July 17, 2014

The Implications of Being Implicit

In my previous post, I presented three "tricky" (read: "inane") math problems from the Internet, the last of which was the following:

I promised I'd come back to this one because it merits special discussion.  Now it's time to do exactly that, as this one (as well as its many variations) has piqued my ire every time I've seen it.

Most people who have had at least a basic prealgebra class tend to agree that the \(1+2\) in parentheses should be evaluated first, reducing the problem to \[6\div 2(3).\]
After that the battlefield gets bloody.

Do we do the multiplication first, or do we do the division first?

Some people say the multiplication needs to come first, valiantly shouting "PEMDAS" as their battle cry, arguing that since "M" comes before "D", the answer is \(1\).

Those who have more experience with order of operations and don't just rely on a silly (and wrong) mnemonic say that the division comes first, since multiplication and division are really the same under the hood - after all, division is equivalent to multiplication by the reciprocal - and by convention* are performed from left to right in the order they appear.  For these more seasoned warriors, the answer is "obviously" \(9\).

Those in the latter camp definitely are applying better mathematical reasoning than those in the former.  But do they have "the" correct answer?  If you're like me, though you know that multiplication and division are supposed to happen in order from left to right, there's just something about that \(2(3)\) that catches your eye, that makes you feel like for some reason it "should" come first.

And that's why we need to talk about implicit (or implied) multiplication.

When we first learn multiplication, we write it with a cross (\(\times\)).  But once variables like \(x\) start to come into play, we have to find new, less confusing ways to write multiplication.  So instead of writing \(a\times b\), we have a few options.
  • We can use a dot: \[a\cdot b\]
  • We can use parentheses: \[a(b)\]
  • Or as long as both factors aren't numerals, we can just concatenate (attach) them:  \[ab\]
There's actually a subtle difference between multiplication with a cross or a dot and multiplication with parentheses or concatentation.  The former two are called explicit multiplication, because we've explicitly indicated our operation using a symbol.  The latter two are called implicit multiplication - we know the intended operation is multiplication because no explicit symbol was provided.

Why does this matter?  As it turns out, in some conventions, implicit multiplication may actually take precedence over explicit multiplication and therefore division!  For example, the "Style and Notation Guide" for Physical Review, an American scientific journal, specifies that implicit multiplication should come before division when submitting manuscripts**.  Of course there are other conventions in which this is not the case, but the point to understand here is that multiple conventions do exist.

What's more, we can't even turn to our trusty calculators to tell us which way is "the" correct way, because different calculators may follow different conventions!
Two Casio calculators

Two TI calculators

(Try this on your own calculator and see which convention it uses!)

I'd like to posit one more reason that the \(2(3)\) may feel like it "should" come first.  One unfortunate side effect of trying to use existing punctuation when possible to represent mathematics is that certain symbols become overloaded - the same symbols can represent different things.  In this case, the notation \(2(3)\) for multiplication bears a very strong resemblance to the notation \(f(3)\) for function evaluation!  If the question were \[6\div f(1+2),\] even though we have no idea what function \(f\) is, there's no question that it would be evaluated before the division took place!  This may be a possible reason that the implicit-trumps-explicit convention is followed in some circles.

The inevitable conclusion is that there is no single correct answer - it all depends on what convention you're using.  At this point you may be ready to throw your hands up in despair.  But there is hope.  The best way to solve this kind of problem is ... you guessed it ... to use better notation in the first place!  (I mean who uses the obelus (\(\div\)) anymore past 5th grade anyway?)
  • If you mean for multiplication to be done first, then say so!  \[\frac{6}{2(1+2)}=1\]
  • If you mean for division to be done first, then say so!  \[\frac{6}{2}(1+2)=9\]
And if you're using a calculator, it never hurts to have too many parentheses.***

It all comes back to the point of the previous article, which I will make explicit one more time:  Math isn't about symbols.  Math is about ideas.  If your symbols don't unambiguously convey your ideas, then use better symbols.

* The left-to-right convention is probably so because those who established the convention spoke the sorts of European languages for which \(6\div 2\) would be vocalized in that order - not always the case if you've ever heard how fractions are read out loud in Japanese or Korean!

** The guide may seem like it's claiming that all multiplication should come before division, but this is because they don't use explicit multiplication at all except in specific contexts such as indicating dimensions and performing operations on vectors.

*** If you have a newer calculator, you may have a nifty fraction template that you can use to clear up confusion even further!  If you're using a TI-84+, and you've got the newest operating system on your calculator, try hitting [ALPHA] and then [Y=].  If you see a little menu come up, choose "n/d" and voilĂ  - you can now do fractions without getting lost in a sea of parentheses!


  1. The commentary here which compares implicit multiplication with function notation brings to mind another possible adjustment of notation. In particular, as Eric Schecter pointed out long ago (http://www.math.vanderbilt.edu/~schectex/commerrs/), the fact that these two things appear so similar leads to even more insidious errors.

    For instance, 2(x+y)=2*x+2*y is just fine. But sin(x+y)=sin(x)+sin(y) is not. It's no surprise that this mistake is common among math students though, on account of how the parentheses are, as you noted, overloaded.

    I would propose that the obviously solution is simply to quit enclosing function arguments with parentheses. Just stop overloading them, and the problem should go away.

    But how? We could switch to square brackets. In fact, we already use them for this in probability and statistics for some reason. For instance, with expectation: E[X+Y]. Oddly enough, though, expected value is the one commonly used named function that IS linear. E[X+Y]=E[X]+E[Y] is true. I have no idea why we've decided not to use parentheses in this case. Anyway, it sets a precedent that we should follow with all functions.

    Right now, a student could be quite nonplussed when his teacher goes bananas over his trying to do things like sin(x+y)=sin(x)+sin(y), but if it were sin[x+y], perhaps he would be more cautious, thinking, "OMG SQUARE BRACKETS I BETTER BE CAREFUL IT MAY NOT BE A LINEAR FUNCTION!"

    There is one place where his caution would still not help him, however. He would still be susceptible to the Freshman's Dream. Exponents are not currently written using function notation, even though they perform a decidedly non-linear operation. Because of this even if we committed to using square brackets for function arguments, we'd still write (x+y)².

    Come to think of it, isn't exponentiation the only operator in high school math books that is applied on the right? The only one that is right-associative? I'm starting to doubt the sanity and utility of exponent notation. But that's a completely different argument...

  2. As the author wisely noted at the end of his article, it never hurts to have too many parentheses!
    /Bravo, --problem eliminated!


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