# Arguing Semantics: the obelus, or division symbol: ÷

We’ve all seen it, and we all know what it means… right?  Commonly referred to as the division symbol, I’m talking about the obelus, ÷.

Now we’re used to seeing the obelus in elementary mathematics problems of the form:

$20\div4=5$

No surprises thus far, I hope.  We use this symbol almost interchangeably with the idea of division and it even exists as the symbol on our calculators for the division operation.  When we punch the button on the calculator, though, it will generally display as a solidus (or slash, “/”).  The interesting thing to note here is that we treat these as interchangeable symbols, and there should be no reason to think otherwise.

Recently, however, a discussion came on a forum that I frequent focusing on the potential ambiguity of the obelus.  More specifically, a problem was posed.  To what does the following expression reduce?

Too simple, right?  We just use our order of operations, PEMDAS (or BEMDAS), so that it reduces as follows:

Well, here’s the interesting part.  We are making the assumption that the obelus is used exactly like the solidus and that it is representing the basic division operation.  To look at why this assumption might not be founded in truth, let’s consider the origins of the obelus.

A friend of mine named Adrian, after doing some reaching, found an old German book called Teutsche Algebra by Johann H. Rahn, dated 1659.  It’s entirely in German, but the beauty of mathematics is that it’s a universal language.  Without being able to read any of the words, I was able to decipher many of the symbols.  It starts with an elementary introduction to the notation to be used in the book, which includes addition, subtraction, multiplication, exponentiation, and of course division.  Interestingly the symbol for exponentiation was a swirl:
At any rate, the obelus is introduced to represent division on page 8.  Previously, the obelus had been used to mark ancient manuscripts that were believed to be corrupt.  As it is used in Teutsche Algebra, however, was the first time it is used in a more modern sense.  There are pages of sample uses for division and all of them are analogous to how it used today.  This is the first known use of the obelus to represent division.
Things get interesting on page 76, however, when the following assertion is made:
For clarity, and since this is provided without context, this can be likened to the following, which is more relate-able:
Now, this clearly doesn’t obey the aforementioned order of operation rules, which suggests something about the nature of the obelus as a symbol for division.  As presented here, it is a line operation that divides the expression entirely (without need for parenthesis or bracketing), pun intended.
When presented with  this information, my first instinct was to defend the original answer to the problem above as a more modern interpretation of the obelus symbol.  My thought was that the use of symbols and their inherent interpretations evolve over time.  I considered that parenthesis were not used as a form of grouping, so the inherent system is slightly devolved from what it is today.  Hell, as I mentioned, this text even uses a swirl to indicate exponentiation, and we clearly don’t use a swirl anymore.
I was content with this way of looking at the situation until that same friend located a more recent book called A First Book in Algebra, by Wallace C. Boyden, dated 1895.  Adrian’s Google-fu is very strong, apparently.  In this text, the following is presented:
Which is clearly aligned with the usage in Teutsche Algebra.  Now, I don’t know where this wild-goose-chase will take me next, but it seems that when the obelus is used as a line operation, it splits the line evenly into numerator and denominator, which is removed from my previous assumptions.  Returning to the original problem, then, yields:
So, which is it?  288 or 2?
My TI-84 thinks it’s 288, as does Wolfram Alpha.  Now, before you start arguing that Wolfram is the infallible machine that we often assume, check out this result.  My intuition even says that it has to be 288, but my (and mostly Adrian’s) research leaves me wondering.  What do you guys think?
I know one thing is for certain… for my own sanity, I will not be using the obelus in my future classrooms.  There’s just one thing that bothers me, though:
Um…what does that button do?
Bonus Points: There is a calculation error on page 15 of Teutsche Algebra.  Can you find it?