Vinculum (symbol)

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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]}.

Uses

The vinculum can be used to express division. The numerator appears above the vinculum and the denominator beneath it. Vulgar fractions are a common special case of this usage.

${\frac {1}{3}}$

In a repeating decimal, the vinculum is used to indicate the group of repeating digits:

${\frac {1}{3}}=0.333\dots =0.\overline {3}$ ; ${\frac {1}{11}}=0.0909\dots =0.\overline {09}.$

It is used as part of the notation of a radical to indicate the radicand whose root is being indicated. In the next case, the quantity $ab+2$ is the radicand, and thus has a vinculum over it:

${\sqrt[ {n}]{ab+2}}$

It is used to show the repeating terms in a periodic continued fraction. Quadratic irrational numbers are the only numbers that have these.

It can be used in signed-digit representation to represent negative digits, such as the following example in balanced ternary:

$\pi \approx 10.011\overline {1}111\overline {1}000\overline {1}011\overline {1}1101\overline /11111100\overline {1}0000\overline {1}1\overline {1}\overline {1}\overline {1}\overline {1}0\overline {1}$

The vinculum is sometimes used in Boolean algebra, where it serves to indicate a group of expressions whose logical result is to be negated, as in:

$\overline {AB}.$

It is also used to refer to the conjugate of a complex number:

${\bar {z}}=\overline {x+iy}={x-iy}$

It can even be used as a notation to indicate a group (bracket similar to parenthesis):

$(a-\overline {b+c})$

meaning to add b and c first and then subtract the result from a.

In statistics the vinculum can be used to indicate the mean of series of values.^[1]

In particle physics, the vinculum is used to indicate antiparticles. For example, p and p are the symbols for proton and antiproton, respectively.

The vinculum should not be confused with a similar-looking vector notation, e.g. $\overrightarrow {AB}$ "vector from A to B", or ${\vec {a}}$ "vector named a", though an overline or underline without the arrowhead is sometimes used instead (e.g., $\overline {a}$ or $\underline {AB}$ ).

Roman numerals

In 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.

Computer entry of the symbol

The vinculum can be typed using the combining overline (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 CSS rule text-decoration: overline, although this does not carry over when pasting onto a plain text editor. In LaTeX, use 33.\overline{3} to give $33.\overline {3}$ .

References

↑ Hayslett, H. T.; Murphy, P. (1968). Statistics made Simple (2nd ed.). W. H. Allen and Co. p. 18. ISBN 0-491-00680-2.

External links

Weisstein, Eric W., "Periodic Continued Fraction", MathWorld.
Weisstein, Eric W., "Vinculum", MathWorld.

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