Sunday 15 December 2019

The Agony and the Obelus, or Much Ado about Notation

In November of 2019 I combined several blogs on Division to create some notes about the history of the topic.  I did not include notes on the history of the symbols commonly used for division, and today I corrected that oversite by including this post from 2015 about those symbols.  This information is now included in that much longer post. 

Recently(2015) James Tanton posted a short article about problems that are circulating on the internet such as (and this is the one he used) "What is the value of the following expression: 62 ÷ 2(3)+4, and then asked, "Is the answer 10 or is the answer 58?" (my personal choice for historical reasons explained below is 3.6)

I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.

The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols
Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunatly, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.

The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write \( \underline {2x+3} \> 5 \) . Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.

 Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).

The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking.  Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, . The root also gives rise to the word obelisk for a pointed pillar of stone.
 The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659. 
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.

Let me make it clear I am not an authority on math history and do not read German,  but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog.  Instead it seems to be used exclusively for a shorthand in explaining the operations used.  

Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.

The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.

Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 62 ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)

So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means \(\frac{a}{bc} \) . He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean \(\frac{a}{b} (c) \)

So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first."  So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that.  Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."

So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven.  The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets.  What was not there was a section on order of operations, or any problems that went beyond " number ÷ number."  No long strings of numbers and operations strung together.

Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher
Read the part in Italics again.... multiplication first, then division, without regard to the order.  That is not what you are telling your students today (I hope).  So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.

Better, right?  then turn the page, and find
So there is our old friend the obelus used exactly as I suspect Pell and Rahn had intended (if they intended it to be used as an operator at all), and lower down the solidus in the manner that Stokes suggested, but apparently used in a way the users thought distinguished it from the use of the obelus.  And you wonder why your students are confused?

I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem.  Anyone who has more information, please share. 
 I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.

1 comment:

Smartman Apps said...

The issue isn't with the obelus, but with not correctly parsing Terms. Terms are separated by operators and joined by grouping symbols. So in the example i÷rt vs. i÷rxt, in the former rt is a single term entirely in the denominator, whereas in the latter r and t have been split into 2 separate terms, which has the effect of taking the t out of the denominator and flipping it into the numerator. Since rt is a single Term, the only way you can add a multiplication sign to it is if you also put it in brackets, to keep it as 1 term. i.e. i÷(rxt).

And yes, more than 100 years ago it was the case that everything following the obelus was in the denominator, but that changed, not sure when, but certainly by Lennes' time (1917) such that only the first term following the obelus was in the denominator, such that you could then have multiple divisions within the same line (instead of only one, which was the limitation with the previous usage) - this is the same usage we have today.

I have a whole thread about the common order of operations mistakes at