Monday 11 November 2013

Vinculum is a Collective Noun

I'm not a pedant. I'm really not... really!

Ok let me explain. Murray Bourne, who writes a really nice math blog called Square Circle Z, as part of his Interactive Math site, (Both are excellent, if you haven't been there, go there) wrote a tweet a couple of days ago that got me started. He wrote, "A 'vinculum' is a horizontal line indicating grouping. E.g. over the '14' in 14/99=0.141414... "

I could have shook my head a little on gone on to the next post, but I like Murray, and think he is probably an excellent teacher. So I sent a brief quibble, which on twitter ran to about three tweets. But I think he missed my point, or perhaps he just didn't think the distinction I was trying to make was important. So here I am trying to tell the world to change to my way of thinking, and I hope with more than 140 characters, I can explain what I mean, and why I think the distinction is important.
The fact is that the overbar in the notation of repeating decimals is the only reference students have for vinculum. I will suggest (encourage/plead) that teachers add the common uses of parentheses and brackets as part of their description of a vinculum .
Many US teachers know of no other representation of repeating decimal fractions, yet they seem to have been the last application of the bar, and seem not to have occurred until after 1930 in the US. In F. Cajori's A History of Mathematical Notations (1929) he points out two forms of marking repeating sequences in decimals but does not mention the overbar. Cajori credits John Marsh [Decimal Arithmetic Made Perfect, (London, 1742)] with being the first to use a symbol to indicate the repeat sequence. Marsh sometimes placed a single dot over the first number in the repeat sequence, and sometimes placed one on the first and last.

This was one of the most frequent in the early arithmetics in the US, possibly due to the fact that many of them were by British authors,  or near verbatim copies of their books. John Bonnycastle and other British came early to the country to work in the early universities.

Like many terms of mathematical interest, vinculum(vincula) is a term that used to be better known. It seems many teachers have only a very limited knowledge of the history of even common arithmetic notations.   I've previously wrote responses to a teacher who gave me flack because I used "reduce", leading me to write this blog, "On Reducing Fractions". And another complained when I suggested that the number one, has been, and could be, labeled a prime leading me to write, "One is Prime if we Wish it to Be."

As the title says, vinculum is a collective noun, like truck, or variable. My old Dodge is a truck. It's not the only kind of truck. Some people have new trucks. Some people even have Ford trucks. I know; but what can you say to them. It's not that I'm prejudice. My own sister drives a Ford and I still love her like a, well, like a sister. And x is a variable, but it's not the only variable, and it is not always a variable, sometimes it is just a letter at the beginning of xenophobe. And if you had a student who argued that y can't be the variable because x is the variable, you would want to give them a more complete explanation.

The horizontal bar above a repeating decimal, such as $.\overline{14}$ is an example of a vinculum. It is now almost the only term that people use that term for, I think because they think it is a name for the bar, rather than a description of it's role in that situation. Horizontal bars were once commonly used beneath repeating decimals, and in fact beneath algebraic expressions in the same way we use grouping symbols today.
In "The Constructive Arithmetic" by James A Christie (1865) he writes, "The bracket { }, or [ ], or horizontal bar (such as sometimes separates the numerator of a fraction from its denominator,) is sometimes employed as a vinculum." Later he writes :

His interpretation of vinculum is a little unusual, as it is generally interpreted as something like binder. One dictionaries etymology gives "from vincire, vinctum, to bind." I have read that it was the name used frequently for a hobble for the legs of cattle in the field to keep them from wandering off. It was something like manacles and meant to allow the animal to move but keep it from moving quickly.
In "A Treatise on Arithmetic: Through which the Entire Science Can be Most Expeditiously and Perfectly Learned, Without the Aid of a Teacher." By Noble Heath he gives :

On another web site I have written, "In the same year as the 29th NCTM yearbook(1964), Irving Adler obtained a copyright for A New Look At Arithmetic, and on page 220 he writes, 'To indicate a repeating decimal with a minimum of writing, it is customary to write only enough decimal places to include the repeating part once, and to identify the repeating part by underlining it. Thus the repeating decimal for $ \frac{211}{990}$  is therefore represented by $.\underline{213}$. '. It is worth mentioning that William Oughtred, the 16th Century mathematician indicated all decimals by underlining. "

Another example, or rather a hybrid of two of the former, also appeared in a book with a 1964 copyright. A A Klaf's Arithmetic Refresher was published a few years after his death by his family. The book is written in a question and answer style somewhat reminiscent of the classic dialogs of antiquity. On page 188 it asks, "How are recurring, circulation, or repeating decimals denoted?" It then goes on to answer, "b) by dots placed over the first and last figures of the recurring group." This is described exactly like the more common earlier usage, but the figure that follows includes dots, and then an arc above them, similar to what I have shown here. Similar arcs were used over groups of three numbers to indicate the periods (thousands, millions, etc) in some early use of Hindu-Arabic numerals. Gerber(980), who later became Pope Sylvester, referred to them as "Pythagorean Arcs."

A popular author of arithmetics in the US in the 19th century was Charles Davies. He was one of the original instructors at the US Military at West Point. In his New University Arithmetic (1860) he uses yet a different type of vinculum than all the others I have mentioned. Davies sets off the repeating digits with a pair of single quotes, so 1/6 would be written .1'6'.

The more general definition may be slipping from use, but I think it is worthwhile to preserve the distinction.  When a symbol is used to bind together other numbers or operations, it is acting as a vincula, whether it is the fraction bar, $\frac{a}{b}$, or the diagonal solidus between fractions, a/b, a parenthesis ln[4{3+2(x+y)}] or brackets.  And when I type two dollar signs around an expression in Latex to make it print it as pretty math, those dollar signs form a vinculum to bind that expression together so that the computer knows, "This is math, print it using the math library I mentioned in the header." 

ADDENDUM:  In the comments, Murray writes with about a problem many teachers have encountered, how do you write repeating decimals on a typewriter or word processor if you don't have $LaTex$ or an equation editor (often not available to middle school teachers and others who teach repeating decimals)? He suggest using a square bracket vinculum (in the manner of James A Christie) to set off the repeating part.  It seems a wonderful idea.  They are distinct, and in  this usage, not easily confused with other potential uses of brackets at that (or any other?) level.  So 1/11 would be .[09], and 1/6 would be .1[6]  and no special typesetting needed.  I think if middle school teachers all over the country started using this it might force the higher school teachers to adapt, or by that time, maybe a different notation wouldn't be a problem for the students.  Being told to switch, they would begin to realize that the notations used in math are matters of choice, after all, we didn't always use = for equal.  Well after I received this suggestion from Murray, one of my brilliant ex students (brilliant and my ex-student, no causative effect suggested), Jacob Coakwerll told me that in Russia, where he lived for an extended period of time, the repeating part is included in parenthesis, so 7/12=  0,58(3)  where the comma is used for the decimal seperatrix. 

I also just noticed that some of the "old" symbols mentioned here may not be extinct.    In answer to a question on Yahoo Answers asking, "what is the name of the repeating sign over decimals?"  The answers included, "I don't know if it has a name, i just call it a dot, cos my teacher taught me to put a dot over each number that repeats."    Another seemed to suggest that something like Murray's practice was already in use, "It's just called a bar. Sometimes you will see (6) instead of a bar above or underneath the number."  A Wikipedia article suggested that the parenthetical use is mostly in Europe.

I would love for folks in different area around the world to write and tell me how they do repeating decimals. I sent a twitter question out and here are some of the responses:
Thony Christie ‏@rmathematicus England, "Bar over the repeat period and a period after the last digit."  $0.\overline{23}.$
MathsEnVideo ‏@MathsEnVideo " In France: same as in the USA or with points over the period's digits."  $0.\dot{2}\dot{3}$
Dong Suk Smith, an ex-student of Korean origin remembers that there they use a dot over the repeating period digits.
A teaching friend who has lived and retired in Japan writes that his wife has never seen the over-line and that the Japanese seem to use a repetition of the repeating period followed by an ellipsis. 

Friday 1 November 2013

Mobius Double Cross

As a grandfather, I always love being able to take time over the holidays to share entertaining math enrichments with the grandkids. Last Christmas I showed them this one and it was a big hit
 This, to me, is the greatest Mobius related activity I have ever seen.  I wrote about it briefly as part of a longer blog, but wanted to focus one on just this neat activity. (I have also included a link at the bottom to a nice Matt Parker video with a view others.   I got this from an Ivars Peterson article in the New York Times
Don't read the article until after you have tried it, He offers teasers of what the outcome could be. 

Start by cutting out a cross of paper.  Make the sides wide enough to do some cutting, one of them into thirds (sort of).

Now take ends of one cross and give them the standard half-twist to make a Mobius strip with a cross piece hanging on.  Now take the other crossing pair and fold them away from the Mobius loop to make a regular (non-twisted) loop.  It should look sort of like a twisted figure eight. 

Here is an image of what it may look like from one of his pages. 

Now draw a line trisecting the Mobius branch.  One line 1/3 of the way across the page should loop back around the loop and eventually make something like three paths on both sides of the strip.
Now cut along the trisected loop, then bisect the non-Mobius loop. 

Shake out all the twists and turns to be amazed.

A while after I wrote this, I came upon a video of a talk by Matt Parker in which he includes several demonstrations kids (of all ages) would enjoy using Mobius strips of different numbers of twists, including zero twists.

So if you are looking for a way to share your love of math over the holidays with young people, you could work up a nice routine with some of these.