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| A '''vinculum''' is a horizontal line used in mathematical notation. It may be placed over a [[mathematical expression]] to indicate that it is to be considered grouped together, or alternatively it may function as a binary connective between arguments appearing above and below it. ''Vinculum'' is [[Latin]] for "bond", "fetter", "chain", or "tie", which is roughly suggestive of some of the uses of the symbol. The vinculum was developed in the 12th century by the Moroccan mathematician [[Abu Bakr al-Hassar]]{{Citation needed|date=January 2014}}.
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| ==Uses==
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| The vinculum can be used to express [[division_(mathematics)#Notation|division]]. The numerator appears above the vinculum and the denominator beneath it.
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| [[Fraction_(mathematics)#Writing_fractions|Vulgar fractions]] are a common special case of this usage.
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| :<math>\frac{1}{3}</math> | |
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| In a [[repeating decimal]], the vinculum is used to indicate the group of repeating digits:
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| :<math>\frac{1}{3} = 0.333\dots = 0.\overline{3}</math> ; <math>\frac{1}{11} = 0.0909\dots = 0.\overline{09}.</math>
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| It is used as part of the notation of a [[Nth root|radical]] to indicate the [[radicand]] whose root is being indicated. In the next case, the quantity <math>ab+2</math> is the radicand, and thus has a vinculum over it:
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| :<math>\sqrt[n]{ab+2}</math>
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| It is used to show the repeating terms in a periodic continued fraction. [[Quadratic irrational]] numbers are the only numbers that have these.
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| It can be used in [[signed-digit representation]] to represent negative digits, such as the following example in [[balanced ternary]]:
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| :<math>\pi \approx 10.011\overline{1}111\overline{1}000\overline{1}011\overline{1}1101\overline/11111100\overline{1}0000\overline{1}1\overline{1}\overline{1}\overline{1}\overline{1}0\overline{1}</math>
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| The vinculum is sometimes used in [[Boolean algebra (logic)|Boolean algebra]], where it serves to indicate a group of expressions whose logical result is to be [[negation|negated]], as in: | |
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| :<math>\overline{AB}.</math>
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| It is also used to refer to the [[Complex number#Conjugation|conjugate of a complex number]]:
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| :<math>\bar{z} = \overline{x+iy} = {x-iy}</math>
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| It can even be used as a notation to indicate a group (bracket similar to parenthesis):
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| :<math>(a-\overline{b+c})</math>
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| meaning to add ''b'' and ''c'' first and then subtract the result from ''a''.
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| In statistics the vinculum can be used to indicate the mean of series of values.<ref>{{cite book |title=Statistics made Simple |edition=2nd |first=H. T. |last=Hayslett |first2=P. |last2=Murphy |publisher=W. H. Allen and Co |year=1968 |page=18 |isbn=0-491-00680-2 }}</ref>
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| In particle physics, the vinculum is used to indicate [[antiparticle]]s. For example, p and {{Overline|p}} are the symbols for proton and antiproton, respectively.
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| The vinculum should not be confused with a similar-looking [[vector (geometric)|vector]] notation, e.g. <math>\overrightarrow{AB}</math> "vector from A to B", or <math>\vec{a}</math> "vector named ''a''", though an overline or underline without the arrowhead is sometimes used instead (e.g., <math>\overline{a}</math> or <math>\underline{AB}</math>).
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| ==Roman numerals== | |
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| In [[Roman numerals|Roman numeral]] notation, a vinculum indicated that the numerals under the line represented a thousand times their unmodified value. Other interstitial markings could be used for similar and even larger place value modifications.
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| ==Computer entry of the symbol==
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| The vinculum can be typed using the combining overline ([[Unicode|U+]]0305) after the character that one wishes to add it to. For example, typing ‘33.333...’ with combining overlines over the final three ‘3’s produces: ‘33.3̅3̅3̅...’. It can also be added over any given character or run of characters by using the [[Cascading Style Sheets|CSS]] rule <code>text-decoration: overline</code>, although this does not carry over when pasting onto a [[plain text]] [[Text editor|editor]]. In [[LaTeX]], use <code>33.\overline{3}</code> to give <math>33.\overline{3}</math>.
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| ==See also==
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| *[[Macron]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|urlname=PeriodicContinuedFraction|title=Periodic Continued Fraction}}
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| *{{MathWorld|urlname=Vinculum|title=Vinculum}}
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| {{DEFAULTSORT:Vinculum (Symbol)}}
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| [[Category:Mathematical symbols]]
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| [[Category:Elementary mathematics]]
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