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to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of parentheses.[1] Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal[2][3] is a significant exception and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.

The vinculum, in its general use, was introduced by Francis 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.[4]

Usage

Modern

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...

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

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

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):

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.[5]

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 $ab+2$ 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.[6]

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).[7]

References

Cajori, Florian (2012) [1928]. A History of Mathematical Notations. I. Dover. p. 384. ISBN 978-0-486-67766-8.
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

External links

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[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, p. 386

[5] Cajori 2012, pp. 390–391

[6] Cajori 2012, p. 208

[7] Abbott, Jacob (1847) [1847], Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II), p. 27