This is the last of a series on our discussions, since I closed comments at the end of 2021, of Implied Multiplication First (IMF), the idea that multiplications written by juxtaposition, rather than with a symbol, are to be done before other multiplications or divisions. Last time, we saw that there is no “official” answer. Here, we’ll look at attempts to **prove** one view or the other. Most of them are variations on the misuse of the distributive property that was discussed in the original post; but there are some new things to learn from them.

## For Strict PEMDAS: “You just need to add brackets”

Some people argue against IMF, and in favor of “strict PEMDAS”, according to which all multiplications and divisions are done as written, from left to right.

Samuel, from Tennessee, argued in July 2022 that, since we can indicate that a multiplication is to be done first by putting parentheses around it, there is no need for the IMF rule:

The answer given ignores specific other operators that are made specifically for those instances. Namely the bracket or braces. Specifically

if you’re wanting a multiplier (implied or otherwise) to apply firstbefore other operatorsthat is why the bracket and braces were specifically made. The confusion of the issue is most schools skip the brackets and braces as unless you going into high level math operations they are never used. As suchmost people never remember them or learn them. Pemdas is just mnemonic device not the be all end all of the rule. And adding a new rule instead of taking advantage of the already written rulesbreaks conventionthat convention has already accounted for.Specific examples:

6 / 2(1+2) = X

In that equation X should equal 9. If however you want x = 1 you would simply write it

6 / [2(1+2)] = X

Anyways hope that helps clear the confusion.

It’s unclear what he thinks students are not taught; perhaps he means that only round parentheses, rather than square brackets and curly braces, are used in most math classes, and he feels that those different forms help here. He’s right that using extra parentheses lets us say what we want (regardless of which rule you go by); but which kind you use doesn’t really matter.

I replied, referring to the post he had read,

It’s certainly true that adding parentheses to clarify what you want to write is a solution to the problem

; alternatively, one can avoid the horizontal format for division and always write it as a fraction, as we recommend. But that doesn’t remove the problemfrom the writer’s perspectivefor awho comes across such an expression, becausereaderhe can’t be sure the writer sees things as you do. In that case, the main thing we need is to be aware that not everyone means it the same way, so that we need toask the writerwhat rules they were using.I also showed in the following post why the higher precedence of implicit multiplication might be considered correct. (But I still said we should avoid depending on it, because both readers and writers will always vary in their interpretations, even if rules were uniformly taught.)

He went on to argue that it isn’t “subjective”, a discussion I showed in the first of this series. In the end, he understood that the rules are not as well established as he had assumed (see the second of the series).

## For IMF: “The factor *belongs to* the parentheses”

More people seem to write to us defending IMF, often not having noticed that I (tentatively) supported it.

An anonymous reader from Romania, in September 2022, started with a seemingly innocuous question, and eventually revealed that he was arguing for IMF. He initially wrote “4 ÷ x^2 = 4 ÷ x*x”, which is false unless you do *all* multiplications before division (as we’ve seen that some do). Then he wrote,

But if it is written in the form 4 ÷ x(x) with the first x being

the common factor removedfrom x^2?Is there a rule in mathematics that requires parentheses to be inserted before the common factor?

I replied,

I’m not sure what you mean.

The expression 4 ÷ x

^{2}means that you are dividing 4 by the square of x. There is no “common factor” to be “removed”.Now, if you follow the (non-universal) “implicit multiplication first” rule, as explained in

Order of Operations: Implicit Multiplication?

then 4 ÷ x(x)

wouldmean the same thing as 4 ÷ x^{2}, because you would be doing the multiplication first, though this would be quite an unnatural thing to write, and risky because of the ambiguity. This would be different from what you originally wrote, with an explicit multiplication: 4 ÷ x^2 = 4 ÷x*x.

One would never do this “simplification”; but he evidently thinks that it should not be necessary to write 4 ÷ [x(x)]. He answered,

I don’t remember there being any rulethat tells me that if I take out the common factor before division,I must put the square bracket. That’s why consider that if there is no multiplication sign between the factored number and the parenthesis, it means thatthe number belongs to that parenthesis, that’s how we say it, it belongs. So the square bracket in front of the factor is implicit and I do those operations first. I don’t want to create a controversy, I just want to understand the logic, because I can give my children the wrong advice by telling them that it would be interesting to impose a rule that no matter what type of equation you have, when you take out the common factor you put before the parenthesis factor.

I responded,

Again, you are using the term “common factor” in a way that doesn’t fit its meaning; I don’t think the term means what you think it means! A common factor of

twonumbers (or expressions) means a factor theybothhavein common; for example, 6 and 8 share the factor 2, and x and x^{2}share the factor x, so 2 is a common factor of 6 and 8, and x is a common factor of x and x^{2}. None of what you write includes two different things with a common factor. So I’m struggling to understand what you are saying. Probably you just mean “factor”; you are “factoring out” an x from x^2.A general rule I recommend is to

put parentheses around anything that you change, just in case it might change the meaning. That may be more or less what you are saying. For example, when you replace a variable with a number, it is wise to put it in parentheses: When you replace x in “2x – 3” with -1, you need parentheses, “2(-1) – 3″ so that you are still multiplying by 2. If you just wrote “2 -1 – 3”, it would not have the same meaning. There are many other examples.In the case you are asking about, writing “4 ÷ x^2 = 4 ÷ (x*x)” is appropriate, because it ensures that x*x is taken as a unit. In addition, this is part of the reason we usually use the fraction bar rather than the division sign you are using; it is clear that \(\displaystyle\frac{4}{x^2}=\frac{4}{x\cdot x}\). No one could misunderstand that!

I do think you will benefit from reading the page I referred to, which is, in part, about when you need parentheses to make a meaning clear. The point there is that some people do teach that “if there is no multiplication sign between the factored number and the parenthesis, it means that the number belongs to that parenthesis”, while

others do not, so that using parentheses prevents misunderstanding. Furthermore, in your example, this prevents the error of taking it as you do in the first equality, but then changing to the explicit multiplication in the second, which changes the order. This doesn’tcreatea controversy, but works around anexistingone.

We’ll see more about this idea of **protective grouping** as we proceed. Even when parentheses are not necessary (by whatever rules you are using), they can keep you from silly mistakes.

## For IMF: “Absurd results for common factors”

This May, David from Australia argued for IMF, both catastrophizing and giving a faulty reason much like that last one:

In a discussion on this meme a posted article from you suggests that there is no good reason for completing the bracketed expression 8(14-8) before applying the division

I put it to you that there is a very good reason for doing so.

If we don’t, mathematics as we know it collapses.Consider this: 48 / (14*8 – 8*8) = 1.

Extracting the common factor, we get the equation 48 / 8(14 – 8) = 1if you do the division first you get the absurd result that 48 / 8(14 – 6) = 36

To get the correct answer i.e. 1,

you need to accept that implicit multiplication has a higher precedencethan Multiplication or Division.Would you like to comment and perhaps clarify the online article?

He had only read a snippet from Implied Multiplication, and not the following post, Historical Caveats, where I give arguments in favor of IMF. (Be careful about correcting someone when you don’t know what they’ve actually said.)

I responded, first discussing the proper way to manipulate expressions (which we’ll see later in an improved form), and eventually got to this:

In the same way, when you factor the denominator in

48 / (14*8 – 8*8), you need to put the entire factored denominator in parentheses, as48 / (8(14 – 8)), if you want to be sure to preserve the meaning.So if mathematics as we know it has collapsed, it is not the order of operations rule that did it, but your own carelessness with parentheses.

You must always think about meaning, rather than deriving meaning from an unthinking rewrite.beforeyou rewrite an expressionAs I said in the post, “you unknowingly apply an alternative interpretation when you think you are just applying the distributive property.” By not adding parentheses, you are implicitly

assumingthat the multiplication will be done before the division (so that the parentheses have no effect, and can be omitted), which is what you are claiming to prove.I hope that clarified things. The order of operations is a

choicewe make, not something we canproveto be mathematicallynecessary. As I said in Order of Operations: Historical Caveats, therearesome good reasons (human more than mathematical) for preferring the implicit-multiplication-first approach; but what we actually do has to beagreed upon by mathematicians, rather than proved, and that agreement is not currently universal (at least in the U.S.).

One “human” reason for preferring IMF is that it is easy to accidentally do what he does! But it is better to consciously express the correct meaning.

## For IMF: “Substitution requires it”

Pete from Georgia, like David, in July argued that substitution of a subexpression implies IMF:

Your article piqued my interest and I read it in its entirety. It explained the dichotomy of solutions very well. What caught my attention was your reference to what you call “the

sticky parentheses.” I would have preferred you call it “the sticky GCF or coefficient” though both appellations nicely describe how I view the problem.MY QUESTION IS THIS: IS NOT THE 2 OUTSIDE THE PARENS THE GCF/COEFFICIENT OF THE GROUP AND AS SUCH

IS IT NOT “TIED” TO THE EXPRESSION?Here’s my reasoning: Let us take two expressions: (2*1+2*2) and 2(1+2). By themselves, they will both compute to 1 and therefore they are equivalent values in either form. Factoring a binomial does not change its value. If we were to

substitutethe first one as in 6÷(2*1+2*2) it will solve to 1 using PEMDAS and the order of operations. And of course, if we were to simplify that expression by factoring out the 2 it will of course be the YouTube form of 6÷2(1+2). It is problematic (for me) that when two expressions of equivalent values when taken alone seem tochange their value when put into the formula with other operands! In order to arrive at an answer of 9 the 2 would have to be ripped away from the remaining factor of (1+2). I guess it wasn’t sticky enough!

I replied:

Did you miss the fact that the whole “sticky” idea is an

error? My point there is that parentheses are “sticky” only on the inside (that is, they hold together what isinsidethem, so that it has to be evaluated as a single entity), and the “P” rule says nothing about how they relate to whatever is on theoutside. That is covered by the other rules; and the whole issue is whether there is a rule that implied multiplication is to be done before division. All you are doing is relating the idea to factoring, which is an application of distribution; it is stillwrong. Whether you relate it to the parentheses or to the factor outside them, the a(b+c) construction does not “stick together” merely because of distribution/factoring.

He later referred to a description of “equality-preserving transformations” of algebraic expressions, which says,

The simplest equality-preserving transformations involve arithmetic. For example, you know that (2)(14) = 28. Transforming expression (2)(14) into 28 is an equality-preserving transformation.

You also know that

x(y+z) =xy+xz. That means(5

w) + (x(y+z)) = (5w) + (xy+xz)since you can perform an equality-preserving transformation on a subexpression.

Applying this to 6÷2(1+2), he said,

This proof manifests itself in this way: By

substituting a sub-expression with an equal sub-expression. So if we were to replace the 2(1+2) with the equal (2*1+2*2) it would have to resolve to 1 using PEMDAS.

So he is asserting that we can directly replace \(2(1+2)\) in \(6\div2(1+2)\) with \(2\cdot1+2\cdot2\) to get \(6\div(2\cdot1+2\cdot2)=6\div(2+4)=6\div6=1\).

### Preemptive parentheses

I responded:

No reasoning can provethat itmustbe interpreted one way or another; as I have said over and over, this is not a matter of mathematics, but of linguistic convention.Your reference to “equality-preserving transformations” actually

agrees with what I said, and not with your claim. You evidently missed the details of their example:You also know that

x(y+z) =xy+xz. That means(5

w) +(x(y+z))= (5w) +(xy+xz)since you can perform an equality-preserving transformation on a subexpression.

Note the

parentheseshe put around the sub-expression in both expressions, just as I said you need to do. He could, in this case, have just said5

w+x(y+z) = 5w+xy+xzbecause the parentheses

in this casedon’t change anything (because of a combination of order of operations and the associative property); buthe carefully put parentheses around each term on the LHS before doing the substitutionin order to demonstrate the level of caution that is needed in doing a substitution. You have to at leastthinkthose parentheses, and omit them only when youknowthey are not needed. This same practice would prevent all the errors I demonstrated in making mechanical substitutions — and would also prevent your mistake.

I had previously quoted what I’d said to David about substitution; now I restated that with the preemptive parentheses:

Your thinking is equivalent to what I discussed in the section headed “Misapplying the distributive property”. You are just factoring (undistributing) rather than distributing as that person was doing.

In mechanically replacing (2*1+2*2) with 2(1+2), you are ignoring the order of operations, or rather,

assuming that a literal substitution never affects the order. Here are some other examples in which such a substitution is inappropriate:Given 2x, if I mechanically replace x with y+3, I get 2y+3, which is wrong; I have to see it as 2

(x)and then replace(x)with(y+3), using parentheses, to get 2(y+3)in order to get an expression that means 2 times y+3. Without the parentheses, we are only multiplying the y by 2, and not also the 3.Given 3

^{2}, if I mechanically replace 3 with 1+2, I get 1+2^{2}, which is wrong; the original expression has the value 9, but the new one is only 5, because we are now using only the 2 as a base. I need, again, to use parentheses to preserve the intended order:(3)^{2}=(1+2)^{2}= 9.Given (2*1+2*2)

^{2}, if you factor 2 out of the base and write 2(1+2)^{2}, you will be wrong; the original expression has the value 6^{2}= 36, while the new expression has the value 2(3^{2}) = 18. In order to make an equivalent expression, you need to write(2*1+2*2)^{2}=(2(1+2))^{2}, replacing a parenthesized expression with an equivalent parenthesized expression, so that the multiplication will be done before the exponentiation.In the same way, when you factor the denominator in 6÷(2*1+2*2), you need to put the entire factored denominator in parentheses, as 6÷

(2*1+2*2)= 6÷(2(1+2)), if you want to be sure to preserve the meaning.

I added,

Comparing what you say with your reference, you would need to replace, not just 2(1+2), but (2(1+2)), with (2*1+2*2), in order to be sure not to change the order of operations.

But in our expression 6÷2(1+2),

, then 2(1+2) is not properly aif strict PEMDAS is followedsubexpression, as it is not evaluated as a unit. (And we aren’t changing its value to 3.) In the same way, I wouldn’t call “2 + 3” a subexpression of “1*2+3*4”, and expect to be able to replace it with 5. Why? Because I couldn’t write the entire expression as “1*(2+3)*4″.So the problem with your attempted proof is that you are

implicitly assuming one interpretation of the order of operationsin order to prove that only that order makes sense. This is circular reasoning.

So the bottom line is as I have always said: Rather than try to prove one interpretation “mathematically correct”, we need to recognize that all three rules make some sense, and have been taught, so that caution is the only rule.

JohnWhile this seems to be a US problem, as only in the US and maybe some parts of Australia does 6 / 2(1+2) =9

The rest of the world would say that the question is very poorly written and is the equivalent of “I seen my neighbour with binoculars”

But after reading many posts and comments, it does seem that Dave Peterson likes to go on about “enlightenment” but is very closed minded about the issue and looks to be an apologist for the US PEMDAS method that the vast majority of mathematicians and the rest of the world doesn’t agree with.

When you know how the underlying maths work, you can throw the order of operations out of the window and do it in any order you like, want to start with addition no problem, you can. The reason for the order of operations is the encoding of the problem, which save on brackets and lots and lots of terms.

For 6 / 2(1+2) =9 you always see this done as:

Apart from viral maths problems, I wonder if Dave Peterson can cite a use where there is a division with an implied multiplication that could be either in the denominator or as a multiplier that isn’t expressly denoted. You know some real world use cases.

I’ve seen lots of uses where you have a formatted depiction using the current example of 6 / 2(1+2) =

would have it as

but in text later have it inline as 6 / 2(1+2) =

But what I haven’t seen is any real world uses where 6 / 2(1+2) =

is depicted as

As I am not from the US, you might find uses like this in the US, but mathematics isn’t just a US thing, it is universal.

It would be interesting to see how many non US published works follow the US method.

Dave PetersonThis really should have been a comment on part 2, which is about evidence that either rule is

standard, rather than here, since you are not claiming to prove that it ismathematically required.But I am baffled, because while your first paragraph

represents what I have said(that such expressionsshould not be written, and that many people do seem to follow “IMF” in practice), your second paragraph utterlymisrepresents what I have said, as I frequently express an opinion that IMFcan be reasonably supported, and sometimes evenagree with its supporters. (When I respondto Americans who have been taught strict PEMDAS, I tell them to do what they are taught, even though I consider it inappropriate. But that does not make me an “apologist”.) I can only conclude that you have not fairly read what I wrote, especially Part 1.In fact, however, I am not at all sure that I have seen an American textbook that

explicitly teachesstrict PEMDAS (in the sense of giving examples using it in cases of juxtaposition). In my own experience, authors either don’t give any such examples, or write all divisions as fractions so that it couldneverarise. I have suggested that it may be (some) American teachers who (ignorantly?) give such examples and follow the rules they were told to teach even when they should not apply.I have several times asked for evidence of what is actually (explicitly)

taughtin various places, rather than mere examples of what isdone(which could easily be mistakes or accidental assumptions). This includes both examples of American (or other) textbooks that apply strict PEMDAS to juxtaposition, and examples of recent (American or other) textbooks that explicitly teach IMF. I ask this not because I doubt the existence of either, but because I want concrete evidence of the actual state of teaching, worldwide. I want to know whether those who respond to these memes with strict PEMDAS are doing what they were actually taught, or merely extending what they were taught to a special case that they were not taught (or forgot), where it is inappropriate. As far as I recall, I only have anecdotal evidence (from students who have written to us) that strict PEMDAS is taught, and personal experience that typical American textbooksdon’t mentionimplied multiplication one way or the other with regard to order of operations.JohnPeople on the PEMDAS side always argue a “Follow PEMDAS” argument, the problem with that is, it’s not an order of operations issue as to why there are 2 answers, it is that it is a reading issue, in both cases people ARE using the correct order of operations, the problem is the question it is the mathematic equivalent of “I saw a man with binoculars” you can’t tell who had the binoculars.

In the 6/2(1+2) the grouping for the vinculum is lost when you change it from multiline to single line and brackets should be added to remove any ambiguity which all computer based calculators do they will either evaluate it as (6/2)(1+2) as is the case for the US, or 6/(2(1+2)) as is the case for places that don’t follow PEMDAS. The issue is the assumption of what the input was meant to be, because the problem isn’t explicit.

Using explicit multiplication would be the simplest and quickest way to resolve it to be 9, 6/2*(1+2) but using implied multiplication reads that it is intended to be 6/(2(1+2)) and should be read as 6/((1+2)+(1+2)).

So there are 3, all perfectly valid answers to this problem 1, 9, or undefined.

Dave PetersonHi, John.

I don’t want to go quite so far as to say that everyone is “using

thecorrect order of operations” and that they are all “perfectly validanswers“, but I agree with your main point, that the expression is simplyambiguous(so that different reasonable perspectives can lead to different answers). In Part 1, I used a similar example of ambiguity in English, “I saw her duck”: Did she move down quickly, or did she own a waterfowl? Both are valid interpretations, because the sentence is worded poorly, so that we can’t be sure what role each word was intended to play.The ambiguity in these expressions, however, is due to the fact that there are different formulations of the “grammar” of algebraic notation, each of which either claims to remove ambiguity, or simply feels natural. My sense is that no one rule can fully resolve these issues, so, as you say, the best we can do is to rewrite defensively. I used my binoculars to see the duck that woman was holding, which is \(\frac{6}{2(1+2)}\) year old.

Doug HendrieOn the 6/2(1+2) problem, I see a factored parenthetical expression to be divided into the term 6.

I would distribute the common factor 2 back into the parentheses to give (2*1+2*2) to (2+4) to

6

6/6=1

For the answer to be 9, the (1+2) must be multiplied by the 6/2. That cannot happen mathematically. Factors are whole numbers. They cannot be fractions. And as factors are separated by multiplication and division, the 6 is separated from the 2 by explicit division.

The ONLY way the 6/2 can multiply the (1+2) is for the 6/2 to be in its own parentheses (6/2) thereby changing the whole essence of the calculations from a term divided by a factored parenthetical expression, to a straightforward 2 parenthetical expressions with implied multiplication.

And finally, using the golden rule of algebra,

what you do to one side of the equation, you must do to the other side.

(6/2)(1+2)=X one implied multiplication so divide both sides by the expression (1+2) to give

(6/2)=X/(1+2) simplify both sides to give

3=X/3 multiply both sides by 3 to give

9=X

6/2(1+2)=X one explicit division so multiply both sides by 2(1+2) to give

6=X*2(1+2) simplify both sides to give

6=X*6 divide both sides by 6 to give

1=X

I hope the above makes some sense.

I’m not a mathematician, just interested in numbers.

Thank you for reading this far.

Dave PetersonHi, Doug.

Unfortunately, no — it doesn’t make any sense.

First, when you say, “Factors are whole numbers. They cannot be fractions,” you appear to be confusing the idea of factors

of a whole numberwith factorsof an algebraic expression. Factors in this sense can beany number(or variable expression) at all. (That’s irrelevant anyway, since 6/2 = 3 is in fact a whole number.)Second, this has nothing to do with solving equations. It’s all about

how we interpret the meaningof an expression. You are simply doing as many people do,assumingthat the expression means what it appears to be at first glance, and everything you do is determined by that tacit assumption. It proves nothing, which was the point of this whole post.But, for the record, the only reason I, in some sense, defend the “strict PEMDAS” view is that it is (as far as I know) taught in some places. I don’t recommend it (in fact, I think it’s silly and unnatural); rather, I recommend

never writing anything like this, because, whatever rule you are taught, it is easy to misread it. And I also recommendnot responding to argumentsyou don’t even claim to know anything about, as so many people do.