Dear Jean-Luc,
These are good points.
As you say, the vinculum expresses aggregation. If we have an aggregate
consisting of one item, it is natural that it should sit above a single
character. Another example is the horizontal line in $\sqrt 2$ (square
root of two). The radix symbol consists of two parts: a vinculum and a
symbol that looks like a V, actually a capital U in Latin. This V stands
for "radix universalis", an earlier way to indicate aggregation in the
argument of a root; its scope is clarified by the vinculum. The
historical discussion is interesting but may not be of interest to most
members of this list.
As you suspected, typesetters make a difference in length between the
vinculum and the macron. Try compiling a LaTeX file with $\overline{m}$
and $\bar m$ : the lines have clearly different lengths.
The term "vinculum" is rather rare in contemporary mathematical usage,
because the vinculum was displaced by brackets, especially round
brackets. The horizontal line occurs in restricted situations: in
Boolean algebra, or to express complex conjugation, or the closure in
topology, to name a few of them. In the notation of the interior (in
topology) you do see a curved line that only means aggregation, with a
small circle on top of it that indicates "interior". The vinculum is
also found in elementary teaching in some countries, especially in
India. "Bar", "horizontal line (over...)" are much more common ways of
describing it.
In Philology, it may be useful to distinguish "vinculum", "macron" and
"bar" in general, even though mathematical usage does not always do so.
Thus, Charles Wells writes on p. 51 his "A Handbook of Mathematical
Discourse", version 0.9 (Apr. 25, 2002):
bar : A line drawn over a single-symbol identifier is pronounced "bar".
For example, $\overline x$ is pronounced "x bar". Other names for this
symbol are "macron" and "vinculum".
He adds: "Acknowledgments: Atish Bagchi"..
Kind regards,
Satyanad
Le 17/03/2023 à 08:26, Jean-Luc Chevillard a écrit :
Dear Manu, dear Satyanad, dear Charles and dear list members,
if I understand correctly,
the nature of a vinculum is that it "ties together", by means of an
overline, several items,
like in the example provided by Satyanad,
namely \overline{x+y}
However, sometimes, there is ONLY ONE element which is kept prisoner
by the vinculum, as in the examples provided as screenshots taken from
one of my books
+ the definition of a "rising factorial power"
(on page x of Graham-Knuth-Patashnik[1994]
+ the binomial theorem for factorial powers
(on page 245 of the same)
I include also a scan of the cover of that book, for easy identification
(ISBN 0-201-55802-5)
That would mean that a "vinculum" over a single letter
is longer that a "macron" over a single letter.
அன்புடன்
-- Jean-Luc
https://htl.cnrs.fr/equipe/jl-chevillard/
On 16/03/2023 21:25, Manu Francis wrote:
It is also not clear to me what is the difference between "vinculum"
and "macron".
--
**********************************************
Satyanad KICHENASSAMY
Professeur des Universités
Laboratoire de Mathématiques (LMR, CNRS, UMR9008)
Université de Reims Champagne-Ardenne
F-51687 Reims Cedex 2
France
Web: http://phare.normalesup.org/~kichenassamy
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