Vinculum (symbol)

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

line segment from A to B

1⁄7 = 0.142857

repeated 0.1428571428571428571...

${\displaystyle {\overline {a+bi}}}$

complex conjugate

${\displaystyle Y={\overline {\rm {AB}}}}$

boolean NOT (A AND B)

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

radical ab + 2

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

bracketing function

Vinculum usage

A vinculum (from

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:

• 1⁄7 = 0.142857 = 0.1428571428571428571...

A vinculum can indicate the complex conjugate of a complex number:

• ${\displaystyle {\overline {2+3i}}=2-3i}$

Logarithm of a number less than 1 can conveniently be represented using vinculum:

• ${\displaystyle \log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699}$

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

• ${\displaystyle Y={\overline {\rm {AB}}},}$

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):

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

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.^[6]

In India, the use of this notation is still tested in primary school.^[7]

As a part of a radical

The vinculum is used as part of the notation of a

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

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).[9]

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