml"><i>x</i> = 3</span> in the following steps: </p><p><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 4(3)^{2}+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>4</mn> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 4(3)^{2}+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd22cef91639f5bc8561c3b5f5d14f54c43df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.191ex; height:3.009ex;" alt="{\textstyle 4(3)^{2}+3}"></span>, (replace x with 3) </p><p><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 4\cdot (3\cdot 3)+8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot (3\cdot 3)+8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b861b55da883dd661c08b0f1dec38fb6f125557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.658ex; height:2.843ex;" alt="{\displaystyle 4\cdot (3\cdot 3)+8}"></span> (use definition of <a href="/wiki/Exponentiation" title="Exponentiation">exponent</a>) </p><p><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 4\cdot 9+8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mn>9</mn> <mo>+</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 9+8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f7af3c79e32ee0646b3e2ba16d3797b90fc71a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.007ex; height:2.343ex;" alt="{\displaystyle 4\cdot 9+8}"></span> (simplify) </p><p><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 36+8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mo>+</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36+8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b51b048521e1866800b5352b4ce8e99c554245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 36+8}"></span> </p><p><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 44}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>44</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 44}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007b371fb10b4cfe6e52ef5102e0a13eab5d46d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 44}"></span> </p><p>A <i>term</i> is a constant or the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of a constant and one or more variables. Some examples include <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 7,\;5x,\;13x^{2}y,\;4b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>5</mn> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mn>13</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mn>4</mn> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7,\;5x,\;13x^{2}y,\;4b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b732ed99120f7edfea839ffb91fe2e643e146df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.716ex; height:3.009ex;" alt="{\displaystyle 7,\;5x,\;13x^{2}y,\;4b}"></span> The constant of the product is called the <a href="/wiki/Coefficient" title="Coefficient">coefficient</a>. Terms that are either constants or have the same variables raised to the same powers are called <i><a href="/wiki/Like_terms" title="Like terms">like terms</a></i>. If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable. </p><p><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 4x+7x+2x=15x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>15</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x+7x+2x=15x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f462a80542d7754585e28617b1bbc6e4e4da5699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.91ex; height:2.343ex;" alt="{\displaystyle 4x+7x+2x=15x}"></span> </p><p>Any variable can be classified as being either a <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free variable</a> or a <a href="/wiki/Bound_variable" class="mw-redirect" title="Bound variable">bound variable</a>. For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be <a href="/wiki/Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a>. Thus an expression represents an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> over constants and free variables and whose output is the resulting value of the expression.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>For a non-formalized language, that is, in most mathematical texts outside of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, for an individual expression it is not always possible to identify which variables are free and bound. For example, in <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 _{i<k}a_{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i<k}a_{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfead271ac3b0eb595afb83c5d6123b767a72d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.662ex; height:3.009ex;" alt="{\textstyle \sum _{i<k}a_{ik}}"></span>, depending on the context, the variable <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f327332497b21a059c479e7b2ce098baa1a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\textstyle i}"></span> can be free 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="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"></span> bound, or vice-versa, but they cannot both be free. Determining which value is assumed to be free depends on context and <a href="/wiki/Semantics_of_logic" title="Semantics of logic">semantics</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equivalence">Equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=6" title="Edit section: Equivalence"><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/Identity_(mathematics)" title="Identity (mathematics)">Identity (mathematics)</a></div> <p>An expression is often used to define a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, or denote <a href="/wiki/Function_composition" title="Function composition">compositions</a> of funtions, by taking the variables to be <a href="/wiki/Argument_of_a_function" title="Argument of a function">arguments</a>, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression.<sup id="cite_ref-Codd19702_26-0" class="reference"><a href="#cite_note-Codd19702-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> 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 x\mapsto x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8840cd9ef722e99165d3d2456a8f26b62dc98ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle x\mapsto x^{2}+1}"></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 f(x)=x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060789c792de89c492dc74c0f3e74edd0cbf87c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.903ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+1}"></span> define the function that associates to each number its <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square</a> plus one. An expression with no variables would define a <a href="/wiki/Constant_function" title="Constant function">constant function</a>. In this way, two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. <sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The equivalence between two expressions is called an <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a> and is sometimes denoted with <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 \equiv .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc7f5b0b842a98b4f618a19d8760648568c338b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:1.676ex;" alt="{\displaystyle \equiv .}"></span> </p><p>For example, in 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 \sum _{n=1}^{3}(2nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{3}(2nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ed6c96512896854b698debfc20880e6f93f57d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.116ex; height:3.509ex;" alt="{\textstyle \sum _{n=1}^{3}(2nx),}"></span> the variable <span class="texhtml"><i>n</i></span> is bound, and the variable <span class="texhtml"><i>x</i></span> is free. This expression is equivalent to the simpler expression <span class="texhtml">12 <i>x</i></span>; that is <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 \sum _{n=1}^{3}(2nx)\equiv 12x.}"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>12</mn> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{3}(2nx)\equiv 12x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9973f385d27db4c9b779b2a134c2747ed77637b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.451ex; height:7.176ex;" alt="{\displaystyle \sum _{n=1}^{3}(2nx)\equiv 12x.}"></span> The value for <span class="texhtml"><i>x</i> = 3</span> is 36, which can be denoted <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 \sum _{n=1}^{3}(2nx){\Big |}_{x=3}=36.}"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>36.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{3}(2nx){\Big |}_{x=3}=36.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40bc8094293bd91ad80c5eb86c86a4208740c81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.041ex; height:7.176ex;" alt="{\displaystyle \sum _{n=1}^{3}(2nx){\Big |}_{x=3}=36.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Polynomial_evaluation">Polynomial evaluation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=7" title="Edit section: Polynomial evaluation"><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/Polynomial_evaluation" title="Polynomial evaluation">Polynomial evaluation</a></div> <p>A polynomial consists of variables and <a href="/wiki/Coefficient" title="Coefficient">coefficients</a>, that involve only the operations of <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> to <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integer</a> powers, and has a finite number of terms. The problem of <a href="/wiki/Polynomial_evaluation" title="Polynomial evaluation">polynomial evaluation</a> arises frequently in practice. In <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>, polynomials are used to compute function approximations using <a href="/wiki/Taylor_polynomials" class="mw-redirect" title="Taylor polynomials">Taylor polynomials</a>. In <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> and <a href="/wiki/Hash_table" title="Hash table">hash tables</a>, polynomials are used to compute <a href="/wiki/K-independent_hashing" title="K-independent hashing"><i>k</i>-independent hashing</a>. </p><p>In the former case, polynomials are evaluated using <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>, which is not exact. Thus different schemes for the evaluation will, in general, give slightly different answers. In the latter case, the polynomials are usually evaluated in a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, in which case the answers are always exact. </p><p>For evaluating the <a href="/wiki/Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">univariate polynomial</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="{\textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db57513467585952ea3b3fcb6de006b9956a9a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.369ex; height:2.843ex;" alt="{\textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},}"></span> the most naive method would use <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> multiplications to compute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f506c33004e8dc037782a620e6fe94df16110e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.996ex; height:2.676ex;" alt="{\displaystyle a_{n}x^{n}}"></span>, use <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 n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960c88fa1831b7505d9672de66058532fa5d4053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\textstyle n-1}"></span> multiplications to compute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n-1}x^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n-1}x^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fb9264640577eeb57fe89f6c74e5278ddf46b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.197ex; height:3.009ex;" alt="{\displaystyle a_{n-1}x^{n-1}}"></span> and so on for a total 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="{\textstyle {\frac {n(n+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {n(n+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36688a2a9c1eb70db60203211854e8088cddd55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\textstyle {\frac {n(n+1)}{2}}}"></span> multiplications 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> additions. Using better methods, such as <a href="/wiki/Horner%27s_rule" class="mw-redirect" title="Horner's rule">Horner's rule</a>, this can be reduced 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="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> multiplications 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> additions. If some preprocessing is allowed, even more savings are possible. </p> <div class="mw-heading mw-heading3"><h3 id="Computation">Computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=8" title="Edit section: Computation"><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/Computation" title="Computation">Computation</a></div> <p>A <a href="/wiki/Computation" title="Computation">computation</a> is any type of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> or non-arithmetic <a href="/wiki/Calculation" title="Calculation">calculation</a> that is "well-defined".<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the <a href="/wiki/1600s_(decade)" title="1600s (decade)">1600s</a>,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> but agreement on a suitable definition proved elusive.<sup id="cite_ref-Davis_Davis_2000_31-0" class="reference"><a href="#cite_note-Davis_Davis_2000-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> A candidate definition was proposed independently by several mathematicians in the 1930s.<sup id="cite_ref-Davis_32-0" class="reference"><a href="#cite_note-Davis-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The best-known variant was formalised by the mathematician <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Couldn't find string "well-defined" in the paper. (October 2024)">page needed</span></a></i>]</sup> Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed <a href="/wiki/Equations" class="mw-redirect" title="Equations">algebraic statements</a>, and all statements written in modern computer programming languages.<sup id="cite_ref-Davis_Davis_2000_p._34-0" class="reference"><a href="#cite_note-Davis_Davis_2000_p.-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes <a href="/wiki/The_halting_problem" class="mw-redirect" title="The halting problem">the halting problem</a> and <a href="/wiki/Busy_beaver" title="Busy beaver">the busy beaver game</a>. It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> All statements characterised in modern programming languages are well-defined, including <a href="/wiki/C%2B%2B" title="C++">C++</a>, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>, and <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>.<sup id="cite_ref-Davis_Davis_2000_p._34-1" class="reference"><a href="#cite_note-Davis_Davis_2000_p.-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Common examples of computation are basic <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> and the <a href="/wiki/Execution_(computing)" title="Execution (computing)">execution</a> of computer <a href="/wiki/Algorithms" class="mw-redirect" title="Algorithms">algorithms</a>. A <a href="/wiki/Calculation" title="Calculation">calculation</a> is a deliberate mathematical process that transforms one or more inputs into one or more outputs or <i>results</i>. For example, <a href="/wiki/Multiplying" class="mw-redirect" title="Multiplying">multiplying</a> 7 by 6 is a simple algorithmic calculation. Extracting the <a href="/wiki/Square_root" title="Square root">square root</a> or the <a href="/wiki/Cube_root" title="Cube root">cube root</a> of a number using mathematical models is a more complex algorithmic calculation. </p> <div class="mw-heading mw-heading4"><h4 id="Rewriting">Rewriting</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=9" title="Edit section: Rewriting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Expressions can be computed by means of an <i><a href="/wiki/Evaluation_strategy" title="Evaluation strategy">evaluation strategy</a>.</i><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> To illustrate, executing a function call <code>f(a,b)</code> may first evaluate the arguments <code>a</code> and <code>b</code>, store the results in <a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">references</a> or memory locations <code>ref_a</code> and <code>ref_b</code>, then evaluate the function's body with those references passed in. This gives the function the ability to look up the original argument values passed in through dereferencing the parameters (some languages use specific operators to perform this), to modify them via <a href="/wiki/Assignment_(computer_science)" title="Assignment (computer science)">assignment</a> as if they were local variables, and to return values via the references. This is the call-by-reference evaluation strategy.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Evaluation strategy is part of the semantics of the programming language definition. Some languages, such as <a href="/wiki/PureScript" title="PureScript">PureScript</a>, have variants with different evaluation strategies. Some <a href="/wiki/Declarative_language" class="mw-redirect" title="Declarative language">declarative languages</a>, such as <a href="/wiki/Datalog" title="Datalog">Datalog</a>, support multiple evaluation strategies. Some languages define a <a href="/wiki/Calling_convention" title="Calling convention">calling convention</a>. </p><p>In <a href="/wiki/Rewriting" title="Rewriting">rewriting</a>, a <a href="/wiki/Reduction_strategy" title="Reduction strategy">reduction strategy</a> or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. A rewriting strategy specifies, out of all the reducible subterms (<a href="/wiki/Redex" class="mw-redirect" title="Redex">redexes</a>), which one should be reduced (<i>contracted</i>) within a term. One of the most common systems involves <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Well-defined_expressions">Well-defined expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=10" title="Edit section: Well-defined expressions"><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/Well-defined_expression" title="Well-defined expression">Well-defined expression</a></div> <p>The <a href="/wiki/Language_of_mathematics" title="Language of mathematics">language of mathematics</a> exhibits a kind of <a href="/wiki/Grammar" title="Grammar">grammar</a> (called <a href="/wiki/Formal_grammar" title="Formal grammar">formal grammar</a>) about how expressions may be written. There are two considerations for well-definedness of mathematical expressions, <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a> and <a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">semantics</a>. Syntax is concerned with the rules used for constructing, or transforming the symbols of an expression without regard to any <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> or <a href="/wiki/Meaning_(linguistics)" class="mw-redirect" title="Meaning (linguistics)">meaning</a> given to them. Expressions that are syntactically correct are called <a href="/wiki/Well-formedness" title="Well-formedness">well-formed</a>. Semantics is concerned with the meaning of these well-formed expressions. Expressions that are semantically correct are called <a href="/wiki/Well-defined_expression" title="Well-defined expression">well-defined</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Well-formed">Well-formed</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=11" title="Edit section: Well-formed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The syntax of mathematical expressions can be described somewhat informally as follows: the allowed <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operators</a> must have the correct number of inputs in the correct places (usually written with <a href="/wiki/Infix_notation" title="Infix notation">infix notation</a>), the sub-expressions that make up these inputs must be well-formed themselves, have a clear <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a>, etc. Strings of symbols that conform to the rules of syntax are called <a href="/wiki/Well-formedness" title="Well-formedness"><i>well-formed</i></a>, and those that are not well-formed are called, <i>ill-formed</i>, and are do not constitute mathematical expressions.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>For example, in <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, the expression <i>1 + 2 × 3</i> is well-formed, but </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 \times 4)x+,/y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times 4)x+,/y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f457f81fa524c195ef553b23529efcf5f0e3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.365ex; height:2.843ex;" alt="{\displaystyle \times 4)x+,/y}"></span>.</dd></dl> <p>is not. </p><p>However, being well-formed is not enough to be considered well-defined. For example in arithmetic, 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 {\frac {1}{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>0</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35722de5e8f4391bce55f8fb984325a122a8341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{0}}}"></span> is well-formed, but it is not well-defined. (See <a href="/wiki/Division_by_zero" title="Division by zero">Division by zero</a>). Such expressions are called <a href="/wiki/Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Well-defined">Well-defined</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=12" title="Edit section: Well-defined"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Semantics" title="Semantics">Semantics</a> is the study of meaning. <a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics</a> is about attaching meaning to expressions. An expression that defines a unique <a href="/wiki/Value_(mathematics)" title="Value (mathematics)">value</a> or meaning is said to be <a href="/wiki/Well-defined_expression" title="Well-defined expression">well-defined</a>. Otherwise, the expression is said to be ill defined or ambiguous.<b><sup id="cite_ref-MathWorld2_40-0" class="reference"><a href="#cite_note-MathWorld2-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup></b> In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an <a href="/wiki/Equation" title="Equation">equation</a> that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving 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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" 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 \oplus }"></span> to designate an internal <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a>. </p><p>In <a href="/wiki/Algebra" title="Algebra">algebra</a>, an expression may be used to designate a value, which might depend on values assigned to <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> occurring in the expression. The determination of this value depends on the <a href="/wiki/Semantics" title="Semantics">semantics</a> attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression <i>1 + 2 × 3</i> can have different values (mathematically 7, but also 9), depending on the <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> implied by the context (See also <a href="/wiki/Order_of_operations#Calculators" title="Order of operations">Operations § Calculators</a>). </p><p>For <a href="/wiki/Real_number" title="Real number">real numbers</a>, the product <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\times b\times 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\times b\times c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8dc3b01443c2edcec21c58d0c3bcca2ea99ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.915ex; height:2.176ex;" alt="{\displaystyle a\times b\times c}"></span> is unambiguous because <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\times b)\times c=a\times (b\times c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>×</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\times b)\times c=a\times (b\times c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74ad4422b4ae5d85f956edfe3696c5a07311c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a\times b)\times c=a\times (b\times c)}"></span>; hence the notation is said to be <i>well defined</i>.<sup id="cite_ref-MathWorld_41-0" class="reference"><a href="#cite_note-MathWorld-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> This property, also known as <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted. The <a href="/wiki/Subtraction" title="Subtraction">subtraction</a> operation is non-associative; despite that, there is a convention 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 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/f313f7e025e1cb202bdf127abf99decec5d0d954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.915ex; height:2.343ex;" alt="{\displaystyle a-b-c}"></span> is shorthand for <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"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <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/661ed8f03a576cb502ef97f6b6b4a71ff8ddec0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle (a-b)-c}"></span>, thus it is considered "well-defined". On the other hand, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a> is non-associative, and in the case 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 a/b/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <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/1461068682a3429e7b15ae9fe65b230f7a8bd20d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.559ex; height:2.843ex;" alt="{\displaystyle a/b/c}"></span>, parenthesization conventions are not well established; therefore, this expression is often considered ill-defined. </p><p>Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of <a href="/wiki/Operator_precedence" class="mw-redirect" title="Operator precedence">precedence</a>, associativity of the operator). For example, in the programming language <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a>, the operator <code>-</code> for subtraction is <i>left-to-right-associative</i>, which means that <code>a-b-c</code> is defined as <code>(a-b)-c</code>, and the operator <code>=</code> for assignment is <i>right-to-left-associative</i>, which means that <code>a=b=c</code> is defined as <code>a=(b=c)</code>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In the programming language <a href="/wiki/APL_(programming_language)" title="APL (programming language)">APL</a> there is only one rule: from <a href="/wiki/APL_(programming_language)#Design" title="APL (programming language)">right to left</a> – but parentheses first. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=13" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term 'expression' is part of the <a href="/wiki/Language_of_mathematics" title="Language of mathematics">language of mathematics</a>, that is to say, it is not defined <i>within</i> mathematics, but taken as a <a href="/wiki/Primitive_notion" title="Primitive notion">primitive</a> part of the language. To attempt to define the term would not be doing mathematics, but rather, one would be engaging in a kind of <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a> (the <a href="/wiki/Metalanguage" title="Metalanguage">metalanguage</a> of mathematics), usually <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>. Within mathematical logic, mathematics is usually described as a kind of <a href="/wiki/Formal_language" title="Formal language">formal language</a>, and a well-formed expression can be <a href="/wiki/Recursive_definition" title="Recursive definition">defined recursively</a> as follows:<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">alphabet</a> consists of: </p> <ul><li>A set of individual <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>: Symbols representing fixed <a href="/wiki/Mathematical_object" title="Mathematical object">objects</a> in the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a>, such as <a href="/wiki/Numeral_system" title="Numeral system">numerals</a> (1, 2.5, 1/7, ...), <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</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 \varnothing ,\{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing ,\{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646a617656abbe96621e49c0ae2eeec8f35df555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.722ex; height:2.843ex;" alt="{\displaystyle \varnothing ,\{1,2,3\}}"></span>, ...), <a href="/wiki/Truth_values" class="mw-redirect" title="Truth values">truth values</a> (T or F), etc.</li></ul> <ul><li>A set of individual variables: A <a href="/wiki/Countable_set" title="Countable set">countably infinite</a> amount of symbols representing <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> used for representing an unspecified object in the domain. (Usually letters like <span class="texhtml mvar" style="font-style:italic;">x</span>, or <span class="texhtml mvar" style="font-style:italic;">y</span>)</li></ul> <ul><li>A set of operations: <a href="/wiki/Function_symbols" class="mw-redirect" title="Function symbols">Function symbols</a> representing <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> that can be performed on elements over the domain, like addition (+), multiplication (×), or set operations like union (∪), or intersection (∩). (Functions can be understood as <a href="/wiki/Unary_operations" class="mw-redirect" title="Unary operations">unary operations</a>)</li></ul> <ul><li>Brackets ( )</li></ul> <p>With this alphabet, the recursive rules for forming a well-formed expression (WFE) are as follows: </p> <ul><li>Any constant or variable as defined are the <a href="/wiki/Atomic_formula" title="Atomic formula">atomic expressions</a>, the simplest well-formed expressions (WFE's). For instance, the constant <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span> or the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> are syntactically correct expressions.</li></ul> <ul><li>Let <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> be a <a href="/wiki/Metavariable" title="Metavariable">metavariable</a> for any <a href="/wiki/N-ary_operation" class="mw-redirect" title="N-ary operation">n-ary operation</a> over the domain, and let <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 \phi _{1},\phi _{2},...\phi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{1},\phi _{2},...\phi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287a7ecf7f89f5d9dad8b203e56747cc1c709dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.653ex; height:2.509ex;" alt="{\displaystyle \phi _{1},\phi _{2},...\phi _{n}}"></span> be metavariables for any WFE's.</li></ul> <dl><dd>Then <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(\phi _{1},\phi _{2},...\phi _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\phi _{1},\phi _{2},...\phi _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e847c0d2c59d08cbfab12a632c67313ddee95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.203ex; height:2.843ex;" alt="{\displaystyle F(\phi _{1},\phi _{2},...\phi _{n})}"></span> is also well-formed. For the most often used operations, more convenient notations (like <a href="/wiki/Infix_notation" title="Infix notation">infix notation</a>) have been developed over the centuries.</dd> <dd>For instance, if the domain of discourse is the <a href="/wiki/Real_number" title="Real number">real numbers</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> can denote the <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> +, then <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 \phi _{1}+\phi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{1}+\phi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666e6cfbe03d79725e123f3eb1ec556efd9879bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.72ex; height:2.509ex;" alt="{\displaystyle \phi _{1}+\phi _{2}}"></span> is well-formed. Or <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> can be the unary operation <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 \surd }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">√</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \surd }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9d637675e4ee00572431a0e42fa556901a4ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.046ex; width:1.982ex; height:2.843ex;" alt="{\displaystyle \surd }"></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 {\sqrt {\phi _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\phi _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69580a36e294843b91c918be662c5bf15f4e2456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.763ex; height:3.509ex;" alt="{\displaystyle {\sqrt {\phi _{1}}}}"></span> is well-formed.</dd> <dd>Brackets are initially around each non-atomic expression, but they can be deleted in cases where there is a defined <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a>, or where order doesn't matter (i.e. where operations are <a href="/wiki/Associative_property" title="Associative property">associative</a>).</dd></dl> <p>A well-formed expression can be thought as a <a href="/wiki/Abstract_syntax_tree" title="Abstract syntax tree">syntax tree</a>.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Node_(computer_science)" title="Node (computer science)">leaf nodes</a> are always atomic expressions. Operations <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> 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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"></span> have exactly two child nodes, while operations <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 {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f01419d50f8331ed8f948d3b0dce7d5bd75950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.266ex; height:2.843ex;" alt="{\textstyle {\sqrt {x}}}"></span>, <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 {\text{ln}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ln</mtext> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\text{ln}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4c47e83bad0065cc133ee406ef49e83fc2ddd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\textstyle {\text{ln}}(x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ce97a0649139023076f33481001a29d8d4ea4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.636ex; height:3.843ex;" alt="{\textstyle {\frac {d}{dx}}}"></span> have exactly one. There are countably infinitely many WFE's, however, each WFE has a finite number of nodes. </p> <div class="mw-heading mw-heading3"><h3 id="Lambda_calculus">Lambda calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=14" title="Edit section: Lambda calculus"><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/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></div> <p>Formal languages allow <a href="/wiki/Formal_system" title="Formal system">formalizing</a> the concept of well-formed expressions. </p><p>In the 1930s, a new type of expressions, called <a href="/wiki/Lambda_calculus#Definition" title="Lambda calculus">lambda expressions</a>, were introduced by <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> and <a href="/wiki/Stephen_Kleene" class="mw-redirect" title="Stephen Kleene">Stephen Kleene</a> for formalizing <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> and their evaluation. <sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> They form the basis for <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>, a <a href="/wiki/Formal_system" title="Formal system">formal system</a> used in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and the <a href="/wiki/Programming_language_theory" title="Programming language theory">theory of programming languages</a>. </p><p>The equivalence of two lambda expressions is <a href="/wiki/Decision_problem" title="Decision problem">undecidable</a>. This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (<a href="/wiki/Richardson%27s_theorem" title="Richardson's theorem">Richardson's theorem</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_expressions">Types of expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=15" title="Edit section: Types of expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algebraic_expression">Algebraic expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=16" title="Edit section: Algebraic expression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i><a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expression</a></i> is an expression built up from <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic constants</a>, <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, and the <a href="/wiki/Algebraic_operation" title="Algebraic operation">algebraic operations</a> (<a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> by a <a href="/wiki/Rational_number" title="Rational number">rational number</a>).<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> For example, <span class="texhtml">3<i>x</i><sup>2</sup> − 2<i>xy</i> + <i>c</i></span> is an algebraic expression. Since taking the <a href="/wiki/Square_root" title="Square root">square root</a> is the same as raising to the power <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .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><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, the following is also an algebraic expression: </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 {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad9f581ad795fb737a643d9520f087858110527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.547ex; height:7.676ex;" alt="{\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}"></span></dd></dl> <p>See also: <a href="/wiki/Algebraic_equation" title="Algebraic equation">Algebraic equation</a> and <a href="/wiki/Algebraic_closure" title="Algebraic closure">Algebraic closure</a> </p> <div class="mw-heading mw-heading3"><h3 id="Polynomial_expression">Polynomial expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=17" title="Edit section: Polynomial expression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Polynomial_expression" class="mw-redirect" title="Polynomial expression">polynomial expression</a> is an expression built with <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> (numbers of elements of some field), <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminates</a>, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers; 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(x+1)^{2}-xy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3(x+1)^{2}-xy.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b080ce8b76f38445dda813fbb7203f8f9d3ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.331ex; height:3.176ex;" alt="{\displaystyle 3(x+1)^{2}-xy.}"></span> </p><p>Using <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a>, <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> and <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>, every polynomial expression is equivalent to a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, that is an expression that is a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of products of integer powers of the indeterminates. For example the above polynomial expression is equivalent (denote the same polynomial as <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 3x^{2}-xy+6x+3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}-xy+6x+3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a15c08017c1b68b086bcec78ef16048c22afc1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.854ex; height:3.009ex;" alt="{\displaystyle 3x^{2}-xy+6x+3.}"></span> </p><p>Many author do not distinguish polynomials and polynomial expressions. In this case the expression of a polynomial expression as a linear combination is called the <i>canonical form</i>, <i>normal form</i>, or <i>expanded form</i> of the polynomial. </p> <div class="mw-heading mw-heading3"><h3 id="Computational_expression">Computational expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=18" title="Edit section: Computational expression"><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/Expression_(computer_science)" title="Expression (computer science)">Expression (computer science)</a></div> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, an <i>expression</i> is a <a href="/wiki/Syntax_(programming_languages)" title="Syntax (programming languages)">syntactic</a> entity in a <a href="/wiki/Programming_language" title="Programming language">programming language</a> that may be evaluated to determine its <a href="/wiki/Value_(computer_science)" title="Value (computer science)">value</a><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> or fail to terminate, in which case the expression is undefined.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> It is a combination of one or more <a href="/wiki/Constant_(programming)" class="mw-redirect" title="Constant (programming)">constants</a>, <a href="/wiki/Variable_(programming)" class="mw-redirect" title="Variable (programming)">variables</a>, <a href="/wiki/Function_(programming)" class="mw-redirect" title="Function (programming)">functions</a>, and <a href="/wiki/Operator_(programming)" class="mw-redirect" title="Operator (programming)">operators</a> that the programming language interprets (according to its particular <a href="/wiki/Order_of_operations" title="Order of operations">rules of precedence</a> and of <a href="/wiki/Associative_property" title="Associative property">association</a>) and computes to produce ("to return", in a <a href="/wiki/State_(computer_science)" title="State (computer science)">stateful</a> environment) another value. This process, for mathematical expressions, is called <i>evaluation</i>. In simple settings, the <a href="/wiki/Return_type" title="Return type">resulting value</a> is usually one of various <a href="/wiki/Primitive_data_type" title="Primitive data type">primitive types</a>, such as <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a>, <a href="/wiki/Boolean_expression" title="Boolean expression">Boolean</a>, or numerical (such as <a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">integer</a>, <a href="/wiki/Floating-point_number" class="mw-redirect" title="Floating-point number">floating-point</a>, or <a href="/wiki/Complex_data_type" title="Complex data type">complex</a>). </p><p>In <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a>, formulas are viewed as expressions that can be evaluated as a Boolean, depending on the values that are given to the variables occurring in the expressions. 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 8x-5\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8x-5\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c798bbe752dacb43f38bc6c5d12da206ce867b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.756ex; height:2.343ex;" alt="{\displaystyle 8x-5\geq 3}"></span> takes the value <i>false</i> if <span class="texhtml mvar" style="font-style:italic;">x</span> is given a value less than 1, and the value <i>true</i> otherwise. </p><p>Expressions are often contrasted with <a href="/wiki/Statement_(computer_science)" title="Statement (computer science)">statements</a>—syntactic entities that have no value (an instruction). </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cassidy.1985.015.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/400px-Cassidy.1985.015.gif" decoding="async" width="400" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/600px-Cassidy.1985.015.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/800px-Cassidy.1985.015.gif 2x" data-file-width="1087" data-file-height="357" /></a><figcaption>Representation of the expression <span class="texhtml">(8 − 6) × (3 + 1)</span> as a <a href="/wiki/Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a> tree, from a 1985 Master's Thesis<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Except for <a href="/wiki/Number" title="Number">numbers</a> and <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, every mathematical expression may be viewed as the symbol of an operator followed by a <a href="/wiki/Sequence" title="Sequence">sequence</a> of operands. In computer algebra software, the expressions are usually represented in this way. This representation is very flexible, and many things that seem not to be mathematical expressions at first glance, may be represented and manipulated as such. For example, an equation is an expression with "=" as an operator, a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> may be represented as an expression with "matrix" as an operator and its rows as operands. </p><p>See: <a href="/wiki/Computer_algebra#Expressions" title="Computer algebra">Computer algebra expression</a> </p> <div class="mw-heading mw-heading3"><h3 id="Logical_expression">Logical expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=19" title="Edit section: Logical expression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, a <i>"logical expression"</i> can refer to either <a href="/wiki/Term_(logic)" title="Term (logic)">terms</a> or <a href="/wiki/Well-formed_formula#Predicate_logic" title="Well-formed formula">formulas</a>. A term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. </p><p>A <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> term is <a href="/wiki/Recursive_definition" title="Recursive definition">recursively constructed</a> from constant symbols, variables, and <a href="/wiki/Function_symbol_(logic)" class="mw-redirect" title="Function symbol (logic)">function symbols</a>. An expression formed by applying a <a href="/wiki/Predicate_(logic)" class="mw-redirect" title="Predicate (logic)">predicate symbol</a> to an appropriate number of terms is called an <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formula</a>, which evaluates to <a href="/wiki/Truth#Truth_in_mathematics" title="Truth">true</a> or <a href="/wiki/False_(logic)" title="False (logic)">false</a> in <a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">bivalent logics</a>, given an <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a>. For example, <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 (x+1)*(x+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)*(x+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9735d976c75be5ae6e180644f2609ca5150092a" 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 (x+1)*(x+1)}"></span>⁠</span> is a term built from the constant 1, the variable <span class="texhtml mvar" style="font-style:italic;">x</span>, and the binary function symbols <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 +}"> <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>⁠</span> and <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 *}"> <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/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span>⁠</span>; it is part of the atomic formula <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 (x+1)*(x+1)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)*(x+1)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5211c9623e8567fc98f2d2b7dc93e2bf5929744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.739ex; height:2.843ex;" alt="{\displaystyle (x+1)*(x+1)\geq 0}"></span>⁠</span> which evaluates to true for each <a href="/wiki/Real_number" title="Real number">real-numbered</a> value of <span class="texhtml mvar" style="font-style:italic;">x</span>. </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=Expression_(mathematics)&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Analytic_expression" class="mw-redirect" title="Analytic expression">Analytic expression</a></li> <li><a href="/wiki/Closed-form_expression" title="Closed-form expression">Closed-form expression</a></li> <li><a href="/wiki/Formal_calculation" title="Formal calculation">Formal calculation</a></li> <li><a href="/wiki/Functional_programming" title="Functional programming">Functional programming</a></li> <li><a href="/wiki/Infinite_expression" title="Infinite expression">Infinite expression</a></li> <li><a href="/wiki/Number_sentence" title="Number sentence">Number sentence</a></li> <li><a href="/wiki/Regular_expression" title="Regular expression">Regular expression</a></li> <li><a href="/wiki/Rewriting" title="Rewriting">Rewriting</a></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature (logic)</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=21" title="Edit section: Notes"><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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">The study of non-computable statements is the field of <a href="/wiki/Hypercomputation" title="Hypercomputation">hypercomputation</a>.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">For a full history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006).</span> </li> </ol></div></div> <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=Expression_(mathematics)&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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"><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, s.v. “<a href="//doi.org/10.1093/OED/4555505636" class="extiw" title="doi:10.1093/OED/4555505636">Expression (n.), sense II.7</a>,” "<i>A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function.</i>"</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"><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 id="CITEREFStoll1963" class="citation book cs1">Stoll, Robert R. (1963). <i>Set Theory and Logic</i>. San Francisco, CA: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.place=San+Francisco%2C+CA&rft.pub=Dover+Publications&rft.date=1963&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary,</a> s.v. "<a href="//doi.org/10.1093/OED/3423541985" class="extiw" title="doi:10.1093/OED/3423541985">Evaluate (v.), sense a</a>", "<i>Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).</i>"</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"><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, s.v. “<a href="//doi.org/10.1093/OED/1018661347" class="extiw" title="doi:10.1093/OED/1018661347">Simplify (v.), sense 4.a</a>”, "<i>To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.</i>"</span> </li> <li id="cite_note-Codd1970-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Codd1970_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCodd1970" class="citation journal cs1"><a href="/wiki/Edgar_F._Codd" title="Edgar F. Codd">Codd, Edgar Frank</a> (June 1970). <a rel="nofollow" class="external text" href="https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf">"A Relational Model of Data for Large Shared Data Banks"</a> <span class="cs1-format">(PDF)</span>. <i>Communications of the ACM</i>. <b>13</b> (6): 377–387. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F362384.362685">10.1145/362384.362685</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207549016">207549016</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2004-09-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-04-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=A+Relational+Model+of+Data+for+Large+Shared+Data+Banks&rft.volume=13&rft.issue=6&rft.pages=377-387&rft.date=1970-06&rft_id=info%3Adoi%2F10.1145%2F362384.362685&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207549016%23id-name%3DS2CID&rft.aulast=Codd&rft.aufirst=Edgar+Frank&rft_id=https%3A%2F%2Fwww.seas.upenn.edu%2F~zives%2F03f%2Fcis550%2Fcodd.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCoy1960" class="citation book cs1">McCoy, Neal H. (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=%22purely+formal+expression%22"><i>Introduction To Modern Algebra</i></a>. Boston: <a href="/wiki/Allyn_%26_Bacon" title="Allyn & Bacon">Allyn & Bacon</a>. p. 127. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/68015225">68015225</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Modern+Algebra&rft.place=Boston&rft.pages=127&rft.pub=Allyn+%26+Bacon&rft.date=1960&rft_id=info%3Alccn%2F68015225&rft.aulast=McCoy&rft.aufirst=Neal+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontomo00mcco%2Fpage%2F126%2Fmode%2F2up%3Fq%3D%2522purely%2Bformal%2Bexpression%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraleigh2003" class="citation book cs1">Fraleigh, John B. (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinabs07edfral/page/198/mode/2up?q=%22formal+sum%22"><i>A first course in abstract algebra</i></a>. Boston : Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-76390-4" title="Special:BookSources/978-0-201-76390-4"><bdi>978-0-201-76390-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+abstract+algebra&rft.pub=Boston+%3A+Addison-Wesley&rft.date=2003&rft.isbn=978-0-201-76390-4&rft.aulast=Fraleigh&rft.aufirst=John+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffirstcourseinabs07edfral%2Fpage%2F198%2Fmode%2F2up%3Fq%3D%2522formal%2Bsum%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-Marshack2-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Marshack2_8-0">^</a></b></span> <span class="reference-text">Marshack, Alexander (1991). <i>The Roots of Civilization</i>, Colonial Hill, Mount Kisco, NY.</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">Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hrRPAAAAMAAJ&pg=PA314">314</a></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">Mathematical Excursion, Enhanced Edition: Enhanced Webassign Edition By Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Cleg. Pg <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GTgTnSGMukgC&pg=PA186">186</a></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">Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AKJZvXOS7n4C&pg=PA14">14</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html">Diophantine Equations</a>. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Boyer (1991). "Revival and Decline of Greek Mathematics". pp. 180-182. "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with approximate solutions of determinate equations as far as the third degree, the Arithmetica of Diophantus (such as we have it) is almost entirely devoted to the exact solution of equations, both determinate and indeterminate. [...] Throughout the six surviving books of Arithmetica there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ {\displaystyle \zeta } (perhaps for the last letter of arithmos). [...] It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."</span> </li> <li id="cite_note-Boyer-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer_14-0">^</a></b></span> <span class="reference-text">Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg <a href="//archive.org/details/bub_gb_7DDQAAAAMAAJ/page/n472" class="extiw" title="iarchive:bub gb 7DDQAAAAMAAJ/page/n472">456</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg <a href="//archive.org/details/bub_gb_7DDQAAAAMAAJ/page/n474" class="extiw" title="iarchive:bub gb 7DDQAAAAMAAJ/page/n474">458</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Banna.html">"al-Marrakushi ibn Al-Banna"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=al-Marrakushi+ibn+Al-Banna&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FAl-Banna.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-Gullberg2-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gullberg2_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGullberg1997" class="citation book cs1"><a href="/wiki/Jan_Gullberg" title="Jan Gullberg">Gullberg, Jan</a> (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull"><i>Mathematics: From the Birth of Numbers</i></a></span>. W. W. Norton. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull/page/298">298</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-04002-X" title="Special:BookSources/0-393-04002-X"><bdi>0-393-04002-X</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%3A+From+the+Birth+of+Numbers&rft.pages=298&rft.pub=W.+W.+Norton&rft.date=1997&rft.isbn=0-393-04002-X&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-Qalasadi2-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Qalasadi2_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Qalasadi.html">"Abu'l Hasan ibn Ali al Qalasadi"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abu%27l+Hasan+ibn+Ali+al+Qalasadi&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FAl-Qalasadi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=k0U1AQAAMAAJ">Der Algorismus proportionum des Nicolaus Oresme</a>: Zum ersten Male nach der Lesart der Handschrift R.40.2. der Königlichen Gymnasial-bibliothek zu Thorn. <a href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a>. S. Calvary & Company, 1868.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><i>Later <a href="/wiki/Early_modern" class="mw-redirect" title="Early modern">early modern</a> version</i>: <a href="//archive.org/details/anewsystemmerca04walsgoog" class="extiw" title="iarchive:anewsystemmerca04walsgoog">A New System of Mercantile Arithmetic</a>: Adapted to the Commerce of the United States, in Its Domestic and Foreign Relations with Forms of Accounts and Other Writings Usually Occurring in Trade. By <a href="/w/index.php?title=Michael_Walsh_(1801)&action=edit&redlink=1" class="new" title="Michael Walsh (1801) (page does not exist)">Michael Walsh</a>. <a href="/wiki/Edmund_M._Blunt" class="mw-redirect" title="Edmund M. Blunt">Edmund M. Blunt</a> (proprietor.), 1801.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="#CITEREFDescartes2006">Descartes 2006</a>, p.1xiii "This short work marks the moment at which algebra and geometry ceased being separate."</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarecekMathis2020" class="citation web cs1">Marecek, Lynn; Mathis, Andrea Honeycutt (2020-05-06). <a rel="nofollow" class="external text" href="https://openstax.org/books/intermediate-algebra-2e/pages/1-1-use-the-language-of-algebra#term-00018">"1.1 Use the Language of Algebra - Intermediate Algebra 2e | OpenStax"</a>. <i>openstax.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=openstax.org&rft.atitle=1.1+Use+the+Language+of+Algebra+-+Intermediate+Algebra+2e+%7C+OpenStax&rft.date=2020-05-06&rft.aulast=Marecek&rft.aufirst=Lynn&rft.au=Mathis%2C+Andrea+Honeycutt&rft_id=https%3A%2F%2Fopenstax.org%2Fbooks%2Fintermediate-algebra-2e%2Fpages%2F1-1-use-the-language-of-algebra%23term-00018&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC.C._ChangH._Jerome_Keisler1977" class="citation book cs1"><a href="/wiki/Chen_Chung_Chang" title="Chen Chung Chang">C.C. Chang</a>; <a href="/wiki/H._Jerome_Keisler" class="mw-redirect" title="H. Jerome Keisler">H. Jerome Keisler</a> (1977). <i>Model Theory</i>. Studies in Logic and the Foundation of Mathematics. Vol. 73. North Holland.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Model+Theory&rft.series=Studies+in+Logic+and+the+Foundation+of+Mathematics&rft.pub=North+Holland&rft.date=1977&rft.au=C.C.+Chang&rft.au=H.+Jerome+Keisler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span>; here: Sect.1.3</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Sobolev, S.K. (originator). <a rel="nofollow" class="external text" href="http://encyclopediaofmath.org/index.php?title=Free_variable&oldid=46988"><i>Free variable</i></a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1402006098" title="Special:BookSources/1402006098"><bdi>1402006098</bdi></a>.</span> </li> <li id="cite_note-Codd19702-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Codd19702_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCodd1970" class="citation journal cs1"><a href="/wiki/Edgar_F._Codd" title="Edgar F. Codd">Codd, Edgar Frank</a> (June 1970). <a rel="nofollow" class="external text" href="https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf">"A Relational Model of Data for Large Shared Data Banks"</a> <span class="cs1-format">(PDF)</span>. <i>Communications of the ACM</i>. <b>13</b> (6): 377–387. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F362384.362685">10.1145/362384.362685</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207549016">207549016</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2004-09-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-04-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=A+Relational+Model+of+Data+for+Large+Shared+Data+Banks&rft.volume=13&rft.issue=6&rft.pages=377-387&rft.date=1970-06&rft_id=info%3Adoi%2F10.1145%2F362384.362685&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207549016%23id-name%3DS2CID&rft.aulast=Codd&rft.aufirst=Edgar+Frank&rft_id=https%3A%2F%2Fwww.seas.upenn.edu%2F~zives%2F03f%2Fcis550%2Fcodd.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Equation. Encyclopedia of Mathematics. URL: <a rel="nofollow" class="external free" href="http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613">http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613</a></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: <a rel="nofollow" class="external free" href="https://plato.stanford.edu/entries/algebra/#Laws">https://plato.stanford.edu/entries/algebra/#Laws</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</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.merriam-webster.com/dictionary/computation">"Definition of COMPUTATION"</a>. <i>www.merriam-webster.com</i>. 2024-10-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-12</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.merriam-webster.com&rft.atitle=Definition+of+COMPUTATION&rft.date=2024-10-11&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fcomputation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCouturat1901" class="citation book cs1">Couturat, Louis (1901). <i>la Logique de Leibniz a'Après des Documents Inédits</i>. Paris. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0343895099" title="Special:BookSources/978-0343895099"><bdi>978-0343895099</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=la+Logique+de+Leibniz+a%27Apr%C3%A8s+des+Documents+In%C3%A9dits&rft.pub=Paris&rft.date=1901&rft.isbn=978-0343895099&rft.aulast=Couturat&rft.aufirst=Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-Davis_Davis_2000-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Davis_Davis_2000_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisDavis2000" class="citation book cs1">Davis, Martin; Davis, Martin D. (2000). <i>The Universal Computer</i>. W. W. 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Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61471-7" title="Special:BookSources/978-0-486-61471-7"><bdi>978-0-486-61471-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+%26+Unsolvability&rft.pub=Courier+Corporation&rft.date=1982-01-01&rft.isbn=978-0-486-61471-7&rft.aulast=Davis&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1937" class="citation news cs1">Turing, A.M. (1937) [Delivered to the Society November 1936]. <a rel="nofollow" class="external text" href="http://www.comlab.ox.ac.uk/activities/ieg/e-library/sources/tp2-ie.pdf">"On Computable Numbers, with an Application to the Entscheidungsproblem"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the London Mathematical Society</i>. 2. Vol. 42. pp. 230–65. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-42.1.230">10.1112/plms/s2-42.1.230</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=On+Computable+Numbers%2C+with+an+Application+to+the+Entscheidungsproblem&rft.volume=42&rft.pages=230-65&rft.date=1937&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-42.1.230&rft.aulast=Turing&rft.aufirst=A.M.&rft_id=http%3A%2F%2Fwww.comlab.ox.ac.uk%2Factivities%2Fieg%2Fe-library%2Fsources%2Ftp2-ie.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-Davis_Davis_2000_p.-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-Davis_Davis_2000_p._34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Davis_Davis_2000_p._34-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisDavis2000" class="citation book cs1">Davis, Martin; Davis, Martin D. (2000). <i>The Universal Computer</i>. W. W. Norton & Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04785-1" title="Special:BookSources/978-0-393-04785-1"><bdi>978-0-393-04785-1</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+Computer&rft.pub=W.+W.+Norton+%26+Company&rft.date=2000&rft.isbn=978-0-393-04785-1&rft.aulast=Davis&rft.aufirst=Martin&rft.au=Davis%2C+Martin+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis,_Martin2006" class="citation journal cs1">Davis, Martin (2006). 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Cambridge, MA: <a href="/wiki/MIT_Press" title="MIT Press">The MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0262062794" title="Special:BookSources/978-0262062794"><bdi>978-0262062794</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essentials+of+Programming+Languages&rft.place=Cambridge%2C+MA&rft.edition=third&rft.pub=The+MIT+Press&rft.date=2008&rft.isbn=978-0262062794&rft.au=Daniel+P.+Friedman&rft.au=Mitchell+Wand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1963" class="citation book cs1">Stoll, Robert R. (1963). <i>Set Theory and Logic</i>. San Francisco, CA: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.place=San+Francisco%2C+CA&rft.pub=Dover+Publications&rft.date=1963&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-MathWorld2-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-MathWorld2_40-0">^</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="http://mathworld.wolfram.com/Well-Defined.html">"Well-Defined"</a>. From MathWorld – A Wolfram Web Resource<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-01-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Well-Defined&rft.pub=From+MathWorld+%E2%80%93+A+Wolfram+Web+Resource&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FWell-Defined.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-MathWorld-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-MathWorld_41-0">^</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="http://mathworld.wolfram.com/Well-Defined.html">"Well-Defined"</a>. From MathWorld – A Wolfram Web Resource<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-01-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Well-Defined&rft.pub=From+MathWorld+%E2%80%93+A+Wolfram+Web+Resource&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FWell-Defined.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</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.geeksforgeeks.org/operator-precedence-and-associativity-in-c/">"Operator Precedence and Associativity in C"</a>. <i>GeeksforGeeks</i>. 2014-02-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-10-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GeeksforGeeks&rft.atitle=Operator+Precedence+and+Associativity+in+C&rft.date=2014-02-07&rft_id=https%3A%2F%2Fwww.geeksforgeeks.org%2Foperator-precedence-and-associativity-in-c%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC.C._ChangH._Jerome_Keisler1977" class="citation book cs1"><a href="/wiki/Chen_Chung_Chang" title="Chen Chung Chang">C.C. Chang</a>; <a href="/wiki/H._Jerome_Keisler" class="mw-redirect" title="H. Jerome Keisler">H. Jerome Keisler</a> (1977). <i>Model Theory</i>. Studies in Logic and the Foundation of Mathematics. Vol. 73. North Holland.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Model+Theory&rft.series=Studies+in+Logic+and+the+Foundation+of+Mathematics&rft.pub=North+Holland&rft.date=1977&rft.au=C.C.+Chang&rft.au=H.+Jerome+Keisler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span>; here: Sect.1.3</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermes1973" class="citation book cs1"><a href="/wiki/Hans_Hermes" title="Hans Hermes">Hermes, Hans</a> (1973). <i>Introduction to Mathematical Logic</i>. Springer London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540058192" title="Special:BookSources/3540058192"><bdi>3540058192</bdi></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/1431-4657">1431-4657</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.pub=Springer+London&rft.date=1973&rft.issn=1431-4657&rft.isbn=3540058192&rft.aulast=Hermes&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span>; here: Sect.II.1.3</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1932" class="citation journal cs1"><a href="/wiki/Alonzo_Church" title="Alonzo Church">Church, Alonzo</a> (1932). "A set of postulates for the foundation of logic". <i>Annals of Mathematics</i>. Series 2. <b>33</b> (2): 346–366. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968337">10.2307/1968337</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968337">1968337</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=A+set+of+postulates+for+the+foundation+of+logic&rft.volume=33&rft.issue=2&rft.pages=346-366&rft.date=1932&rft_id=info%3Adoi%2F10.2307%2F1968337&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968337%23id-name%3DJSTOR&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorris1992" class="citation book cs1">Morris, Christopher G. (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/academicpressdic00morr"><i>Academic Press dictionary of science and technology</i></a></span>. Gulf Professional Publishing. p. <a rel="nofollow" class="external text" href="https://archive.org/details/academicpressdic00morr/page/74">74</a>. <q>algebraic expression over a field.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Academic+Press+dictionary+of+science+and+technology&rft.pages=74&rft.pub=Gulf+Professional+Publishing&rft.date=1992&rft.aulast=Morris&rft.aufirst=Christopher+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Facademicpressdic00morr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="/wiki/John_C._Mitchell" title="John C. Mitchell">Mitchell, J.</a> (2002). Concepts in Programming Languages. Cambridge: Cambridge University Press, <i>3.4.1 Statements and Expressions</i>, p. 26</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Maurizio Gabbrielli, Simone Martini (2010). Programming Languages - Principles and Paradigms. Springer London, <i>6.1 Expressions</i>, p. 120</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCassidy1985" class="citation thesis cs1">Cassidy, Kevin G. (Dec 1985). <a class="external text" href="https://commons.wikimedia.org/wiki/File:The_feasibility_of_automatic_storage_reclamation_with_concurrent_program_execution_in_a_LISP_environment._(IA_feasibilityofaut00cass).pdf"><i>The Feasibility of Automatic Storage Reclamation with Concurrent Program Execution in a LISP Environment</i></a> <span class="cs1-format">(PDF)</span> (Master's thesis). Naval Postgraduate School, Monterey/CA. p. 15. ADA165184.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=The+Feasibility+of+Automatic+Storage+Reclamation+with+Concurrent+Program+Execution+in+a+LISP+Environment&rft.inst=Naval+Postgraduate+School%2C+Monterey%2FCA&rft.date=1985-12&rft.aulast=Cassidy&rft.aufirst=Kevin+G.&rft_id=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AThe_feasibility_of_automatic_storage_reclamation_with_concurrent_program_execution_in_a_LISP_environment._%28IA_feasibilityofaut00cass%29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Works_Cited">Works Cited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expression_(mathematics)&action=edit&section=23" title="Edit section: Works Cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><p><cite id="CITEREFDescartes2006" class="citation book cs1">Descartes, René (2006) [1637]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9FOC5F6nVaAC&pg=PR60"><i>A discourse on the method of correctly conducting one's reason and seeking truth in the sciences</i></a>. Translated by Ian Maclean. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-282514-3" title="Special:BookSources/0-19-282514-3"><bdi>0-19-282514-3</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+discourse+on+the+method+of+correctly+conducting+one%27s+reason+and+seeking+truth+in+the+sciences&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=0-19-282514-3&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9FOC5F6nVaAC%26pg%3DPR60&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpression+%28mathematics%29" class="Z3988"></span></p><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</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/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</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/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></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/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</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/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</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/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></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/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</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/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a class="mw-selflink selflink">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group 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