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This question made the rounds recently -

$8÷2(2+2)=?$

Now, I glanced at this, answered "1" and then saw the full article printed in the New York Times, The Math Equation That Tried to Stump the Internet.

The article concludes that 16 is the correct answer, citing that

$8÷2×4$

when approached from left to right, results in 16.

My disagreement lies in the dismissal of the parentheses, as my explanation would be that

$8÷2(x+x)$ would, as a first step, simplify to

$8÷2(2x)$ and then

$8÷4x$

in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first.

Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?

(If you agree with the articles' '16', I am happy to hear the reasoning.)

The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.

Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter.

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    $\begingroup$ This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority? $\endgroup$ – Joseph O'Rourke Sep 7 at 21:42
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    $\begingroup$ For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division. $\endgroup$ – JTP - Apologise to Monica Sep 7 at 21:58
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    $\begingroup$ You are missing option #4: bad typography and shoddy proof-reading. Something like this should simply never be encountered, period. $\endgroup$ – Jörg W Mittag Sep 8 at 11:17
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    $\begingroup$ I agree with @JörgWMittag You should never see this and you should never write this. If you do see it, you may assume it was written by people who do not know what they are doing. $\endgroup$ – Gerald Edgar Sep 8 at 12:39
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    $\begingroup$ please also see math.stackexchange.com/questions/3313038/…, and in particular Bill Dubuque's answer (which I wrote based on his comment to keep a trace): math.stackexchange.com/a/3313474/96639. The linked article is excellent to answer this question. $\endgroup$ – WoJ Sep 8 at 16:09

13 Answers 13

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If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right).

48dvt12

But an expression such as $48\div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.

48d4p12

If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.

As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.

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    $\begingroup$ "that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues? $\endgroup$ – Joseph O'Rourke Sep 8 at 1:09
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    $\begingroup$ At the P stage of PEMDAS, it is the expression inside the parentheses that is evaluated. Parentheses have no effect outside of them. In "4(12)", there are both parentheses and an implicit multiplication. The evaluation is done as follows: $\endgroup$ – Jake Sep 8 at 11:45
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    $\begingroup$ (0) 48/4(12); (1) P: 48/4*12; (2) D: 12*12; (3) M: 144. $\endgroup$ – Jake Sep 8 at 11:46
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    $\begingroup$ Calculators do not an authority make! $2(4) = 2\times 4$. $w\div x(y+z) = w \div x \times (y+z) = \frac wx \times (y+z)$. Using fractions rather than elementary school-level $\div$ helps folks clarify what they mean. If one means $w \div [x(y + z)]$, one writes $\frac{w}{x(y+z)}$. If one intends $w\div x \times (y + z) = (w\div x) \times (y+z)$, by the equivalence in precedence between $\div$ and $\times$, reading left to right, we have $\frac wx \times (y+z)$ $\endgroup$ – Namaste Sep 8 at 15:50
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    $\begingroup$ @EricDuminil: It's certainly not multiplication. Nor any elementary operation under discussion. Your suggestion further down that it should come after exponents is definitely madness. $\endgroup$ – Daniel R. Collins Sep 9 at 2:13
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The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.

If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.

I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)

Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?

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    $\begingroup$ Thank-you. Yes, this - "I hate these examples, because they lead people to think math is very silly." Which is the issue with many Common Core examples taken out of context. To be fair, the Times author did note "No professional mathematician would ever write something so obviously ambiguous." $\endgroup$ – JTP - Apologise to Monica Sep 8 at 14:35
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    $\begingroup$ Just curious: why should 8÷2(2𝑥) be any different than 8/2(2+2)? $\endgroup$ – Eric Duminil Sep 8 at 17:01
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    $\begingroup$ I don't think anyone thinks it should be. I think @JoeTaxpayer thought it would clarify why his way of thinking seemed sensible. $\endgroup$ – Sue VanHattum Sep 8 at 17:28
  • $\begingroup$ @Eric And how is $8/2(2+2)$ any different than $$8/2 \times (2+2) = 8/2 \times 4 = \frac 82 \times 4 = 4\times 4 = 16$$ When there is the potential for misunderstanding of the intended order of operations, use parentheses, and this asked question would disappear. $\endgroup$ – Namaste Sep 8 at 21:06
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    $\begingroup$ There is no such thing as "implied multiplication" in mathematics. All there is, is an implied multiplication sign in mathematical notation. If in doubt, put the implicit sign in the equation explicity and confusion vanishes. $\endgroup$ – Gnudiff Sep 9 at 16:28
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I am with you on this one. I feel like concatenation (implied multiplication) is of higher precedence than explicit division. For me $8:2x$ means "8 divided by 2 x'es" - $8:(2x)$ not $4x$. Replacing $x$ with $(2+2)$ shouldn't change anything.

But the formal answer is that it's undefined. There is no C for concatenation in PEMDAS. For me it should be PECMDAS.

People in comments are suggesting juxtaposition (positioning side by side) instead of concatenation (joining strings together). PEJMDAS then. IMO both terms are very similar, but juxtaposition is indeed more common in the math world.

I definitely don't agree that concatenation should be resolved to multiplication before applying PEMDAS. By such logic you could also start with resolving E and get $8:2^2=8:2×2=8$ which is wrong. You are supposed to apply PEMDAS before resolving notation otherwise you are mixing order.

A final but necessary note- just never write division inline, it's too ambiguous unless you divide by one term only at the very end of line. I always either divide under the bar or for simple cases just multiply by the inverse.

EDIT: Rethinking again I am convincing myself even more that this is not and shouldn't be answerable in terms of PEMDAS. PEMDAS is a tool to resolve sequence of operations out of a notation. If you add notation that is neither of PEMDAS, there is no answer. Doesn't even matter if it's concatenation, derivative, logarithm, applying function... Neither of that is answered by PEMDAS.

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    $\begingroup$ Sorry for the repetitive comment with the other thread, I agree that saying it is undefined is an arguably reasonable stance. I disagree however with the "By that logic", since E has higher precedence the $2^2$ must be replaced by $(2 \times 2)$, and the problem goes away. The implied addition of those parenthesis is the very point of the precedence rule. $\endgroup$ – quid Sep 8 at 18:29
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    $\begingroup$ @quid just like we could replace $ab$ with $(a \times b)$ if we agreed on precedence of that. And the problem goes away. $\endgroup$ – Džuris Sep 9 at 1:04
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    $\begingroup$ @JoeTaxpayer: I think the term is juxtaposition, not concatenation. Concatenating $12$ and $34$ would yield $1234$. $\endgroup$ – Eric Duminil Sep 9 at 10:18
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    $\begingroup$ @Namaste: JoeTaxpayer was explicitely talking about the word "concatenation", I merely mentioned the existence of another word which might be more suitable. No need to try to make your point in yet another comment. $\endgroup$ – Eric Duminil Sep 9 at 18:53
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    $\begingroup$ @JoeTaxpayer: I agree with Eric that "juxtaposition" is the correct term (not "concatenation"). en.wikipedia.org/wiki/Juxtaposition#Mathematics $\endgroup$ – Daniel R. Collins Sep 9 at 19:23
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Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?

The conventional order-of-operations in textbook math (or any math) simply don't prioritize juxtaposed multiplication over other types of multiplication. The ordering is simply (elementary operations only):

  1. Exponents and radicals
  2. Multiplication and division
  3. Addition and subtraction

Each stage associates left-to-right, and grouping symbols (parentheses) always prioritize whatever's inside them. One can also observe all kind of useful symmetries built in this list: Operations are in pairs, with inverses together on each line (really two sides of the same coin in each case). Operations on same-base-powers downshift one place in all cases (to exponentiate powers, multiply the exponents; to multiply powers, add the exponents, etc.). Operations distribute over any operation on line lower. Etc., etc. See more.

If this is the agreed-upon ordering, then the value of the expression in the OP is 16.

Now let's say that we really, really want the value to be 1 instead. In this case we must add the extra, perceived (by the OP) rule that juxtaposed multiplication should come before other types of multiplying/division. We must insert a new stage:

  1. Exponents and radicals
  2. Juxtaposed multiplications
  3. Other multiplications and division
  4. Addition and subtraction

With that, the result produced in the OP is 1. But note that all the previous symmetries are destroyed! We no longer dependably (and memorably) have two operations on each line, inverses are not entirely collected on one line, the "shift down one place" shortcut for operations on same-base powers is gone, etc.

So: you can design a rule which forces juxtaposed multiplications to happen at a special earlier stage. But do you really want to destroy all the associated symmetries for that?

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    $\begingroup$ Well said, Daniel! $\endgroup$ – Namaste Sep 9 at 0:05
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    $\begingroup$ Just playing devil's advocate here : couldn't you add functions in the second line to get two "operations" again on each line? f(2x) and 2(2x) with the same priority? $\endgroup$ – Eric Duminil Sep 9 at 0:52
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    $\begingroup$ If your argument were valid, it would mean that $a/bc$ means $(a/b)c$, i.e. $ac/b$. Is this what you see in mathematical texts? Because I don't. I see e.g. $1/2x$ used to mean $1/(2x)$, not $(1/2)x$. You argue against your second order (the one with juxtaposition listed 2nd in a list of 4) but IME that is the order that mathematical writers actually use. $\endgroup$ – Rosie F Sep 9 at 9:24
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    $\begingroup$ I think the question regarding $/$ came up maybe on this site or some other SEsite at some point. It seems that in some circles there indeed was the convention that $some / thing$ would mean $\frac{some}{thing}$ back in the day when fraction where hard to typeset. Somebody then even mention an exact convention maybe it was until the next $+,-,=$ or even next $=$. That said I think it's a bit tangential to the question at hand. cc @RosieF $\endgroup$ – quid Sep 9 at 18:57
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    $\begingroup$ @quid you are right about $some/thing$. Here's a famous Feynman's paper and, among simpler $e^2/\pi i$, it even contains $(p_1+m/2m)$ in (29) which means $\frac {p_1+m}{2m}$. That's a bit extreme but any physicist would take $/2x$ to mean $/(2x)$. $\endgroup$ – Džuris Sep 10 at 13:10
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My disagreement lies in the dismissal of the parentheses, as my explanation would be that

$8÷2(x+x)$ would, as a first step, simplify to

$8÷2(2x)$ and then

$8÷4x$

It seems to me that you are:

  1. ignoring the Left To Right rule,
  2. not making the implied multiplication between $2$ and $(2+2)$
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    $\begingroup$ No, I want to make that multiplication, 2(2+2), first. That's why I got 1. And yes, I was giving that preference over left to right. Džuris offered the word I was looking for, "concatenation", which, whether my conclusion was correct or not, at least offered the vocabulary for what I was trying to express. $\endgroup$ – JTP - Apologise to Monica Sep 9 at 8:58
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    $\begingroup$ @JoeTaxpayer you might wish to do it, and you might be taught so at school. But there are no such mathematical operations as "juxtaposition" or "implicit multiplication". $\endgroup$ – Gnudiff Sep 9 at 16:45
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    $\begingroup$ You can want to make $8\div 2(2+2) = 8\div [2(2+2)] = 1$ by diverging from the classic PEDMAS by insisting multiplication ALWAYS take precedence over division. But that's contradictory to the accepted understanding in the math community of PEDMAS (PEMDAS). $\endgroup$ – Namaste Sep 9 at 16:59
  • $\begingroup$ ^^^^ @JoeTaxpayer $\endgroup$ – Namaste Sep 9 at 20:06
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    $\begingroup$ @Gnudiff There are no such mathematical operations as "parentheses" or "brackets" either. This is not a story about mathematical operations but about notation - the tools to convey intended order of operations. And juxtaposition is a notational tool, just one that is not strictly defined in PEMDAS. $\endgroup$ – Džuris Sep 9 at 20:44
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This is one of the cases where a linguist has to admit that we don't actually make the rules for a language. We discover them, and codify them. And there's always exceptions like

I before E
Except when your foreign neighbor Keith receives eight beige counterfeit sleighs from feisty caffeinated weightlifters

English is weird.

PEMDAS is a linguistic rule in the same vein. There is no worldwide governing board that defines the typography and interpretation for these equations. It's just how we got used to communicating with one another. If you don't use PEMDAS, you increase the likelihood of miscommunication.

However, when it comes to multiplication after a division, in particular multiplication with an elided operator there are dialectical disagreements about the meaning. Remember, all of these rules are designed to make communication easier. Communication is easier without all those parentheses in the way, so if there is indeed an unambiguous way of parsing expressions without adding parentheses, we like to do so. That's why we have PEMDAS in the first place. It got rid of the need to write (2*x) + (3*y) = 0.

So what about 1/xy? There's an easier way to write (1/x)y, mainly y/x, so that strongly suggests that the only reason one would write in this order is to imply 1/(xy). But then what about 2/3x? That could be 2/(3x) or (2/3)x. Writing fractions in front of a variable is an awfully common thing to do, so we typically don't burden it with extra parenthesis. Some prefer to write that case as 2/3*x.

Elided operators seem to bind "tighter" than others, but not everyone agrees on this. This tends to be the fundamental problem with this entire class of "trick" questions. Most are very comfortable with agreeing upon $8\div 4\times x$, but get less comfortable with $8\div 4x$. Situations like $\frac{1}{2x}$ show up often enough to want to be able to phrase them with a short parentheses-free notation. Whether this want is strong enough to change your interpretation is up to you.

So what's the correct value of $8\div 2(2+2)$? The correct value is a friend who's trying to be mean using ambiguous syntax's to look smart. Alternatively, the correct answer is to go back to whoever wrote it and ask them to fix the ambiguous phrasing.

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    $\begingroup$ So anyone writing $\frac 12 \pi$ must, invariably, mean $\frac 1{2\pi}$, because if they mean $\frac 12 \times \pi$, they would have written $\frac \pi 2$?? That a psychological argument, appealing to what you presume all rational people will do, when writing $\frac 12 \pi$ (i.e., your answer is opinionated, without support from Cog Sci, both out of the purview of math or math/ed.). Please stick to mathematical precedence, like PEMDAS/aka PEDMAS. and or other such mandated rules in math. $\endgroup$ – Namaste Sep 9 at 19:32
  • $\begingroup$ @Namaste "mandated rules"? As a professional mathematician and college teacher of mathematics, I read ${1\over 2}\pi$ as being equal in value to ${\pi\over 2}$. $\endgroup$ – paul garrett Sep 9 at 20:18
  • $\begingroup$ Yes, @paulgarrett. My bad. I meant to write "So anyone writing $1/2 \pi$ must invariably mean $\frac 1{2\pi},$ because if they mean $\frac 12 \pi,$ they would have written $\frac \pi 2$? Very sorry for my inadvertent mistake/typo in formatting. $\endgroup$ – Namaste Sep 9 at 20:22
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    $\begingroup$ @Namaste, I think your point is clear, though I disagree with it. $\endgroup$ – paul garrett Sep 9 at 21:29
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    $\begingroup$ @Namaste Actually, my argument is that the grammar is ambiguous because people can think along the way you don't believe they can. And I'd love to see what body can mandate rules in math, if you can point me at it. And the mere fact that people get this question "wrong" often enough to make a meme out of it suggests that I am not crazy in suggesting that people think this way. $\endgroup$ – Cort Ammon Sep 9 at 22:05
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I think (other than the fact that it's pretty much a deliberately ambiguous question) the thing that is missing is the concept of things being single terms, which ought to be evaluated first.

I would interpret the expression as 1, because I'd consider 2(2 + 2) to be a single term that should be evaluated first before the answer is substituted back into the rest of the expression. If it had been written with a multiplication sign it wouldn't be a single term, so the answer would end up being 16. Similar to how 1 ÷ 2 would be treated differently to 1/2.

This can also be expressed as implicit operations having higher priority than explicit ones.

(I'm based in the UK, so likely to have come from a different perspective - I was certainly taught, when we got that far, that implicit had priority and thus you should treat terms as single items)

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    $\begingroup$ This is very interesting, it appears that the concept of "single term", which only appears as a separate math article in Simple English Wikipedia, is something coined specifically for teaching elementary math. It might be responsible for a lot of confusion, because it appears to explicitly apply to polynomials. However, expression 8/2(2+2) is not a polynomial, since it contains division, therefore concept of single term doesn't seem to be applicable here. $\endgroup$ – Gnudiff Sep 9 at 19:26
  • $\begingroup$ Why do you say that given $8\div 2(2+2) = 8/2(2+2)$, that $2(2+2) is a single term. I've yet to encounter "single term" defined in mathematics, save for the single term "x", or "y" or "z". Please provide me with your working definition of single term. I suspect you created it "on the fly" when trying to answer this question. $\endgroup$ – Namaste Sep 9 at 19:47
  • $\begingroup$ @Namaste I actually found "single term" with some difficulty: it appears to be some sort of elementary math education definition, apparently in the US(?) whatcom.edu/home/showdocument?id=1772 $\endgroup$ – Gnudiff Sep 9 at 20:08
  • $\begingroup$ "a term is a single mathematical expression", which in this case, is not $2(2+2)$, but actually $8 \div 2(2+2) = 8/2(2+2) = 8/2\times 4$ Because division and multiplication have equal precedence, in which case one evaluates from left to right, your single term evaluates, $(8/2) \times 4 = 16$. You clearly misunderstand what the link means: that "a term is a single mathematical expression", as is $8 \div 2(2+2)$. $\endgroup$ – Namaste Sep 9 at 20:15
  • $\begingroup$ @Namaste, "Term" is a phrase that you wouldn't argue about if any of those numbers was replaced by a variable, it clearly applies to constant terms in an expression which also contains a variable, and it is a natural use that I have heard frequently in my schooling to also apply it to elements within an expression such as this one in the obvious manner, as a shorthand for talking about groupings via implicit operations vs explicit operations. Maybe that's not common wording worldwide. $\endgroup$ – meta Sep 10 at 12:14
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This would be better as a comment but I don't have sufficient reputation. Daniel R. Collins asked for an example of a mathematical text writing something like $a/bc$ to mean $a/(bc)$, and it might be useful as a counterexample to the strong claims that a mathematician would never write such a thing.

On page 2 of Geometry Revisited by Coxeter and Greitzer, in the context of the Law of Sines, they write that $\sin A=a/2R$, where $R$ is the circumradius of the triangle $ABC$. On the next page, they ask the reader to prove that the area of said triangle is $abc/4R$.

So examples in pure math books by well-known mathematicians can be found, but the only reason I could easily point to this example is because I think it's the only one I've personally seen and I remember noticing the ambiguity right away.

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  • $\begingroup$ Indeed, I've seen a few such things, and/but the stuff in front (with the early division) is somehow a "coefficient" of the other part "$R$" in your example. Also, this broaches the possibility of disambiguating by using "units"... so in engineering, physics, chemistry contexts a good fraction of the ambiguity is removed. $\endgroup$ – paul garrett Sep 9 at 20:20
  • $\begingroup$ Thanks for that example. Admitting that they have a number of other nonstandard typography issues on just the first few pages; e.g., they use $XY$ as the length of line segment from point $X$ to $Y$, they omit the words "show" or "prove" in all exercises for brevity, etc. $\endgroup$ – Daniel R. Collins Sep 9 at 22:09
  • $\begingroup$ @DanielR.Collins "...they use XY as the length of line segment from point X to Y." This has been my standard naming practice. Is this nonstandard? $\endgroup$ – Nick C Sep 10 at 15:58
  • $\begingroup$ @NickC: I would have thought $XY$ is the line segment and $|XY|$ or something the length. Coxeter/Greitzer themselves note they're using $XY$ because of a typographical limitation. $\endgroup$ – Daniel R. Collins Sep 11 at 0:39
  • $\begingroup$ @DanielR.Collins I always used $XY $ for the length, and $\overline {XY} $ for the segment itself, along the lines (no pun intended) of: en.m.wikipedia.org/wiki/Line_segment $\endgroup$ – Nick C Sep 11 at 0:58
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The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.

I'm going to suggest a variant of this which you might consider a fourth option: The $\div$ operator is at best confusing if not ambiguous, and should not be used. For a long time I actually thought it was a nelogism based on a fraction bar and placeholder dots, and derided it on that too, but apparently it does have much deeper history. Still, it's not used whatsoever by working mathematicians, engineers, physicists, etc. for very good reason. The lesson here, in a math-education context, is that it shouldn't be used. Full fraction bars for division make it entirely unambiguous, and use of $/$ is almost universally interpreted as binding looser than implicit multiplication (but should still probably be avoided in ambiguous contexts).

Beyond that, this question makes great lesson for students in the potential ambiguity of notation, disagreements about meaning, and importance of defining any notation you want to use for which there is not an obvious agreed-upon meaning.

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    $\begingroup$ For what "very good" reason is the obelus not used? In programming, it's a lot harder typing Alt+246 than hitting the forward slash, and in hand-written stuff it's easier and clearer to draw the numerator over the denominator. But I use the obelus symbol quite frequently in non-programming contexts (and in comments in code) because it's less confusing than a forward slash. Unless I feel like going through the trouble of formatting a proper fraction slash, MathJax, etc. I also never assume 1/2X means 1÷(2X), and write (1/2)X, ½X, 1/2 X, or ½ X to be clear in that context. $\endgroup$ – MichaelS Sep 9 at 9:58
  • $\begingroup$ I agree that $\div$ is elementary, and not used at the secondary school level, nor at all in college/university. Good point, R.. $\endgroup$ – Namaste Sep 10 at 1:31
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In the order of operations, one multiplication does not take precedence over another; all multiplications and divisions are performed from left to right. A so-called "concatenation" with parentheses a(b) has the exact same meaning as "a times b". The parenthesis operation only takes precedence over other operations to evaluate what is inside. After that, it becomes an M, and the P is done.

It is very unfortunate that Casio wasn't competent enough to parse an unambiguous context-free grammar correctly in one of their models. That model should be recalled, because it is wrong.

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In a comment you write

"multiplying what's in parentheses should take priority"...

But the only expression within parentheses is the sum: $(2+2)$. Hence we have $$8\div 2(4) = 8 \div 2 \times 4.\tag 1 $$ I think you are confusing concatenation of $2$ with $(2+2)$ as meaning that All of $2(2+2) = 2 \times (2+2)$ must be evaluated as the dividend of $8$.

Multiplication and division have equal precedence, so we proceed left to right, meaning, we evaluate $$8\div 2(2+2) = 8\div 2 \times (2+2) = (8 \div 2) \times 4 = 4\times 4 = 16.\tag{2}$$

See also Symbolab's Order of Operations Calculator.

Had the expression actually been written as follows: $$8 \div \big(2(2+2)\big),\tag{3}$$ such that the product of the sum $2\times (2+2)$ is enclosed in parentheses, only then would the answer be $1$.

This question highlights why authors and students are best advised to disambiguate the expressions they write, even though 16 is correct in this case, there's no reason not to write the expression as I did so in $(2)$. If one intends to write the expression in $(3)$, help us all by using the parentheses!

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    $\begingroup$ But most engineers and scientists working with frequencies and angular frequencies would always interpret $x/2\pi$ as $x/(2\pi)$, not $(x/2)\pi$. The real silliness of the question is the implicit assumption that math notation is a pointless context-independent game which kids have to learn to play for no practical reason. $\endgroup$ – alephzero Sep 8 at 15:30
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    $\begingroup$ @Džuris that's just not analogous. The rules regarding precedence I am aware of do not include "concatenation" (while they do include exponentiation) thus if one wants to treat it in that context at all it seems by far most natural to treat it like multiplication which is what it stands for. I'd say if we do anything else we enter uncharted territory and the question becomes meaningless. Except maybe as a speculativeone what a good rule might be or what the writer might have meant. $\endgroup$ – quid Sep 8 at 18:23
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    $\begingroup$ @Džuris since when is $(a + a) = a^2$ for all $a \in \mathbb R$. By precedence of operators, with division and multiplication being equivalent in precedent, we evaluate left to right, after adding $a+a = 2a$: $$b\div a(a+a) = b\div a \times (2a) = \dfrac ba \times 2a= ba$$ $\endgroup$ – Namaste Sep 8 at 21:01
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    $\begingroup$ If I may say so, you appear a bit too sure of yourself. The notation is ambiguous, and it's perfectly fine to say so, instead of insisting that 16 is the one and only answer. $\endgroup$ – Eric Duminil Sep 8 at 21:26
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    $\begingroup$ @Namaste concatenation is not the same notation as $\times$. Just like $a^2$ is not the same notation as $a \times a$ and $2a$, $2 \times a$ and $a + a$ are all different. You can't arbitrarily replace concatenation with multiplication just like you can't do $a - 2 \times a = a - a + a$. $\endgroup$ – Džuris Sep 9 at 1:14
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Part of the difficulty is that order of operations is taught wrong.

The basic rule of order of operations is this:

"Operations are performed from left to right unless..."

The first unless is the existence of parentheses.

Now here's where another factor (pun unintentional) comes into play: we write in lines. Because of that, something has to come first as we read.

What we should say is that multiplication and division are coprecedent: neither takes place before the other, at which point, the "left to right unless" comes into play.

So in $8 \div 2 \times 4$, we do the $8 \div 2$ first, getting $4$, then evaluating $4 \times 4$, getting $16$.

Unfortunately, when we reduce order of operations to an acronym, we lump M and D into the same word, and since it's harder to pronounce PEDMAS, we say PEMDAS.

(A and S have the same coprecedence relationship: $8 - 3 + 5 = 10$, not 0)

Here's a more in-depth overview of order of operations:

https://youtu.be/A3o555TzKb0?list=PLKXdxQAT3tCtutiLV9QYh_2lDuNphhhCP

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  • $\begingroup$ Well said, @Jeff! $\endgroup$ – Namaste Sep 9 at 20:09
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At first I thought comments would suffice, instead of an answer, because there have been a couple good answers already.

However, it appears that there are a number of issues at hand, some of them regarding math education rather than mathematical notation.

Nevertheless, let me first make a couple notes about notation though, as part of the subject.

  • I would like to note, that the original question could only arise in "paper" maths, which has the wonderful opportunity of its notation being able to be interpreted differently.

In computer programming, math expressions usually are not ambiguous at all. Every math expression is calculated the same way in the same programming language, and writing it one way you can only get one and the same answer with respect to operation precedence.

The order of precedence is, as @Daniel-R-Collins wrote:

  1. all operations calculated started from left-to-right, subject to below precedence in order:
  2. operations within parenthesis taken as separate list of operations and calculated first, using this same set of rules.
  3. exponentiation (and some other operations less common in school equations, such as modulus)
  4. multiplication/division (usually denoted by "*" and "/" signs)
  5. addition/subtraction

So, if you wanted 8÷2(2+2) to mean 16, you'd write it: 8/2*(2+2) = 16 (1)

If you wanted it to mean 1, you'd have to write: 8/(2*(2+2)) = 1 (2)

Computers don't care what you want. If you wrote (1), you'd always get 16, regardless of what you wanted [Note that there exist programming languages which do use eg. postfix notation for mathematical operations, such as Reverse Polish Notation, which would write the above as: 8 2 / 2 2 + * to get 16, and 8 2 2 2 + * / to get 1.]

Nevertheless I do not know any general purpose programming language, in which 8/2*(2+2) would be 1. So at least this front has come to an agreement on the subject.

  • Next, as I wrote in comments already, there are five mathematical operations + parenthesis, which are described by the OP's PEMDAS rules. Five. There is no such mathematical operation as "implicit multiplication". The only thing there is, is an implied multiplication _sign_ when writing mathematical equations on paper.

It can also be reasonably inferred, how this could have come about. Writing equations on paper you would like to get rid of ·, which is a pesky little symbol that is taking space, seems to be more frequent than division, and is usually also tightly coupled to the expression, so it would frequently be first in priority of expression. Also, on paper you do have the large horizontal bar for all your division needs, so you could simply stretch a luxurious $\frac{8}{2(2+2)}$ over your implied multiplication and avoid any confusion whatsoever.

As a rule of thumb, if you turn something implicit into explicit, it should show what it was about. If you do not construct any imaginary "implied multiplication", but take the "implicit multiplication" to simply mean "implicit multiplication sign", then what you get left from 8÷2(2+2) is exactly 8÷2·(2+2).

  • Now we have come to the part of the matter, which keeps popping up in many comments. Forget about implicit multiplication, what about the division sign?

As I read that in engineering (at least in some parts of the world), it is frequently expected that 1/something implies 1/(something), using slash sign as a fraction bar. While I can't argue for or against it in particular field, I would be surprised if it didn't only work for values, the correct form of which were expected to be known. If you know that the modulating signal frequency in Hertz is equal to $$\frac{\omega_m}{2\pi}$$ then it is expected that you could often write it as ωm/2π, and indeed it seems that people do.

However, this agains seems to work in particular jargon of the field and on paper. You can't willy nilly put long or short fraction bars when writing computer programs, therefore even engineers will have to use computer notation, when dealing with programming. Computer notation, again, to avoid ambiguity, deals with the matter simply by having a division sign that follows the same above listed PEMDAS rules. If you want to write $\frac{1/2\pi}{3+x}$ on computer, you have to write 1/(2*pi)/(3+x). No shortcuts.

All of this leads me to the following conclusions:

  1. The way to calculate expressions such as 8/2(2+2) will feel more "natural" depending on how you were taught. In addition, in some fields, it is more expected to for division signs in electronic texts to be meant as long fraction bars on paper.

  2. As described here and in @Daniel's answer it might be argued that it is more correct to use the form where 8/2(2+2)=16, simply because there are no no hidden expectations such as "implicit multiplication" and "my division signs are in fact long fraction bars".

  3. I would be surprised if much of the world didn't expect their equations the 8/2(2+2)=16 way. The whole of the former USSR, for example, where nobody ever heard of implicit multiplication or fraction bars [we used ":" as division sign in notation and used parenthesis a lot more than fraction bars. Starting with some 5th grade it was more frequent to see school examples as $15:(3+3)$ rather than $\frac{15}{3+3}$ ]

  4. Nevertheless, when it comes to math education, the considerations should probably include:

    • whatever is taught to people, they must be made aware of the possibilities of different interpretation to notations;

    • especially if they are taught that 8/2(2+2)=1 , they should be made aware that no computers will understand it this way, because exposure to computers is ubiquitous nowadays (I am not even talking programming, try doing this in Excel);

    • whatever is taught to people in school will probably need to be in line with the existing textbooks.

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  • $\begingroup$ You make some good points, e.g. with engineering. But for many programming languages (e.g. Ruby/Python/Java/...), it's not even allowed to write 8/2(2+2) so I'm not really sure what your argument is. Juxtaposition simply isn't considered in those programming languages because f(x) could mean f times x or f function evaluated at x. OP's question is about juxtaposition. $\endgroup$ – Eric Duminil Sep 9 at 19:18
  • $\begingroup$ @EricDuminil No programming language I know of, allows you to omit multiplication sign. It is an indirect confirmation of the fact that when you write 2(2+2) you are not creating any special new operations [such as some strange implicit multiplication with higher priority than the rest of ops], but just using a shorthand, and 2(2+2)= 2*(2+2) which follows the normal rules as if it was written in full. $\endgroup$ – Gnudiff Sep 9 at 19:32
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    $\begingroup$ No programming language I know of, allows you to omit multiplication sign. so why are you talking about programming languages in a question which resolves around the omission of the multiplication sign? It is an indirect confirmation. No, that's just confirmation bias from your part. I explained why juxtaposition cannot be included in programming languages in my previous comment. Otherwise, should abc be a*b*c or a*bc or just the abc variable? The rest of the answer is really good BTW. $\endgroup$ – Eric Duminil Sep 9 at 19:36
  • $\begingroup$ @EricDuminil (1) References to programming languages are relevant, because programming languages have to implement math correctly to be useful, and people have to frequently use computers [as I wrote, not only programming, but also Excel], so they should know what the computer expects. Your mileage may vary, but at least for myself, I see the connection. $\endgroup$ – Gnudiff Sep 9 at 19:50
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    $\begingroup$ The former USSR hadn't heard of fraction bars and implicit multiplication? Is that a joke? Have you read any article by Soviet scientists? Here is one of the Landau's books. It's full of implied multiplication and fraction bars and doesn't contain $:$ or $\div$ at all. Footnote on page 28 contains $~\hbar / mL^2$ which means $~\hbar/(mL^2)$. It is not some Anglo-Saxon notation, it's just how people (outside secondary school) write math. $\endgroup$ – Džuris Sep 10 at 13:49

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