Vinculum
Encyclopedia
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
.
Vulgar fractions are a common special case of this usage.
In a repeating decimal
, the vinculum is used to indicate the group of repeating digits:
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
is the radicand, and thus has a vinculum over it:
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
:
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:
It is also used to refer to the conjugate of a complex number:
It can even be used as a notation to indicate a group (bracket smaller to parenthesis):
meaning to add b and c first and then subtract the result from a.
In particle physics, the vinculum is used to indicate antiparticle
s. For example, p and are the symbols for proton and antiproton, respectively.
The vinculum should not be confused with a similar-looking vector notation, e.g.
"vector from A to B", or 
"vector named a".
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
editor
. In LaTeX
, use
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
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
Abu Bakr al-Hassar
Al-Hassar or Abu Bakr Muhammad ibn Abdallah ibn Ayyash al-Hassar was a mathematician from Morocco, living in the 12th century. He is the author of two books Kitab al-bayan wat-tadhkar , a manual of calculation and Kitab al-kamil fi sinaat al-adad , on the break-down of numbers...
.
Uses
The vinculum can be used to express division. The dividend appears above the vinculum and the divisor beneath it.Vulgar fractions are a common special case of this usage.
In a repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...
, the vinculum is used to indicate the group of repeating digits:
It is used as part of the notation of a radical
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
to indicate the radicand whose root is being indicated. In the next case, the quantity
is the radicand, and thus has a vinculum over it:It is used to show the repeating terms in a periodic continued fraction. Quadratic irrational
Quadratic irrational
In mathematics, a quadratic irrational is an irrational number that is the solution to some quadratic equation with rational coefficients...
numbers are the only numbers that have these.
It can be used in signed-digit representation
Signed-digit representation
Signed-digit representation of numbers indicates that digits can be prefixed with a − sign to indicate that they are negative.Signed-digit representation can be used in low-level software and hardware to accomplish fast addition of integers because it can eliminate carries...
to represent negative digits, such as the following example in balanced ternary
Balanced ternary
Balanced ternary is a non-standard positional numeral system , useful for comparison logic. It is a ternary system, but unlike the standard ternary system, the digits have the values −1, 0, and 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
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
, as in:
It is also used to refer to the conjugate of a complex number:
It can even be used as a notation to indicate a group (bracket smaller to parenthesis):
meaning to add b and c first and then subtract the result from a.
In particle physics, the vinculum is used to indicate antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...
s. For example, p and are the symbols for proton and antiproton, respectively.
The vinculum should not be confused with a similar-looking vector notation, e.g.
"vector from A to B", or "vector named a".Computer entry of the symbol
The vinculum can be typed using the combining overline (U+Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
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
Cascading Style Sheets
Cascading Style Sheets is a style sheet language used to describe the presentation semantics of a document written in a markup language...
rule
text-decoration: overline, although this does not carry over when pasting onto a plain textPlain text
In computing, plain text is the contents of an ordinary sequential file readable as textual material without much processing, usually opposed to formatted text....
editor
Text editor
A text editor is a type of program used for editing plain text files.Text editors are often provided with operating systems or software development packages, and can be used to change configuration files and programming language source code....
. In LaTeX
LaTeX
LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...
, use
3.\overline{333}...