Vinculum (symbol)

A vinculum is a horizontal line used in mathematical notation for a specific purpose. It is most commonly used today to indicate the repetend of a repeating decimal. It may be placed as an overline (or underline) over (under) a mathematical expression to indicate that the expression is to be considered grouped together. For most of its uses it has been replaced by parentheses in modern notational style.^[1]

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

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

• $\overline{\rm AB}.$

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

• ¹⁄₇ = 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):

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

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.^[2]

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 $ab+2$ is the whole radicand, and thus has a vinculum over it:

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

In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.^[3]

The symbol used to indicate a vinculum need not be a line segment (overline or underline), sometimes braces, pointing up or down can be used.^[4]

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:

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

or the bar notation in common logarithms, such as

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

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:

$\overline{AB}.$

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

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

In statistics the overbar can be used to indicate the mean of series of values.^[5]

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

In electronics, the vinculum is used to notate complementary signals. For example, CONTROL pronounced "control-not," would be the same signal as CONTROL but with the opposite polarity.

The vinculum should also 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

It has been stated that in Roman numeral notation, a vinculum indicated that the numerals under the line represented a thousand times their unmodified value. However, mathematical historian David Eugene Smith denies the validity of this statement.^[6]

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 simulated 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

1. Cajori, Florian (2012) [1928], A History of Mathematical Notations I, Dover, p. 384, ISBN 978-0-486-67766-8
2. Cajori 2012, pp. 390–391
3. Cajori 2012, p. 208
4. Abbott, Jacob (1847) [1847], Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II), p. 27
5. Hayslett, H. T.; Murphy, P. (1968). Statistics made Simple (2nd ed.). W. H. Allen and Co. p. 18. ISBN 0-491-00680-2.
6. Smith, David Eugene (1958) [1925], History of Mathematics II, p. 60, ISBN 0-486-20430-8

External links

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