This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?

## Online calculators are misleading

Last October, Julia from Wisconsin had a simple version of the question:

I’m seeking assistance because I’m genuinely curious … as to who is right and who isn’t.

Okay so basically I was on Reddit today and a viral math problem appeared in r/YoungPeopleYouTube. It was quite literally written as

8÷2(2+2). Now me and a fellow redditor are at an impasse:they say the answer is verifiably 1, and I say it is verifiably 16.When I plug in the equation into

MathwayorWolfram Alpha(as it is written, so literally typing it out in order and not changing how I type it or any extra key commands, etc),both services change itinto this: (8)/(2(2+2)) where 2(2+2) is the whole bottom of the fraction. If we were to solve the equation this way, naturally it would equal 1. 2+2=4 (parentheses first), then 4*2 (because with fractions you take care of the denominator first if you have nothing on top to do, correct?) which equals 8, 8/8=1.But how I am understanding the equation as it is written (which I fully understand is a poor example, but it is just a meme after all) is rather that it needs to be processed through (or thought through) our minds as if it was written like this:

(8/2)*(2+2). To my understanding, you do the parentheses first (which would equal 4), then you do the division because it comes first in the problem (M or D, A or S, left to right), so that gets you 8/2=4, then you’re left with 4(4), which naturally gets you the result of 16.Who is right here? And why? Are these calculators inputting it wrong simply due to machine error? Or am I wrong in my thinking? Any help would be greatly appreciated! Thanks!

Julia finds “strict PEMDAS” most natural, but sees that some websites seem to assume IMF. Or do they?

I answered:

Hi, Julia.

This problem, or others like it, have been viral literally for decades. We have discussed the issue in our blog here:

Order of Operations: Implicit Multiplication?

As explained there,

both approaches to evaluating mixed division and multiplication are taught, in different places. Yours, which treats all multiplications alike, with the same precedence as division, iscommonly taught in America. The other, which evaluates multiplication written without a symbol first, seems to becommon in other places, and among actual users of math. I explain in the subsequent post why I half-heartedly support the latter; but my overall advice isnever to write such an expression. (On the other hand, I am troubled by the idea that inserting the multiplication symbol where it was previously only implied would change the value of an expression; that’s why it’s only half-hearted!)People who post such questions likely know it will cause arguments, and enjoy doing so. You should just ignore it. It’s like arguing over British vs American spelling, only worse.

If you believe in IMF, you have to be consistent enough to know reflexively not to rewrite \(8/2(2+2)\) as \(8/2\cdot(2+2)\), because that would change its meaning, according to IMF rules.

I then repeated her experiments with online calculators, with more subtle results.

When I type the expression into

16, rewriting it as (8/2)*(2+2), along with various explanations of the issue, not necessarily correct! And when I type it intoWolfram Alphain“natural language” formatusing either the obelus, 8÷2(2+2) or the slash, 8/2(2+2), it gives16, translating it as (8/2)(2+2); but if I enter it as 8/2(2+2) in“math input” format, it translates it as a fraction with only 8 in the numerator, and gives the answer1. This appears to be due only tohow the math input works(as it makes this change within the input field), and not necessarilyhow they interpret the symbols. Using the obelus in “math input”, it still gives 16.

“Natural language”, on the left, means that it reads questions typed in plain English or symbols; it then shows its **interpretation**, which here shows that they follow the “strict PEMDAS” rule, doing all multiplications and divisions from left to right.

In “math input” mode, they use what you type (or buttons you click) as **formatting instructions**. When you type a slash, it inserts a fraction bar (as if you had selected the fraction button), and stays in the denominator until you explicitly exit; **this can’t be taken as their interpretation of the expression as originally written**, so it does not indicate acceptance of the IMF rule. (There is no obelus button; when you type “÷”, it just enters that without making a fraction, and gives 16.)

I find similar results when I use Mathway. When I **type in** 8/2(2+2), it translates it into a fraction as I go, like “math input” mode, and gives me this:

But when I **paste in** the same expression, 8/2(2+2), I get this, showing that it interprets the entire expression differently, using strict PEMDAS:

The same is true if I paste in 8÷2(2+2):

It would be harder to type that.

Conclusion: All these sites actually **interpret** expressions using strict PEMDAS!

But are they *right*?

## Books that mention IMF

### “Here’s where I learned IMF in the 1970s” … or not

Pete, from Georgia, wrote in early July, while I was writing this, and included an excerpt from his 1969 textbook, “Mathematics for Management and Finance with Basic and Modern Algebra”, 2nd ed. by Stephen P. Shao, Ph.D.:

I replied:

Thanks for the image from the book. I have long searched for examples of explicit teaching in a textbook, new or old, that specifies that implied multiplication is to be done first; this may be the first. Evidently this is American, despite the impression I have that it is American (high-school-level) texts that teach what I call “strict PEMDAS”.

(Actually, on reading more closely,

it says that, not justallmultiplications are done before divisionsimpliedmultiplications – though the latter are all they use. I’ve less often discussed this third option.)

This third option, All Multiplications First (AMF), is mentioned in the Historical Caveats post, where I refer to this explanation by Cajori:

I argued against it, however, in Order of Operations: Common Misunderstandings.

I would welcome examples of books, from anywhere and any time, that *do* explicitly teach about order of operations with implied multiplication, regardless of their view.

### Common Core doesn’t mention IMF … but a book about it agrees with us

Allison (from Canada, whom we met last time) said this:

I have found older generations to do it the previous way and

newer generations to do it the new way(seeing as all one question).

It is quite possible that strict PEMDAS is taught more often today, in Canada as in the U.S.; I simply don’t know.

I replied to this, in part:

I have from time to time tried to find evidence as to what rule has been taught when, and to the best of my knowledge,

it has always been somewhat mixed, to the extent it has been taught at all. As far as I can recall, when I learned algebra in the 60’s, they just didn’t write things like 6/2(1+2). We have heard from students who were taught both ways in their textbooks, and some who used workbooks that followed a different rule in their answers than their textbooks or teachers. We have heard from parents who think that each approach is what they learned long ago.

I don’t think there is any “new math” vs “old math” on this topic.There is, instead,variation.I would agree with your concern about “certain grades teaching it one way and then you end up in the upper grades with older teachers re-teaching it a different way”,

if that is what is actually done; do you have evidence of that? It appears that you are in Canada, judging by your address; I don’t know their standards. TheU.S. Common Core standardsdon’t state the order of operations rules, just saying that students should be taught to follow them! I found this Dummies page (claiming to represent Common Core) that appears to teach the implied-multiplication-first rule while calling itan “unspoken subtlety”, but concludes as I do that itshould be avoided:Maybe these people got it right. There is really no way to know what the person who wrote the expression meant for it to say, and that brings you back to where you started.

In any case, the solution is simply to teach students not to write ambiguous expressions, rather than to ask what is the “right” way.

The Common Core standard 6.EE.2.c, for example, merely says,

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents,

in the conventional orderwhen there are no parentheses to specify a particular order (Order of Operations).For example, use the formulas /(V=s^3/) and /(A=6s^2/) to find the volume and surface area of a cube with sides of length /(s = 1/2/)

The “Dummies” reference, an excerpt from the book *Common Core Math For Parents For Dummies*, was one I had just found in searching for what Common Core might say. Here are more quotes from it:

But PEMDAS misses some subtleties. Among these subtleties are

the following rules, which are often unspoken:

Do implied multiplications before division or explicit multiplication. Implied multiplication refers to something such as the 2(12) seen in the above expression, where there is no multiplication symbol. …The order of operations is an agreement among people that eases communication. If someone writes an ambiguous expression, such as

\(48\div2(9+3)\),then

shame on that person for writing an ambiguous expression. If that person meant\((48\div2)(9+3)\),then that’s what he should have written. If he meant

\(\displaystyle\frac{48}{2(9+3)}\),then he should have written that.

The order of operations doesn’t exist to trick people with ambiguous mathematical expressions; it exists so that people can communicate their ideas more effectively.

Amen.

## Claims of a universal standard

### “Wikipedia says IMF is standard” … or not

David, from Australia, wrote this May; at one point he referred to several sources others have mentioned as supporting IMF:

I am not a mathematician, however I have completed a couple of years of university maths as part of my IT degree. However

I had never heard of PEMDAS or BODMASuntil long after I completed university degree.I haven’t been able to find a serious article describing PEMDAS. My reference for the order of Operations is the Wikipedia page: https://en.wikipedia.org/wiki/Order_of_operations.

The paragraph that is relevant to this discussion is:

“In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n. For example, the

manuscript submission instructions for the Physical Review journalsstate that multiplication is of higher precedence than division, and this is also the convention observed in prominentphysics textbookssuch as the ‘Course of Theoretical Physics byLandau and Lifshitz’and the ‘Feynman Lectures on Physics.”So, if prominent academic sources state that implied multiplication is of higher precedence than division I think it is reasonable to describe this as a rule.

I have to say PEMDAS wasn’t part of my early learning, either. And it’s true that mathematicians don’t talk about it.

The sources quoted in Wikipedia are essentially all that I have found; a similar set were mentioned in a comment on my original post, and had been by others before that, as well as in videos. (It’s curious that they are mostly from physicists rather than mathematicians!) It’s also interesting that the rule these references support is not IMF, but **all** multiplications first, which is definitely not what is typically taught in textbooks today.

He omitted this statement in Wikipedia:

This **ambiguity** is often exploited in internet memes such as “8÷2(2+2)“, for which there are two conflicting interpretations: 8÷[2(2+2)] = 1 and [8÷2](2+2) = 16.

They recognize an ambiguity, rather than declaring that there is just one answer. (Of course, we need to recognize that Wikipedia is not just one person, and can contradict itself; but perhaps that is the point.)

He continued:

My questions for you are:

- Why does PEMDAS get an answer that disagrees with the
accepted order of operations used by mathematicians.Are you seriously suggesting that order of operations is a choicewhere I can choose theinternational standardand get the answer 1 and others can use PEMDAS and get 36?

I replied, in part:

I don’t expect to convince either the American educational establishment, or the calculator manufacturers, to drop the rigid interpretation of “PEMDAS”. So all

I can do is to emphasize the ambiguitysurrounding this, and especially tell people to avoid those silly memes that are intended to get people arguing.Let’s answer your two explicit questions.

To the first: It’s because, obviously,

people have come to at least three different “consensuses”about this. Basic P(E)(MD)(AS) is one of those; all multiplication before division [P(E)MD(AS)] is another; implied multiplication before others [P(E)J(MD)(AS)] is a third. These are different choices that can be made, and have been made, as Wikipedia points out.To the second: Yes, of course it’s a choice.

That’s what a consensus is:Achoicethat many peopleagree on. We need consensus in order to communicate; Any language is a consensus, in which other choices could have been made. I’m not advocating individual, chaotic choice! But we don’t have auniversalconsensus. This is not a good thing, so we need to talk about it. We can’t just pretend that everyone agrees, and ignore those who don’t.

There is no absolute “international standard”. I don’t like it either, but the references here do not prove that there is, and I have found nothing better.

### “ISO standards resolve the question” … or not

David also said this:

Many calculatorsgive different answers depending on whether you use the solidus (/) or obelus. The symbols are interchangeable. Therefore the answers should be the same.The relevant ISO standard says the obelus should not be used.

This last point is one that I have seen increasingly; unfortunately the ISO [International Organization for Standardization] standard is not available to the non-paying public, so it can’t really be considered a public standard; and that means I have struggled to find out whether it actually provides the absolute standards people think it does. This led me to do some more research.

I responded to several points, including what we saw above about Wolfram Alpha (noting that he must be referring to *online* calculators), and closing with this:

Mention of

ISOis interesting. In principle, you’d think that ought to be the final authority I’ve wished for – but it isnot really written for mathematicians, and my impression is that mathematicians don’t pay much attention to it (see here on Stack Exchange, for example, particularly the answer by Zeping Lee). The fact that it is hidden behind a paywall takes away a lot of the authority it might have had, since ordinary people can’t actually use it. I haven’t read it, but have occasionally found snippets of it, and this one (Section 7.1.3 of ISO 80000-1:2009) is interesting for our discussion:“These procedures can be extended to cases where the numerator or denominator or both are themselves products or quotients. In such a combination,

a solidus (/) shall not be followed by a multiplication sign or a division signon the same lineunless parentheses are inserted to avoid any ambiguity.”In other words,

they agree with methat the main solution is tonever write such an expression. They donotsolve it by decreeing what is the proper order of operations; I don’t get the impression that they actually state a complete order of operations as we might wish they would. (I believe their reason for deprecating theobelusis primarily that it has different meanings in different countries.)

I have since found more excerpts from ISO 80000-1 and -2; the following appears to be all that is relevant to the order of operations:

ISO 80000-1:

ISO 80000-2:

What I see here is that they are trying to avoid **any** ambiguity, even recommending that you add a space in \(ab^{-1}\), which is unambiguous as written, and around binary operators except the solidus for clarity. They ban the obelus, “÷”, because it is sometimes used for subtraction or negation, and recommend avoiding the solidus, “/”, because it looks too much like the letters “*I*” and “*l*“. They don’t like \(a/b/c\) or \(a/b\cdot c\), which could be misread by mistake, and don’t even mention \(a/bc\)! The focus is not on how to **interpret** an expression, but how **not** to **write** it – which is our point.

(When they mention “altering the order of operations”, they don’t mean changing the rules, but rewriting “\(\ln x+y\)” as “\(y+\ln x\)”.)

They do say that multiplication and division have precedence over addition and subtraction, but **not** that either multiplication, or implied multiplication, has precedence over division.

I concluded:

So that’s where I leave it. Since there are different opinions in reputable sources, and no one authoritative source,

the best I can do is to point out the ambiguity and recommend avoidance, which is what I’ve been doing for 25 years.The only people who insist on writing such expressions, in my experience, are Internet trolls who want to start arguments. I choose not to join them, nor to agree with people who predict catastrophe if this isn’t fully resolved. This simply isn’t that important.

Pingback: Implied Multiplication 3: You Can’t Prove It – The Math Doctors

Uncle JimHello, Doctor Peterson,

Since you’ve revisited the topic of implicit multiplication in order of operations, I write to share some observations. Before addressing specific topics, let me say that I agree with your overall conclusions:

Expressions like “8÷2(2+2)” are de facto ambiguous, so it’s best not to write such expressions, at least if your goal is to communicate ideas clearly and not simply to start an argument. And if you come across such an ambiguous expression in your reading, you ought to be careful about interpreting it. (Unfortunately, well-meaning authors do sometimes write such ambiguous expressions, perhaps because they imagine that they are following a generally accepted convention, or perhaps because they simply don’t notice the potential ambiguity at the time, and thus follow what I might call the “I knew what it meant when I wrote it” convention.)

Moving on to specific topics, …

* * * * * * * *

Common Core:

You wrote: “The U.S. Common Core standards don’t state the order of operations rules, just saying that students should be taught to follow them!” I noticed the same thing and share your bemusement. The link you supplied for the standard isn’t working for me, but I found these alternative links:

https://www.nctm.org/uploadedFiles/Standards_and_Positions/Common_Core_State_Standards/Math_Standards.pdf

https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf

On page 36, I find these parenthetical remarks about addition and multiplication of fractions:

Page 42 has this about division of fractions:

In all three cases, the intended interpretation for the expression to the right of the “=” sign is implicit multiplication first (IMF). On page 77, however, I find the expression “1/2 ab sin(C)” for the area of a triangle, clearly intended with the strict PEMDAS interpretation (1/2) ab sin(C).

* * * * * * * *

ISO Standards:

You have quoted, from some ISO document, an admonition that “…, a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line unless parentheses are inserted to avoid any ambiguity.” Note the words “multiplication SIGN” (my emphasis). Arguably, the passage you quoted doesn’t say anything one way or the other about multiplication indicated without a sign—that is, implicit multiplication.

While many ISO documents are expensive, ISO does offer a free SI summary brochure, linked from this page:

https://www.bipm.org/en/publications/si-brochure

Looking at the English language portion of the current (9th) edition of the brochure, I find on page 132 the expressions “1/μ₀c²” and “1/μ₀c”, meaning 1/(μ₀c²) and 1/(μ₀c), respectively (IMF interpretation). On page 204, I find “(2π⁵k⁴/15c²h³) T⁴”, meaning (2π⁵k⁴/(15c²h³)) T⁴” (also IMF).

Do the expressions I just cited confirm my speculation that the phrase “… shall not be followed by a multiplication sign …” was never meant to forbid implicit multiplication following a solidus? Or was the intent really to forbid all expressions whose meaning might depend on the precedence of multiplication (whether explicit or implied) vs. division, and was someone connected with ISO not following their own rules when they wrote the expressions I just cited? Also, if I were to look at a lot of ISO documents, would I find that they sometimes include expressions meant to follow strict PEMDAS rather than IMF, similarly to the expression “1/2 ab sin(C)” that I found in the Common Core standard? I wouldn’t be surprised if the answer to that last question were Yes, but I’m not about to spend hundreds of dollars on ISO standards documents in order to find out.

* * * * * * * *

Online Calculators:

Here are some examples of behavior of online calculators to add to the ones you mentioned.

1) While Wolfram Alpha (in “Natural Language” mode) interprets the input “1/2x” as (1/2)x, it interprets “1/xy” as 1/(xy). The input “1/xy, where x = 2, y = 3” thus gives the result 1/6 (displayed with a horizontal fraction bar). I learned about this from comments by Jenni Gorham in discussions about PEMDAS on her “The How and Why of Mathematics” YouTube channel.

2) As far as I know, the Google calculator doesn’t recognize input expressions containing variables, such as “1/2x”. It does, however, recognize some symbolic constants. In particular, although it interprets “1/2(3)” as (1/2)*3, giving the result 1.5, it interprets “1/2pi” as 1/(2*pi), producing the approximate value 0.15915494309. You can also give it the input “1/2π” (getting the “π” symbol by using a keyboard shortcut or by copying the symbol from someplace and pasting it) with a similar result. Similarly, it interprets “1/2e” as 1/(2*e), giving the result 0.18393972058.

I could share a variety of other examples of the behaviors of calculator apps, symbolic algebra systems, and programming languages, but I’ll stop here.

* * * * * * * *

You also asked about “books, from anywhere and any time, that do explicitly teach about order of operations with implied multiplication, regardless of their view.” I have found several examples, but I will save them for another message.

Dave PetersonHi, Jim. Thanks for the additions (and also for information you have provided in the past, which I didn’t include in the new posts for the sake of space — but may in the future).

What you have done with regard to both Common Core and ISO is what I didn’t have the patience to do: looking beyond what they explicitly

teachto examples of what they actuallydo. (By the way, I did include one of the links you provide for Common Core, in addition to the broken link to a specific spot … which I left in because it looked like it may be a temporary issue. It’s probably good to make it more visible.)Examples, of course, are tricky, because we can’t know whether any inconsistencies are intentional, or the result of the same sort of oversight that, to me, makes these expressions

inherentlyambiguous regardless of rules. The use of 1/2 in “1/2 ab sin(C)” is one I frequently point out; I suspect that there is an unstated rule that a space after a fraction acts like parentheses, just because it looks like it should. A descriptive (not prescriptive) grammar of algebra might well mention that! And I suspect that it is this sort of behavior that leads to claims that scientists and mathematicians in practice follow IMF, and also the reason many students do: It just feels natural, whether you’re taught it or not.By the way, it has long interested me that in writing compound

units, like “meter per second”, the official rule is to use negative exponents rather than division (as in “m^{-1}s” rather than “m/s”). I’ve wondered if that was an attempt to avoid these very ambiguities. But I see that used right alongside the examples you found ofexpressionswith division. (And I notice now that section 5.2 of the brochure does allow the solidus, but also requires either an explicit dot or a space for multiplication.)The information on calculators is also interesting. Ultimately, of course, that isn’t really relevant to what the “official rules” might be, as any programming language can set its own rules, which may depend as much on the needs of their medium or accidents of programming as much as on whether or not they try to follow standards.