# Vinculum (symbol) Information

*https://en.wikipedia.org/wiki/Vinculum_(symbol)*

line segment from A to B

repeated 0.1428571428571428571...

boolean NOT (A AND B)

bracketing function

bracketing function

Vinculum usage

A **vinculum** (from
Latin *
vinculum* 'fetter, chain, tie') is a horizontal line used in
mathematical notation for various purposes. 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to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of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]}It was also used to mark Roman numerals whose values are multiplied by 1,000.

^{ [2]}Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal

^{ [3]}

^{ [4]}is a significant exception and reflects the original usage.

## History

The vinculum, in its general use, was introduced by
Frans van Schooten in 1646 as he edited the works of
François Viète (who had himself not used this notation). However, earlier versions, such as using an underline as
Chuquet did in 1484, or in limited form as
Descartes did in 1637, using it only in relation to the radical sign, were common.^{
[5]}

## Usage

### Modern

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

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

- 1⁄7 = 0.142857 = 0.1428571428571428571...

In Boolean logic, a vinculum may be used to represent the operation of inversion (also known as the NOT function):

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.

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

### Historical

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

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.^{
[6]}

### As a part of a radical

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 is the whole radicand, and thus has a vinculum over it:

In 1637
Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.^{
[7]}

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).^{
[8]}

## Encodings

### In Unicode

- U+0305 ◌̅ COMBINING OVERLINE

### TeX

In
LaTeX, a text <text> can be overlined with `$\overline{\mbox{<text>}}$`

. The inner `\mbox{}`

is necessary to
override the math-mode (here invoked by the dollar signs) which the `\overline{}`

demands.

## See also

- Overline § Math and science similar-looking symbols
- Overline § Implementations in word processing and text editing software
- Underline

## References

**^**Cajori, Florian (2012) [1928].*A History of Mathematical Notations*. Vol. I. Dover. p. 384. ISBN 978-0-486-67766-8.**^**Ifrah, Georges (2000).*The Universal History of Numbers: From Prehistory to the Invention of the Computer*. Translated by David Bellos, E. F. Harding, Sophie MENGNIU , Ian Monk. John Wiley & Sons.**^**Childs, Lindsay N. (2009).*A Concrete Introduction to Higher Algebra*(3rd ed.). Springer. pp. 183-188.**^**Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011).*Aide-mémoire*. Mathématiques 9-10-11. LEP. pp. 20–21.**^**Cajori 2012, p. 386**^**Cajori 2012, pp. 390–391**^**Cajori 2012, p. 208**^**Abbott, Jacob (1847) [1847],*Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II)*, p. 27