When Does Nothing Mean Something?

August 1, 2019

I thought I was done writing about this topic, but it just keeps coming back.  The internet just cannot seem to leave this sort of problem alone:

I don't know what it is about expressions of the form \(a\div b(c+d)\) that fascinates us as a species, but fascinate it does.  I've written about this before (as well as why "PEMDAS" is terrible), but the more I've thought about it, the more sympathy I've found with those in the minority of the debate, and as a result my position has evolved somewhat.

So I'm going to go out on a limb, and claim that the answer should be \(1\).

Before you walk away shaking your head and saying "he's lost it, he doesn't know what he's talking about", let me assure you that I'm obviouly not denying the left-to-right convention for how to do explicit multiplication and division.  Nobody's arguing that.*  Rather, there's something much more subtle going on here.

What we may be seeing here is evidence of a mathematical "language shift".

It's easy to forget that mathematics did not always look as it does today, but has arrived at its current form through very human processes of invention and revision.  There's an excellent page by Jeff Miller that catalogues the earliest recorded uses of symbols like the operations and the equals sign -- symbols that seem timeless, symbols we take for granted every day.

People also often don't realize that this process of invention and revision still happens to this day.  The modern notation for the floor function is a great example that was only developed within the last century.  Even today on the internet, you occasionally see discussions in which people debate on how mathematical notation can be improved.  (I'm still holding out hope that my alternative notation for logarithms will one day catch on.)

Of particular note is the evolution of grouping symbols.  We usually think only of parentheses (as well as their variations like square brackets and curly braces) as denoting grouping, but an even earlier symbol used to group expressions was the vinculum -- a horizontal bar found over or under an expression.  Consider the following expression: \[3-(1+2)\] If we wrote the same expression with a vinculum, it would look like this: \[3-\overline{1+2}\] Vincula can even be stacked: \[13-\overline{\overline{1+2}\cdot 3}=4\] This may seem like a quaint way of grouping, but it does in fact survive in our notation for fractions and radicals!  You can even see both uses in the quadratic formula: \[x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\]

Getting back to the original problem, what I think we're seeing is evidence that concatenation -- placing symbols next to each other with no sort of explicit symbol -- has become another way to represent grouping.

"But wait", you might say, "concatenation is used to represent multiplication, not grouping!"  That's certainly true in many cases, for example in how we write polynomials.  However, there are a few places in mathematics that provide evidence that there's more to it than that.

First of all, as a beautifully-written Twitter thread by EnchantressOfNumbers (@EoN_tweets) points out, we use concatenation to show a special importance of grouping when we write out certain trigonometric expressions without putting their arguments in parentheses.  Consider the following identity:
\[\sin 4u=2\sin 2u\cos 2u\] When we write such an equation, we're saying that not only do \(4u\) and \(2u\) represent multiplications, but that this grouping is so tight that they constitute the entire arguments of the sine and cosine functions.  In fact, the space between \(\sin 2x\) and \(\cos 2x\) can also be seen as a somewhat looser form of concatention.  Then again, so does the space between \(\sin\) and \(x\), which represents a different thing -- the connection of a function to its argument.  Perhaps this is why the popular (and amazing) online graphing calculator Desmos is only so permissive when it comes to parsing concatenation:

In contrast, where we do draw the line is with an expression like the following:\[\sin x+y\] We always interpret this as \((\sin(x))+y\), never \(\sin(x+y)\).  To drive home just how much stronger implicit multiplication feels to us than explicit multiplication, just take a look at the following expression: \[\sin x\cdot y\] Does this mean \((\sin(x))\cdot y\) or \(\sin(x\cdot y)\)?  If that expression makes you writhe uncomfortably, while if it had been written as \(\sin xy\) it would be fine, then you might see what I'm getting at.

An even more curious case is mixed numbers.  When writing mixed numbers, concatenation actually stands for addition, not multiplication. \[3\tfrac{1}{2}=3+\tfrac{1}{2}\] In fact, concatenation actually makes addition come before multiplication when we multiply mixed numbers! \[3\tfrac{1}{2}\cdot 5\tfrac{5}{6}=(3+\tfrac{1}{2})\cdot(5+\tfrac{5}{6})=20\tfrac{5}{12}\]

Now, you may feel that this example shows how mixed numbers are an inelegance in mathematical notation (and I would agree with you).  Even so, I argue that this is evidence that we fundamentally view concatenation as a way to represent grouping.  It just so happens that, since multiplication takes precedence over addition anyway in the absence of other grouping symbols, we use concatenation when we write it.  This all stems from a sort of "laziness" in how we write things -- - laying out precedence rules allows us to avoid writing parentheses, and once we've established those precedence rules, we don't even need to write out the multiplication at all.

So how does the internet's favorite math problem fit into all this?

The most striking feature of the expression \(8\div 2(2+2)\) is that it's written all in one line.

Mathematical typesetting is difficult.  LaTeX is powerful, but has a steep learning curve, though various other editors have made it a bit easier, such as Microsoft Word's Equation Editor (which has much improved since when I first used it!).  Calculators have also recognized this difficulty, which is why TI calculators now have MathPrint templates (though its entry is quite clunky compared to Desmos's "as-you-type" formatting via MathQuill).

Even so, all of these input methods exist in very specific applications.  What about when you're writing an email?  Or sending a text?  Or a Facebook message?  (If you're wondering "who the heck writes about math in a Facebook message", the answer at least includes "students who are trying to study for a test".)  The evolution of these sorts of media has led to the importance of one-line representations of mathematics with easily-accessible symbols.  When you don't have the ability (or the time) to neatly typeset a fraction, you're going to find a way to use the tools you've got.  And that's even more important as we realize that everybody can (and should!) engage with mathematics, not just mathematicians or educators.

So that might explain why a physics student might type "hbar = h / 2pi", and others would know that this clearly means \(\hbar=\dfrac{h}{2\pi}\) rather than \(\hbar=\dfrac{h}{2}\pi\).  Remember, mathematics is not about just answer-getting.  It's about communication of those ideas.  And when the medium of communication limits how those ideas can be represented, the method of communication often changes to accomodate it.

What the infamous problem points out is that while almost nobody has laid out any explicit rules for how to deal with concatenation, we seem to have developed some implicit ones, which we use without thinking about them.  We just never had to deal with them until recently, as more "everyday" people communicate mathematics on more "everyday" media.

Perhaps it's time that we address this convention explicitly and admit that concatenation really has become a way to represent grouping, just like parentheses or the vinculum.  This is akin to taking a more descriptivist, rather than prescriptivist, approach to language: all we would be doing is recognizing that this is already how we do things everywhere else.

Of course, this would throw a wrench in PEMDAS, but that just means we'd need to actually talk about the mathematics behind it rather than memorizing a silly mnemonic.  After all, as inane as these internet math problems can be, they've shown that (whether they admit it or not) people really do want to get to the bottom of mathematics, to truly understand it.

I'd say that's a good thing.

* If your argument for why the answer is \(16\) starts with "Well, \(2(2+2)\) means \(2\cdot(2+2)\), so...", then you have missed the point entirely.