Order of Operations: Implicit Multiplication?

I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like \(a\div bc\) or \(8\div 4(3-1)\), where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication and division are done from left to right. We’ll look at this first from the perspective of students and teachers, and then (next time) investigate some historical issues to close out the series.

Two ways to evaluate ax÷by

Let’s first look at one of the earlier questions we had about this issue, in 1999, to set the stage:

Order of Operations

The problem was presented like this:

   a = 1.56
   b = 1.2
   x = 7.2
   y = 0.2

   ax/by = ?

Here are two ways that I solved it:

1) I first rewrote the problem as [1.56(7.2)/ 1.2](0.2). Second, a was multiplied by x. The product was 11.232. Then, since no parentheses were present, I followed the order of operations and divided 11.232 by b, which was 1.2. The quotient was 9.36. Then I multiplied 9.36 by y, which was 0.2. The final answer was 1.872.

2) The other way, the first thing I did was multiply a by x. The product, which was 11.232, was set aside for the time being. Then b was multiplied by y, which gave the product of 0.24. The problem was now solved by dividing 11.232 (or ax) by 0.24 (or by) to reach a final answer of 46.8.

Can you please tell us which answer is correct and why?

(Note that at that time, the only way to type division in our email was to use the slash, \(a/b\), which I generally assume represents an expression actually written as \(a\div b\). I will occasionally be inserting an obelus, ÷, where we made rough attempts to simulate it.)

The first way follows PEMDAS literally, as usually taught and as I’ve presented it here, by evaluating from left to right as \(a\cdot x\div b\cdot y = ((a\cdot x)\div b)\cdot y\).

The second sees it as \(ax\div by = (ax)\div (by)\). This isn’t explained as following any taught rule, but just as doing what looks right, either because the division is read as if it were a fraction bar, or just because “by” looks like it belongs together as a unit. We’ll be seeing several reasons students have given for doing this.

Though I had been with Ask Dr. Math less than a year, this was already a familiar question, which I wanted to answer thoroughly for the sake of the archive:

You are not alone in wondering about this. We have had several other questions about expressions similar to yours, from confused teachers and students who have found that different books or teachers have different answers, and even calculators disagree.

Note that it is not only students doing what feels right, but also some textbooks and calculators that follow the second method.

A new rule, or what looks right?

I elaborated on the two methods, taking the PEMDAS version as correct (though I’ll have some second thoughts on that):

As written, your expression

    ax/by

should be evaluated left to right: a times x, divided by b, times y. The multiplication is not done before the division, but both are done in the order they appear. Your first solution is right.

Some texts make a rule, as in your second solution, that multiplication without a symbol ("implied multiplication") should be done before any other operations in an expression [except exponents], including "explicit multiplication" using a symbol. Following this rule, you would multiply a by x, then multiply b and y, then divide one  by the other. Some (probably most) texts don't mention such a rule - but some of those may use it without saying so, which is far worse.

I think I had made up the term “implied, or implicit, multiplication” when I answered my first question on the topic a few months before, to refer to multiplication indicated by just putting two numbers or variables or parenthesized expressions next to one another – “juxtaposition“, as others call it – like \( ab\) or \( 2b\) or \( a(b+c)\), as opposed to explicitly writing \( a\times b\) or \( a\cdot b\).

We had seen some questions from students whose textbooks taught only the usual PEMDAS, yet evaluated the second way in examples or solutions, without comment. This might have been due to the answers in the back being written by someone other than the author, but it is an inexcusable inconsistency.

Why would an author make this extra rule? I have had different opinions at various times about whether the rule is a good idea, but have always recognized that it is not what is usually taught:

I don't know of a general rule among mathematicians that implied multiplication should be done before explicit multiplication. As far as I'm concerned, all multiplications fit in the same place in the order of operations. It's not an unreasonable rule, though, since it does seem that implied multiplication ties the operands together more tightly, at least visually; but the idea of Order of Operations (or precedence, as it is called in the computer world) is supposed to be to ensure that everyone will interpret an otherwise ambiguous expression the same way - so if some texts change the rules, or if people do what feels natural, the purpose has been lost.

A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read \(ax\div by\) because \(by\) looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.

In particular, many students assume that it represents a horizontal version of \(\displaystyle\frac{ax}{by}\):

The problem here is that the expression looks as if it were meant to be

     ax
    ----
     by

In the Dr. Math FAQ about writing math in e-mail, one of our recommendations is to use parentheses wherever possible to avoid ambiguity, even where the rules should make it clear, because it can be easy to forget them in some situations.

So in e-mail we would write it like this:

    ax/(by)  or  (ax/b)*y

depending on what is intended.

By using parentheses, we can avoid writing something that people who were taught different rules, or who ignore the rules they were taught, might take differently than we intend.

Calculator issues

In my research for another Dr. Math "patient," I found that some calculators have experimented with this rule. Calculators have somewhat different needs than mathematicians, since they have to take input linearly, one character after another, so they are forced to make a decision about it. On the TI Web site I learned that they deliberately put this "feature" into the TI 82, and then took it out of the TI 83, probably because they decided it was not a standard rule and would confuse people.

The link there went bad long ago; but when a specific question about a calculator came up in 2008, I quoted from what TI said in their Knowledge Base:

Implied Multiplication and TI Calculators

...
Solution 11773: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators. 

Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper. 

This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication are given the same priority.

This makes it clear that calculator designers have to decide on their own rules, which don’t have to be the same as rules for writing on paper; but educators seem to have convinced them to keep things as much the same as possible for students’ sake.

In conclusion (back to the 1999 answer):

So to answer your question, I think both answers can be considered right - which means, of course, that the question itself is wrong. I prefer the standard way (your first answer) when talking to students, unless their own text gives the "implicit multiplication first" rule; but in practice if I came across that expression, I would probably first check where it came from to see if I could tell what was intended. The main lesson to learn is not which rule to follow, but how to avoid ambiguity in what you write yourself. Don't give other people this kind of trouble.

Subsequently, we had many more questions about this; I’ll just quote a few unique bits from some of those answers.

Old-fashioned math?

Here is a typical example of a school conflict, from 2000:

Order of Operations Dispute

The problem reads: N ÷ ml where n=12, m=6, and l=3. I believe the correct answer should be .6666, as 12 divided by 18 equals this.  My husband agrees with me.

My son came home very upset from school, with a note from his teacher that the answer was wrong. She indicated that I should have divided the 6 (m) into 12 (n) before I divided the 3 (l) into the equation.  Her answer was 6.  

My son is very upset with me; his teacher told him I was doing "old fashioned math."  Do I need to go back to school?

The problem is \(N\div ml\), and the parents are doing the multiplication first. I replied, in part:

I can give you some good news and some bad news. First, the bad news: according to the usual order of operations rules now taught, your answer is wrong. ...

I explained the standard rules, and added:

BUT...

You are not alone in your opinion. This part of the rule - doing multiplication and division together - is probably the last rule to have stabilized; I know that in the 1920's, at least, there was no agreement. It seems that an agreement developed, but it is unraveling now, as I hear from many students whose texts answer questions like this the way you did. It appears that they are adding an unstated rule, which seems entirely reasonable in this context, that an implied multiplication (indicated by simply placing two variables or expressions together, as in "ml") should be done first. It certainly looks as if it should mean that. The problem is that, although I've heard of this rule being followed frequently, I've hardly ever heard of it being taught, so these texts are not following their own stated rules.

I’ll have more on the history next time.

Since this type of expression is so ambiguous, with people disagreeing on the rules, and the rules being easy to overlook, my own opinion is that neither your answer nor the teacher's is right: the question is wrong. No responsible mathematician would write such an expression; we would just say

     n
    ---
    m l

so there would be no question about its meaning. After all, the purpose of rules is to allow us to communicate clearly, not to help us trick students and start fights among families.

So you may in fact be "old-fashioned"; or you may be on the cutting edge. In any case, I'm afraid you'll just have to learn how they are doing it in class, and follow along. There shouldn't be many more issues like this to worry about.

More recently, the fights tend to be on social media!

Misapplying the distributive property

I’ll close with the most recent archived discussion. This question is from 2017:

Even More on Order of Operations

I'm curious to know what the answer is for this:

   8/4(3 - 1)

Following strictly PEMDAS, the answer is 4:

   8/4(2)
   2*2
   4

However, if you follow the distributive property, you get 1:

   8/((4*3) - (4*1))
   8/(12 - 4)
   8/8
   1

Which one would be correct and why?
   
Both are valid, so I'm conflicted as to what would be the correct answer.

It should be right or wrong, not two different answers being right.

I answered with a collection of my standard answers to this sort of question; even my first archived answer on the topic in 1999 was largely a standard response I had given to others before. Here, I’ll just look at a few points I made that haven’t been fully covered above.

I first summarized what was going on:

The problem is not a conflict between PEMDAS and distribution; it is that strict interpretation of PEMDAS conflicts with one's natural impression of the meaning of the expression, so that you unknowingly apply an alternative interpretation when you think you are just applying the distributive property.

If you recall earlier statements that PEMDAS is (a) in harmony with the properties of operations, and (b) fitting with the visual impression of our notation, then some alarm bells should already be going off!

When you distributed, you ASSUMED that it was the 4, not 8/4, that was multiplying the (3 - 1). In doing so, you were bypassing the rules and just doing what felt right. If you followed the rules AND distributed, you would get this:

   (((8/4)*3) - ((8/4)*1))
   ((2*3) - (2*1))
   6 - 2
   4

It is not really the distributive property that led to the “wrong” result, but the fact that in distributing, the 4 was seen as the multiplier.

Those who say you should distribute first are putting the cart before the horse: you can't apply tricks to evaluate an expression before you first know what it MEANS, but they are thinking that the distributive property affects the meaning. (In fact, the distributive property is a waste of time here, because it makes you do two multiplications where only one is needed!)

The meaning is determined by the order of operations. Is the multiplication supposed to be done before or after the division?

I have observed that many students learn to distribute so well that they automatically do it when it is not helpful, as here, and even when it is not applicable, as when there is no addition at all!

Why students do it

In an unarchived answer in 2012, I listed reasons students have given for doing the multiplication first; on this occasion the question was about the expression \(a^2\div 4b + c\):

In fact, there are several different reasons people have given (this is a very popular question), some of which are better than others.

As your friend argues, the rules as usually taught tell us to do all multiplications and divisions left to right (within any cluster of them), and make no exceptions that would cause 4b to be evaluated first. Many of us here would agree with that, and be done with it.

Some people would evaluate 4b first because of a misunderstanding of PEMDAS, thinking it means multiplication should be done before division. I think you know they are wrong.

Another wrong reason, applied to a slightly different sort of expression, is a misunderstanding of parentheses: the rule that parentheses "come before" everything else leads them to believe that in an expression like 12/4(4-1), the multiplication 4(4-1) has to be done first. But the rule about parentheses really only says that what's INSIDE parentheses has to be evaluated first; the result is treated like any other number. (I sometimes call this the "sticky parentheses" view.)

Another reason given in relation to this second type of expression is the idea that the distributive property forces you to do the multiplication first, because they first evaluate 4(4-1) = 4*4-4*1 = 12 and then divide; but this begs the question, because the only reason they took the 4, rather than the 12/4, as the multiplier on the left, is that that's the way it looked to them. And, of course, the distributive property is only a way you may, if you wish, rewrite an expression to give the same value; it is outside of the question of what the expression in itself MEANS.

Ultimately, most people probably do it just because it feels right: the 4b looks closer together, so we naturally tend to want to do that first. But they can point to no rule that justifies that; and since math is about proof, and about doing what you KNOW is right, not just what feels right, this is not good.

For an example of “sticky parentheses”, see

Is the 2 Related to the Numbers in Parentheses?

For an example of seeing the division sign as a fraction bar (and a long discussion of not being swayed by visual appearance), see

Order of Operations and Fractions

Back to the 2017 answer …

Avoidance is the best policy

In books and handwritten math beyond the elementary level, we hardly ever use the horizontal division symbol, but use fraction bars instead, which leaves no ambiguity. As a result, the math community has never had a need to make a choice on this situation! It's essentially been left undefined, and it is textbook authors who came up with explicit "rules" to describe what is really just a language that developed organically, based not on carefully stated rules but on tacit agreement.

So which is the "right" way to read such an expression depends on what rules are in force in a particular community (math class, journal,  etc.) -- and what was intended by the writer.

I closed with a plea for peace:

As a result, in problems such as this, the error is being made primarily not by those who give "wrong" answers, but by those who post the problem in the first place (or pass it on). Anyone who really wants to do math correctly will want to communicate clearly about it, and will avoid anything ambiguous or uncertain. They should either fully parenthesize, or use the horizontal fraction bar, which makes the order clear:

      6             6
   --------   or   ---(2 + 1)
   2(2 + 1)         2

Arguments on social media about this sort of thing are a waste of time. But thinking about our conventions can be very enlightening. Next time, I’ll close everything out with a look at history, and some solid reasons to think the “new rule” is in fact correct.

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