Implied Multiplication 1: Not as Bad as You Think

We keep getting new questions related to Order of Operations: Implicit Multiplication?, where we discussed expressions like 6/2(1+2) that keep showing up in social media arguments. Since I closed comments on that page some time ago, because of the toxicity of some of them, further questions have come through our Ask a Question page (as they should, when your goal is to learn something rather than just express your opinion in public). In the next few posts, I’ll excerpt some of the interesting paragraphs from the 17,000+ words of discussion we’ve had in the last two years. Here we’ll focus on what the issue really is; next time on whether there is universal agreement; and finally on attempts to prove one answer is mathematically correct.

Rather than take one question at a time as usual, I’ll arrange this topically, putting pieces of questions and answers into categories, with the asker’s main idea (usually incorrect) in quotes as a title. For brevity, I will abbreviate the Implied [or Implicit] Multiplication First rule as “IMF”, and the rule that all multiplications and divisions are done from left to right as “strict PEMDAS”. Recall that the latter interprets $$6\div2(1+2)$$ as $$6\div2\cdot3=\frac{6}{2}\cdot3=3\cdot3=9$$, while the former takes it as $$6\div2(1+2)$$ as $$6\div2(3)=6\div6=1$$. Evidence suggests that strict PEMDAS may be taught and used primarily in American schools (and calculators designed to serve them, and so on), and IMF may be used more commonly around the world; but that is not certain. A third option, “AMF” (all multiplication before division) also comes up, as we’ll be seeing.

Missing the main point

Readers commonly miss our main point, which is not to take sides in the debate between the options, but to emphasize that this sort of expression is inherently ambiguous, and so should never be written. I have offered reasons to support both sides, and have told students who learned either rule to do as they were taught; but the only real solution is recognition of the ambiguity.

“Which is correct?”

Allison, a former teacher from Quebec, wrote us in April, 2021:

Here is the question: 6 / 2 (1+2)

I read all of your post on this type of question about order of operations, new math, and old math, and implied multiplication.

Me, being old, and 44 years, would see it as

= 6/2 (3)
= 6/6
= 1

However others would see it as

= 6/2*3
= (using left to right…which I am not sure why left to right?)
= 6 / 2 * 3
= 9

——–

Then I read on about your implied multiplication when characters, or numbers, are what I used to call, squished together, which has priority over division.

However, I also see it as clearing the brackets (parenthesis), then completing the division.

——-

Which is correct?

Allison sees it as natural to do the implied multiplication, $$2(3)$$, before the division (“IMF”), but sees it as a matter of having to “clear the brackets” (which evidently means eliminating parentheses, either by distributing, or just by doing the multiplication) before any other operations. This is one of the misunderstandings of PEMDAS (BEDMAS in Canada) that was discussed in the original article. Our discussion of that particular idea just went in circles, and I am omitting it.

Doctor Fenton answered, focusing on the most important point, which so many readers miss:

Since you have read the earlier posts, you should now realize that “which is correct?” is the wrong question.  Carrying out the computation requires a decision on the order in which to perform them, and that choice is a matter of convention, not necessity.

Students today are almost always taught the PEMDAS or BEDMAS rule for the order of operations, but that is not a mathematical axiom, and some people use different conventions.  It is more like the convention of whether to drive on the right side of a roadway, or the left side.  There is no law of nature requiring one or the other, it’s just very dangerous not to have a shared understanding of what to do.

The moral of the discussion is not to write expressions such as this one, which depend upon the convention you use. You can always write the expression in an unambiguous manner, regardless of which convention is used, and that is what you should do.  For example, writing 6/(2(3)), 6/(2*3), or (6/2)*3 makes the intention clear, no matter which convention you use.

He is taking the “strict PEMDAS” position, that Parentheses/Brackets, then Exponents, then Multiplication and Division, then Addition and Subtraction, without any special cases, is standard; this seems to be true at least in American schools, though it appears that mathematicians do commonly make a special case.

Doctor Rick added his thoughts (in response to various later statements he’ll quote):

Hi, Allison; I would like to add some thoughts.

You say:

Would implied multiplication take a higher priority over the division?

You do understand, do you not, that the question of whether “implied multiplication” has precedence over division is the core reason for the “ambiguity” that you say you agree exists? Some teach, or practice without saying, that it does have precedence, modifying the basic “order of operations” shown in the image from your friend’s textbook. Others stick with the basic rules and make no exception for implied multiplication. Neither is right, neither is wrong; either way can be justified.

However, until everyone is agreed on this, one way or another, we recommend that students and teachers alike should not write such an expression as 6 / 2(1 + 2) or 6 ÷ 2(1 + 2). It is not a matter of “right” or “wrong” but of clear communication.

The friend’s book she had shown didn’t show any implied multiplication, just standard BEDMAS examples with explicit multiplications.

We’ll hear more from Allison below.

“It should only have one true solution”

Toward the end of a discussion in July, Pete, from Georgia (a defender of IMF), said

A convention is not law and just means that some have settled on a particular method to use. That is my problem though I know that the whole body of Math relies somewhat on following a particular method. The bottom line to me is this: The expression in the video 6÷2(1+2) is either bogus or it should have only one true solution.

Every expression should have one interpretation, but this is a matter of human communication, not of what math requires; and I have found that even if there were a universal convention, people would misread some such expressions. (For example, students wanting to multiply x by 1/2 will write 1/2x, accidentally changing the meaning; or if they take 1/2x as meaning to divide by 2x, they might accidentally make a substitution (like those you discuss) putting 2*x in place of its synonym 2x, and end up with 1/2*x, changing the meaning to 1/2 times x. So even the rule you and I think makes sense can be dangerous. That’s part of why I don’t think decreeing a rule solves the problem.

This fact that, under IMF, replacing an expression with a synonym can change its meaning, is perhaps the strongest argument in favor of strict PEMDAS. But it is really an argument in favor of caution.

Near the end of our discussion, I summarized:

Please understand, if I haven’t already made it clear, that I am not opposed to the implied-multiplication-first convention. I think it makes sense, and I know many people follow it — maybe even most. The two points I have to insist on are simply that (a) this is not something that can be proved to be mathematically necessary, and (b) that because the alternative is both taught and used by many people, and is natural in some cases, it is impossible to impose any one rule. Therefore, my only recommendation is to never write such expressions — which is the rule I learned.

Catastrophizing

Many who write about this engage in “catastrophizing”, that is, “making a mountain out of a molehill” by seeing a small problem as “the end of the world”, sometimes almost literally. As one American teacher wrote to us this year, “And humans not recognizing this is creating a catastropby. [sic] … BUT someone needs to recognize this and FIX it! It’s causing way too much confusion, globally! And is the reason the USA scores so low on international testing!” Here I’ll gather a few other examples, with bits of our replies.

“How can it not matter? It can be a huge issue”

Allison, above, a supporter of IMF, said this late in our discussion:

But if I understand you correctly you’re saying it doesn’t matter as long as it’s all agreed upon rules.

However I see a problem in certain grades teaching it one way and then you end up in the upper grades with older teachers re-teaching it a different way.

And I can see this becoming a huge issue down the line, this one concept, as many other math concepts, branch off, from this one. Or having other teachers, as myself, teaching it differently, and all of sudden, you wonder why a student is not grasping the concepts, or frequently getting stuck.

Doctor Rick replied:

I disagree: this is not a major issue on which many math concepts depend. It is about math notation and nothing more; it is about convention and not about truth. True, it may cause confusion if a teacher uses one convention and a student uses another, so that one misunderstands what the other has written. However, the potential for such miscommunication can be reduced if the teacher simply does not write such potentially troublesome expressions. In the other direction, the teacher needs to be aware that students may not mean what they write (we at The Math Doctors have to deal with this quite frequently). But with awareness of the issues, it becomes not a big deal.

We have heard, over the years, of much arguing on social media over the “correct” way to evaluate this and similar expressions. You are contributing to yet another cycle of this useless argument, which completely misses the point of what is interesting about math.

Others make stronger exaggerations.

“It isn’t subjective – it could get people killed”

Samuel from Tennessee, in July 2022, took the position that strict PEMDAS is the only right way. Here is his response to a comment that we can’t be sure how a reader will interpret what we write:

But the point I was getting at is it isn’t subjective.  There is an objective uniform procedure that was created and taught for over 100 years now.  Specifically to avoid these types of problems and the serious engineering problems different procedures could cause by someone misreading the math.  Specifically saying, oh, different people were taught different ways so both are correct, is literally a dangerous undermining of a standardized method that could get people killed because someone miscalculated the weight limits on a bridge.  When you’re saying both could be correct depending on how you learned in school.  Sure it is nice to make people feel better about themselves by saying, well you’re not wrong, but in my opinion it is better to say you were taught the wrong method in school and contact the publisher of the books they were using as a bad source.  On a side note I am sure when the order of operation was first introduced plenty of people argued with the method and did not want to adopt it.

Are we just saying everyone is right, to protect their self-esteem, while allowing the world to be destroyed? Will people die for this?

Actually, the issue is, in some sense, subjective; I refer to the fact that there are humans involved, and we can’t get them all to do the same thing. Whether we like it or not, things are taught differently in different places (with good reasons on both sides), and, more important, people tend to misinterpret such expressions regardless of which interpretation you consider correct. (For example, “1/2x” can “obviously” mean whatever you want, depending on context, regardless of what you were taught.) We have to work with things as they are.

Moreover, this is not a matter of inherent mathematical correctness, but of human-made grammar used to communicate mathematics. The conflict is largely due to the fact that some teachers (not mathematicians) came up with a set of rules that don’t accurately represent the pre-existing grammar, and, being unnatural, are easy to get wrong.

In particular, it is not a “procedure that was created” 100 years ago; the grammar already existed, and a faulty attempt was made to formalize it. (The same has happened with the English language!)

Prescriptive “rules” of English grammar, like “never split an infinitive“, or “never end a sentence with a preposition“, inadequately describe actual usage. Likewise, oversimplifying the “rules” by which expressions had been written for a couple hundred years led to the misinterpretations we see.

On the other hand, I don’t say “both are correct“. I say either could be appropriate, and both are taught, and as a result, such expressions (without full parenthesization) are ambiguous. So I just tell people to be careful.

And this is not about making people feel better about themselves. It’s about communication. An analogy might be teaching about differences in usage between America and Britain: In acknowledging differences, the emphasis is not on the fact that neither should feel bad about their speech (though that’s true), but that each should be aware of potential misunderstandings.

In fact, the purpose is to prevent the very kind of error you warn about. My answer to the December 5 comment on the history post makes this point:

Consensus about notation doesn’t matter if you are just doing the math yourself; you can make up any notation you want, as long as you mean the right thing. But it is essential for communication, which is what this discussion is all about.

But you chose an interesting example [calculating rocket trajectories]. Have you never heard of the Mars shot that failed because different project groups lacked consensus on the format of data (namely, what units to use)? If two people share a formula but interpret it differently, then everything goes wrong.

If people make the wrong calculation for a bridge as a result of order of operations errors, that would be an error in communication, not just in thinking, because they are trying to do what someone else said to do, and misinterpreting it. And the way to prevent that is (a) to be aware that some people may interpret what you write differently than you do (whether because of what they were taught, or because what they were taught is not natural to them); and (b) to write defensively, using enough parentheses in any case to make it totally clear.

It would be great if there were an international commission to decree the rules, but there never has been one, and, as I have said, either rule can be misused because we’re human. The best course is to recognize the ambiguity and avoid it.

Neither set of units involved in the Mars disaster was wrong; they just didn’t agree on which to use. We’ll look at the nonexistence of an international commission next week!

“Mathematics is in big trouble without regulations!”

Last September, an anonymous adult, apparently from Romania, after a long discussion about parentheses, turned out to be a supporter of IMF, and eventually said this:

Thank you very much for your time and patience. My conclusion is that mathematics currently has a big problem, not because of mathematics, but because of the subjective human factor, because of conventions. We have an unregulated situation, and the PEMDAS regulation creates ambiguity. It is found that if the rule by which implicit operations have priority would cancel the appearance of ambiguities, the rule that exists is not really a rule because it did not go through a democratic vote, so we do not apply it. … Until it is regulated, mathematics is no longer an exact science, and this dilemma can cause errors not only in the mathematics notebook and book, but also in macroeconomic calculations, engineering calculations that must ensure safety, including in energy ones since this dilemma is introduced in calculation software used by people unfamiliar with anticipatory mathematics.

It is possible that certain economic crises are generated by this ambiguity, because in economics we often use division, and stock market operators do not have to be licensed in mathematics.

You are vastly overstating things.

The fact that there is one bit of notation which is taught differently by different people, and is easily misread even if that were not true, does not cause a problem at all in practice, because we avoid using that kind of expression. As I’ve said repeatedly, mathematicians and engineers don’t use the obelus (÷) in the first place, and are careful not to write in ways that will be ambiguous when they are writing for others to read. This ambiguity causes trouble only for non-mathematicians who don’t understand this, and for people who pay attention to internet trolls trying to stir up trouble.

What we are discussing is not mathematics, but how it is written. What is true here is true of any language; there is always ambiguity in communication. So we all learn to be careful, avoiding ambiguous wording whether in English or in math. I mentioned this here, pointing out that human error caused a Mars shot to miss because of failure to communicate what they were assuming (not because of parentheses, but units). The fact that we use different units can cause trouble, but does not mean that physics is not an exact science! It just means we have to be careful to communicate what we mean.

English is not in big trouble because a phrase like “I saw her duck” can be read with multiple meanings. We know enough to use it carefully — and to ask if we are unsure.

Ambiguity is only dangerous when you are not aware of it! So it is actually those who insist one way is right who are most dangerous.

5 thoughts on “Implied Multiplication 1: Not as Bad as You Think”

1. I’m not a mathematician, but I like solving math puzzles.

I believe that the issue is some get confused with the ÷ sign. One gentleman told me that 80 ÷2(2+2) is different than 80/2(2+2) or ⁸⁰⁄₂(2+2).

Then it hit me. Treat 80 ÷2 as a fraction and (2+2) as a fraction.

So rewrite (or rethink) the equation as ⁸⁰⁄₂ × ⁴⁄1. This is why they get the equation wrong. They’re multiplying the denominator (2) by the numerator 4. Check my math. I think this is correct.

1. Hi, Don.

Making up a new rule, no matter how rational, doesn’t solve the problem that people don’t agree on a rule. (Your interpretation amounts to what I call “strict PEMDAS”, which may in fact be the minority opinion.)

The only solution, short of a dictator decreeing what we should all do (and possibly changing all existing books) is for everyone to avoid using the ambiguous form.

2. I feel like you’ve missed the concept of a term. What we appear to be looking at are coefficients of a term, which makes a product. It isn’t multiplication by juxtaposition, or implied multiplication (which I believe is a term that originated from this blog originally).

1÷2a is always interpreted as 1÷(2 x a), and that is because ‘2a’ is a term. If we make a = 3 + b, then we get:

1÷2(3+b), and so ‘2(3+b)’ is also a term, with 2 as the coefficient.

1. I’ve seen a number of people recently claiming that the word “term” (or, more or less equivalently, “monomial”) in itself solves the problem. You seem to be supposing that the word means “numbers and variables written together without an explicit operator”, which is what we mean by juxtaposition or implicit multiplication, and that it is inherently treated as a unit. But that is not what it means (in any source I can find). Perhaps you can show me a source that teaches using this word the way you do? I’m always interested in seeing how things are taught in other parts of the world.

A term is commonly defined as one of the entities that are added or subtracted in an expression, and which themselves contain only multiplication (sometimes division) and powers, as in a polynomial. (It is used in a couple different ways, sometimes a little loosely, so you may find variations on this, commonly including expressions containing addition inside parentheses.) This definition emphasizes what operations are or are not involved, not how they are written; and I have not seen it used in explaining the order of operations in your manner. The key idea is that the additions in a polynomial are done last; divisions are not (at least primarily) in view.

As I’ve said many times, I think it’s reasonable to perform these multiplications first; and it would not be unreasonable to use words like “term” to express that idea. If you can provide a source that teaches it that way, I will happily add it to my list. But to my knowledge, that is not a standard way to express the idea, and in any case, your assertion as to what is “always” done does not make it so around the world. I wish it were! But see part 2 of this series (which is where this comment perhaps belongs).

Incidentally, the term can’t have originated in this blog, which has only existed since 2018; but it is conceivable that I invented it in early answers on Ask Dr. Math in 1998, because it was much harder to find information on such terminology back then (and then it was just a question from students about how to interpret expressions in class, not yet a controversial meme). I don’t recall where I found it.

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