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class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vinculum_(s%C3%ADmbolo)" title="Vinculum (símbolo) – Spanish" lang="es" hreflang="es" data-title="Vinculum (símbolo)" 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/Vinkulo_(superstreko)" title="Vinkulo (superstreko) – Esperanto" lang="eo" hreflang="eo" data-title="Vinkulo (superstreko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B4%84%EC%84%A0" title="괄선 – Korean" lang="ko" hreflang="ko" data-title="괄선" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95_%D7%9E%D7%97%D7%91%D7%A8" title="קו מחבר – Hebrew" lang="he" hreflang="he" data-title="קו מחבר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vinculum_(symbol)" title="Vinculum (symbol) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Vinculum (symbol)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vezna_%C4%8Drta" title="Vezna črta – Slovenian" lang="sl" hreflang="sl" data-title="Vezna črta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B1%D0%BE%D0%BB_%D0%BD%D0%B0%D0%B4%D0%B2%D1%83%D1%87%D0%B5%D0%BD%D0%BE_(%D1%86%D1%80%D1%82%D0%B0)" title="Симбол надвучено (црта) – Serbian" lang="sr" hreflang="sr" data-title="Симбол надвучено (црта)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%95%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81" title="தொகுப்புக்கோடு – Tamil" lang="ta" hreflang="ta" data-title="தொகுப்புக்கோடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Horizontal line used in mathematical notation</div> <div style="float:right; max-width:17em; text-align:center; margin:0 0 0.5em 0.5em;"> <div style="border:1px solid black"> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\rm {AB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\rm {AB}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541161142b0d23906353c32ec3a34951c5c82be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.504ex; height:3.009ex;" alt="{\displaystyle {\overline {\rm {AB}}}}"></span></div> <p><span style="font-size:85%;">line segment from A to B</span> </p> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">7</span></span> = 0.<span style="text-decoration:overline;">142857</span></div> <p><span style="font-size:85%; padding-left:0.2em; padding-right:0.2em;">repeated <span class="nowrap">0.1428571428571428571...</span></span> </p> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {a+bi}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a+bi}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c14ac2f619ca7b1df24d1d5c4c87fc311f1f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.985ex; height:3.176ex;" alt="{\displaystyle {\overline {a+bi}}}"></span></div> <p><span style="font-size:85%; padding-left:0.2em; padding-right:0.2em;">complex conjugate</span> </p> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y={\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y={\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8979b29c9c191650569d6d45b538c4e1ca08994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.494ex; height:3.009ex;" alt="{\displaystyle Y={\overline {AB}}}"></span></div> <p><span style="font-size:85%;">boolean NOT (A AND B)</span> </p> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><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[{n}]{ab+2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{ab+2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ff6c8ff44dc4d5b0d830d004b918b0107920f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{ab+2}}}"></span></div> <p><span style="font-size:85%;">radical ab + 2</span> </p> <div style="padding:0.5em 0 0.5em 0; font-size:big; border-top:1px solid lightgrey; line-height:125%;"><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-{\overline {b+c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-{\overline {b+c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3de24b4bd34ffd56871afb2b697f2a139203273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.03ex; height:3.176ex;" alt="{\displaystyle a-{\overline {b+c}}}"></span> = a − (b + c)</div> <p><span style="font-size:85%;">bracketing function</span> </p> </div> <p>Vinculum usage </p> </div> <p>A <b>vinculum</b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a> <i> <i lang="la"><a href="https://en.wiktionary.org/wiki/vinculum#Latin" class="extiw" title="wikt:vinculum">vinculum</a></i></i> 'fetter, chain, tie') is a horizontal line used in <a href="/wiki/Mathematical_notation" title="Mathematical notation">mathematical notation</a> for various purposes. It may be placed as an <i><a href="/wiki/Overline" title="Overline">overline</a></i> or <i><a href="/wiki/Underline" class="mw-redirect" title="Underline">underline</a></i> above or below a <a href="/wiki/Mathematical_expression" class="mw-redirect" title="Mathematical expression">mathematical expression</a> to group the expression's elements. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics its use for this purpose has almost entirely been replaced by the use of <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</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> It was also used to <a href="/wiki/Roman_numerals#Vinculum" title="Roman numerals">mark Roman numerals</a> whose values are multiplied by 1,000.<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> Today, however, the common usage of a vinculum to indicate the repetend of a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> is a significant exception and reflects the original usage. </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=Vinculum_(symbol)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vinculum, in its general use, was introduced by <a href="/wiki/Frans_van_Schooten" title="Frans van Schooten">Frans van Schooten</a> in 1646 as he edited the works of <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (who had himself not used this notation). However, earlier versions, such as using an underline as <a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Chuquet</a> did in 1484, or in limited form as <a href="/wiki/Renee_Descartes" class="mw-redirect" title="Renee Descartes">Descartes</a> did in 1637, using it only in relation to the radical sign, were common.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=2" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Modern">Modern</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=3" title="Edit section: Modern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vinculum can indicate a <a href="/wiki/Line_segment" title="Line segment">line segment</a> where <i>A</i> and <i>B</i> are the endpoints: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\rm {AB}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\rm {AB}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1226848d970de5f9ea2bf62b9549e3819f64bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.15ex; height:3.009ex;" alt="{\displaystyle {\overline {\rm {AB}}}.}"></span></li></ul> <p>A vinculum can indicate the repetend of a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> value: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">7</span></span> = 0.<span style="text-decoration:overline;">142857</span> = 0.1428571428571428571...</li></ul> <p>A vinculum can indicate the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of a <a href="/wiki/Complex_number" title="Complex number">complex number</a>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {2+3i}}=2-3i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mn>3</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {2+3i}}=2-3i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63927284e421d55d765cf746b291bf5963082514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.149ex; height:3.009ex;" alt="{\displaystyle {\overline {2+3i}}=2-3i}"></span></li></ul> <p>Logarithm of a number less than 1 can conveniently be represented using vinculum: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>=</mo> <mn>0.301</mn> <mo stretchy="false">⇒</mo> <mi>log</mi> <mo>⁡</mo> <mn>0.2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> <mn>.301</mn> <mo>=</mo> <mo>−</mo> <mn>0.699</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984e3ab4271c17039c5afd63aedff88f83b1d590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.574ex; height:3.176ex;" alt="{\displaystyle \log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699}"></span></li></ul> <p>In <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, a vinculum may be used to represent the operation of inversion (also known as the NOT function): </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y={\overline {AB}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y={\overline {AB}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391e9cdd5e1f27c70ce5cddf879243883e8d8d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.141ex; height:3.343ex;" alt="{\displaystyle Y={\overline {AB}},}"></span></li></ul> <p>meaning that Y is false only when both A and B are both true - or by extension, Y is true when either A or B is false. </p><p>Similarly, it is used to show the repeating terms in a <a href="/wiki/Periodic_continued_fraction" title="Periodic continued fraction">periodic continued fraction</a>. <a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational numbers</a> are the only numbers that have these. </p> <div class="mw-heading mw-heading3"><h3 id="Historical">Historical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=4" title="Edit section: Historical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formerly its main use was as a notation to indicate a group (a bracketing device serving the same function as parentheses): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-{\overline {b+c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-{\overline {b+c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f7bf3dfb55b4cabb177fff4eb5b4256a523b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.677ex; height:3.343ex;" alt="{\displaystyle a-{\overline {b+c}},}"></span></dd></dl> <p>meaning to add <i>b</i> and <i>c</i> first and then subtract the result from <i>a</i>, which would be written more commonly today as <span class="texhtml"><i>a</i> − (<i>b</i> + <i>c</i>)</span>. Parentheses, used for grouping, are only rarely found in the mathematical literature before the eighteenth century. The vinculum was used extensively, usually as an overline, but <a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Chuquet</a> in 1484 used the underline version.<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> </p><p>In India, the use of this notation is still tested in primary school.<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> </p> <div class="mw-heading mw-heading3"><h3 id="As_a_part_of_a_radical">As a part of a radical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=5" title="Edit section: As a part of a radical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vinculum is used as part of the notation of a <a href="/wiki/Radical_symbol" title="Radical symbol">radical</a> to indicate the <a href="/wiki/Radicand" class="mw-redirect" title="Radicand">radicand</a> whose <a href="/wiki/Nth_root" title="Nth root">root</a> is being indicated. In the following, the quantity <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 ab+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbbb6a0917eaa468990262294c668cae62f0e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.23ex; height:2.343ex;" alt="{\displaystyle ab+2}"></span> is the whole radicand, and thus has a vinculum over it: </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[{n}]{ab+2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{ab+2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe77255935a5f78a086d28515e50a228ce1b82f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.813ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{ab+2}}.}"></span></dd></dl> <p>In 1637 <a href="/wiki/Renee_Descartes" class="mw-redirect" title="Renee Descartes">Descartes</a> was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The symbol used to indicate a vinculum need not be a line segment (overline or underline); sometimes braces can be used (pointing either up or down).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Encodings">Encodings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=6" title="Edit section: Encodings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Overline#Implementations" title="Overline">Overline § Implementations</a></div> <div class="mw-heading mw-heading3"><h3 id="In_Unicode">In Unicode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=7" title="Edit section: In Unicode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+0305</span> </span><span style="font-size:125%;line-height:1em"><span style="font-family: serif;">◌̅</span></span> <span style="font-variant: small-caps; text-transform: lowercase;">COMBINING OVERLINE</span></li></ul> <div class="mw-heading mw-heading3"><h3 id="TeX">TeX</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=8" title="Edit section: TeX"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>, a text <text> can be overlined with <code>$\overline{\mbox{<text>}}$</code>. The inner <code>\mbox{}</code> is necessary to override the math-mode (here invoked by the dollar signs) which the <code>\overline{}</code> demands. </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=Vinculum_(symbol)&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Overline#Math_and_science" title="Overline">Overline § Math and science</a> similar-looking symbols</li> <li><a href="/wiki/Overline#Implementations" title="Overline">Overline § Implementations</a> in word processing and text editing software</li> <li><a href="/wiki/Underline" class="mw-redirect" title="Underline">Underline</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .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="CITEREFCajori2012" class="citation book cs1">Cajori, Florian (2012) [1928]. <a href="/wiki/A_History_of_Mathematical_Notations" title="A History of Mathematical Notations"><i>A History of Mathematical Notations</i></a>. Vol. I. Dover. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/384">384</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67766-8" title="Special:BookSources/978-0-486-67766-8"><bdi>978-0-486-67766-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Notations&rft.pages=384&rft.pub=Dover&rft.date=2012&rft.isbn=978-0-486-67766-8&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfrah2000" class="citation book cs1">Ifrah, Georges (2000). <i>The Universal History of Numbers: From Prehistory to the Invention of the Computer</i>. Translated by David Bellos, E. F. Harding, Sophie MENGNIU, Ian Monk. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+History+of+Numbers%3A+From+Prehistory+to+the+Invention+of+the+Computer&rft.pub=John+Wiley+%26+Sons&rft.date=2000&rft.aulast=Ifrah&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChilds2009" class="citation book cs1">Childs, Lindsay N. (2009). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/concreteintroduc00chil_612"><i>A Concrete Introduction to Higher Algebra</i></a></span> (3rd ed.). Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/concreteintroduc00chil_612/page/n184">183</a>-188.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concrete+Introduction+to+Higher+Algebra&rft.pages=183-188&rft.edition=3rd&rft.pub=Springer&rft.date=2009&rft.aulast=Childs&rft.aufirst=Lindsay+N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconcreteintroduc00chil_612&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConférence_Intercantonale_de_l'Instruction_Publique_de_la_Suisse_Romande_et_du_Tessin2011" class="citation book cs1">Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011). <i>Aide-mémoire</i>. Mathématiques 9-10-11. LEP. pp. 20–21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aide-m%C3%A9moire&rft.series=Math%C3%A9matiques+9-10-11&rft.pages=20-21&rft.pub=LEP&rft.date=2011&rft.au=Conf%C3%A9rence+Intercantonale+de+l%27Instruction+Publique+de+la+Suisse+Romande+et+du+Tessin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori2012">Cajori 2012</a>, p. 386</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"><a href="#CITEREFCajori2012">Cajori 2012</a>, pp. 390–391</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/middle-school-math-india/x888d92141b3e0e09:bridge-7th/x888d92141b3e0e09:untitled-302/e/b7-bodmas-1">"BODMAS (Basic) (Practice) | Week 1"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=BODMAS+%28Basic%29+%28Practice%29+%26%23124%3B+Week+1&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fmath%2Fmiddle-school-math-india%2Fx888d92141b3e0e09%3Abridge-7th%2Fx888d92141b3e0e09%3Auntitled-302%2Fe%2Fb7-bodmas-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFCajori2012">Cajori 2012</a>, p. 208</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott1847" class="citation cs2">Abbott, Jacob (1847) [1847], <i>Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II)</i>, p. 27</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vulgar+and+decimal+fractions+%28The+Mount+Vernon+Arithmetic+Part+II%29&rft.pages=27&rft.date=1847&rft.aulast=Abbott&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vinculum_(symbol)&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Periodic_Continued_Fraction"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PeriodicContinuedFraction.html">"Periodic Continued Fraction"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Periodic+Continued+Fraction&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPeriodicContinuedFraction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Vinculum"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Vinculum.html">"Vinculum"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Vinculum&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FVinculum.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVinculum+%28symbol%29" class="Z3988"></span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Vinculum_(symbol)&oldid=1257196634">https://en.wikipedia.org/w/index.php?title=Vinculum_(symbol)&oldid=1257196634</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematical_symbols" title="Category:Mathematical symbols">Mathematical symbols</a></li><li><a href="/wiki/Category:Elementary_mathematics" title="Category:Elementary mathematics">Elementary mathematics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 19:04<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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