# Vinculum (symbol)

A vinculum is a horizontal line used in mathematical notation for a specific purpose. It may be placed as an overline (or underline) over (or under) a mathematical expression to indicate that the expression is to be considered grouped together. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of parentheses.[1] Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal[2][3] is a significant exception and reflects the original usage.

Vinculum is Latin for "bond", "fetter", "chain", or "tie", which is suggestive of some of the uses of the symbol.

## Usage

A vinculum can indicate a line segment where A and B are the endpoints:

• ${\displaystyle {\overline {\rm {AB}}}.}$

A vinculum can indicate the repetend of a repeating decimal value:

• 17 = 0.142857 = 0.1428571428571428571...

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

Its main use was as a notation to indicate a group (a bracketing device serving the same function as parentheses):

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

meaning to add b and c first and then subtract the result from a, which would be written more commonly today as a − (b + c). 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 Chuquet in 1484 used the underline version.[4]

The vinculum is used as part of the notation of a radical to indicate the radicand whose root is being indicated. In the following, the quantity ${\displaystyle ab+2}$ is the whole radicand, and thus has a vinculum over it:

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

In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.[5]

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).[6]

## Other notations

There are several mathematical notations which use an overbar that can easily be mistaken for a vinculum. Among these are:

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

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

or the bar notation in common logarithms, such as

${\displaystyle \log _{10}0.2\approx {\bar {1}}.301=-1+0.301=-0.699}$

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

${\displaystyle {\overline {AB}}.}$

In electronics, the overbar is used to notate complementary binary signals. For example, READY pronounced "not ready", would be the same signal as READY but with the opposite polarity. This usage is closely related to the usage in Boolean algebra.

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

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

In statistics the overbar can be used to indicate the mean of series of values.[7]

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

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

## Roman numerals

It has been stated that in Roman numeral notation, a vinculum may indicate that the numerals under the line represented a thousand times the unmodified value.[8] Mathematical historian David Eugene Smith disputes this.[9] The notation was certainly in use in the Middle Ages.

## Computer entry of the symbol

The vinculum can be formed in Unicode by 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...".

In HTML code, the vinculum can be generated over any given character or run of characters by using the CSS rule text-decoration: overline. However, this does not carry over when pasting into a plain text editor, because CSS affects format, not content.

Word processors frequently have an overbar option. In Microsoft Word, this can be achieved through the Equation Editor or, in the 2007 and later versions, the built-in equation editing feature. On a Mac using Pages, you may have to install the Unicode Hex keyboard first in System Preferences|Keyboard. Open the Word Processor and make sure the Unicode keyboard is active. Type the first number to be overbarred, then ALT 0305, then the second number, then ALT 0305, and so on.

In LaTeX, 33.\overline{888} gives ${\displaystyle 33.{\overline {888}}}$.

## References

1. ^ Cajori, Florian (2012) [1928]. A History of Mathematical Notations. I. Dover. p. 384. ISBN 978-0-486-67766-8.
2. ^ Childs, Lindsay N. (2009). A Concrete Introduction to Higher Algebra (3rd ed.). Springer. pp. 183–188.
3. ^ Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011). Aide-mémoire. Mathématiques 9-10-11. LEP. pp. 20–21.
4. ^ Cajori 2012, pp. 390–391
5. ^ Cajori 2012, p. 208
6. ^ Abbott, Jacob (1847) [1847], Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II), p. 27
7. ^ Hayslett, H. T.; Murphy, P. (1968). Statistics made Simple (2nd ed.). W. H. Allen and Co. p. 18. ISBN 0-491-00680-2.
8. ^ Georges Ifrah (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie Wood, Ian Monk. John Wiley & Sons.
9. ^ Smith, David Eugene (1958) [1925], History of Mathematics, II, p. 60, ISBN 0-486-20430-8