Abstract: even in mathematics, ambiguities can
be hard to spot. The phenomenon seen here in arithmetic goes beyond the usual
PEMDAS rule and illustrates an ambiguity which can lead to
heated arguments and discussions.
This document is organized chronological. See especially:

"The problem 2x/3y1 with x=9 and y=2 was actually posed for a 5th grade homework question and it is interesting how much debate has occurred between our parents. The students who had help from parents had the answer of 2. The teacher explained that the correct answer was 11." 
x=9; y=2; (2x/3)y1 gives 11 2x/(3y)1 gives 2It is not clear what the textbook had intended with the 3y. As written, it can be interpreted both ways. Yes, one could argue that without brackets the given order matters. One can however also argue that "3y" is a unit which belongs together. So, everybody is right and that the textbook problem has just been unclear."
x/3x 2 x Out[1]=  3 x/(3x) 1 Out[2]=  3The computer has in this case chosen to do the division first. This is natural if one looks as division and multiplication as operations on the same level and the division is on the left. To play the devils advocate, note however that the famous PEMDAS (parenthesis, exponents, multiplication, division, addition and subtraction) has the M before the D so that we would have to accept the second one ...
gives both 10/3. Try it out. Funny by the way (and totally unrelated to the topic discussed here) that the results are different
3.333333333333333 3.3333333333333335illustrating that computers do not honor basic laws of arithmetic. The mathematics of rounding error arithmetic is subtle. My PhD advisor Oscar Lanford knew much about this. By the way, also Mathematica adhers to BEDMAS and does division before multiplication. It is in line with Excel and javascript as probably most programming languages like Perl.
3+43 = (3+4)3 = 3+(43)This is called associativity. But already this is trickier if one tries to modify this because
34+3 = (34)+3 = 3(4+3)How come, associativity does not hold here? You find out. Since the logarithm gives an isomorphism between the group (R,+) and (R {0}, *), where * is multiplication, we also have ambiguities in the multiplicative case evenso the multiplicative group (R{0},*) is by definition associative. But it is the role of the division which must be made clear:
12/(3*2) = 2 (12/3)*2 = 8gives different answers. This can not be discovered well in algebra situations like
12/3xwhich is ambiguous. Both of the following expressions are clear, but give different results:
12/(3*x) = 4/x (12/3)*x = 4xThis is also often a source of error in addition, as the minus sign has to be carried through:
34+3 (34)+3 = 2 3(4+3) = 5It is this case which makes the argument that the bracket should be placed to the left. The additive case is also a reason which seduces not to write the brackets in the multiplicative case. It always leads to trouble.
x/x/x ?We again have two interpretations
(x/x)/x = 1/x x/(x/x) = xBut since now, we have only divisions and no multiplication, it is even less clear what comes first. When dealing with fractions, we often make this clear by the size of the division lines
x  x = x  xIt has become clear that we first do the division with the smaller division line and
x  x = 1/x  xWhat does Mathematica do? Lets look and enter
2/100/2
2/(100/2) = 1/25 (2/100)/2 = 1/100Mathematica choses the second choice (as does javascript). It makes the first division first.
3^3^3 ?Again we have an ambiguity with two cases:
(3^3)^3 = 19683 3^(3^3) = 7625597484987Leaving out the brackets invites trouble. Yes, one could argue that the first example is more natural, but then, a printed version of a textbook might have the third 3 smaller so that the reader is seduced to first compute 3^3 and put that into the exponent, producing a much larger number. In the following example, this is even less clear what the writer intends because expressions like 10^{2} are often treated together as we write 10^{6} for a million. The 10^{6} has become a unit: so, if we write
2^10^2which of the two answers
(2^10)^2 = 1048576 2^(10^2) = 1267650600228229401496703205376is correct? Maybe we should ask our professional computer algebra system again. Mathematica 9.0 gives the second choice, if 2^10^2 is entered!
For (a/b/c) the algebra system starts evaluating from the left, while for (a^b^c) it starts evaluating from the right. 
There is only one solution: write the parenthesis. 
256 // Sqrt // Exp // Floor // PrimeQwhich takes the square root of 256, then exponentiates it, take the integer part and then answers the question whether this is prime. It is equivalent to
PrimeQ[Floor[Exp[Sqrt[256]]]]but avoids the brackets. I use the first way often when computing interactively, while the second one when writing code like the following which looks for the statistics of primes in a subexponential sequence of integers:
F[n_]:=Floor[Exp[Sqrt[n]]]; s=Table[If[PrimeQ[F[n]],1,0],{n,1000}]; G[n_]:=Sum[s[[k]],{k,n}]*Log[n]/n; ListPlot[Table[G[n],{n,Length[s]}],PlotRange>{0,1}]If you are curious: here is the Plot.
"Internet rumors claim the American Mathematical Society has written "multiplication indicated by juxtaposition is carried out before division," but no original AMS source exists online anymore (if it ever did). Still, some early math textbooks also taught students to do all multiplications and then all divisions, but most, such as this 1907 highschool algebra textbook, this 1910 textbook, and this 1912 textbook, recommended performing all multiplications and divisions in the order they appear first, followed by additions and subtractions. (This convention makes sense as well with the Canadian and British versions of PEMDAS, such as BEDMAS, BIDMAS, and BODMAS, which all list division before multiplication in the acronym.) The most sensible advice, in a 1917 edition of Mathematical Gazette, recommended using parentheses to avoid ambiguity. (Duh!) But even noted math historian Florian Cajori wrote in A History of Mathematical Notations in 192829, "If an arithmetical or algebraical term contains / and x, there is at present no agreement as to which sign shall be used first."The article has links to sources of textbooks. Here is entry 242 in Florian Cajoris book "A history of mathematical notation" (page 274), which is mentioned in that quote
"I make math videos on YouTube on the channel "MindYourDecisions." Some of the most popular videos are ambiguous expressions involving the order of operations. In doing research, I came across your website and the problem: What is 2x/3y  1 if x = 9 and y = 2? I would answer 11, which was what the 5th grade teacher said. I was stunned that none of the 60 students your Harvard calculus class answered 11 (you explained 58 got the answer 2; and then 2 got the answer 18/5).My answer:
"yes, it is an interesting thing. None of the answers is of course "correct" as we know that both the BEDMAS and PEMDAS interpretations can be used without violating any authority. As indicated on the page, the answer 11 is what most computer languages get. You were obviously taught that. It would be interesting to know what percentage of humans say 11. My experiments say that it is very rare. Most do the multiplication before division as PEMDAS seems to be more popular and more taught in schools. BEDMAS of PE(MD)AS appears to be taught much less. The only thing we know is that the claim that one of the answer is the only right answer, is wrong."
"The problem is a picture of a mobile phone gets 9 from the equation : 6÷2(2+1) which according to them says it would be 1. In your argue you define obelus and division slash to have to whole different meanings. Well yes they actually do have two different meanings and that's why you normally never use obelus. Only American may still use it, but this sign has been removed in use of equations of scientific papers due to its historical problematic. First of obelus in Northern Europe means subtraction. Second of all, obelus is recommended removed in the use of science due to that we already have a sign for either of these (division slash ("/") and subtraction ("")). Even though according to your argue that obelus and division slash should imply two different meanings you often only have one option on a calculator to make a divisionsign."
My Answer:
"Thanks for the example 6÷2(2+1). It illustrates the ambiguity too. Yes, depending on whether one is in the PEMDAS or PEDMAS team, one gets 1 or 9. Its also a beautiful example, where one can see heated debates. As pointed out and also previously pointed out by others in the literature list, there is no right answer. It depends on which rule is applied. Both 1 and 9 are correct. I always see the obelus as a synonym for / but it can be even more confusing and so, yes, should be avoided. "
In our Second midterm PDF of November 2017, we had in Problem 9 the expression x^3/3+y. There is no ambiguity here as PEMDAS very clearly makes the expression defined. Still, I should have been more clear when writing the exam as there were many students who asked during the exam whether it could be x^3/(3+y). I was surprised and had to put a clarification on the blackboard: 
1+2x3 = 9 Microsoft calculator in standard view 1+2x3 = 7 Microsoft calculator in programmers viewIt shows that the same vendor, in a framework where no ambiguity exists (nobody has ever questioned that multiplication should go before addition), an ambiguity in the same product. An other example from that article mentions Texas Instruments calculators
1/2x = 1/(2x) in TI82 calculator 1/2x = (1/2)x in TI83 calculatorSelfproclaimed rules like this are hardly helpful.
PEMDAS  BEDMAS  PE(MD)AS  
6/2*(1+2)  1  9  9 
(1+2)*6/2  9  9  9 
How on earth can you say it is ambiguous when it is AXIOMATIC that multiplication and division are inverse operations? How can you say it is ambiguous when ANY division can be expressed as multiplication by the reciprocal? Shame on you for perpetuating bullshit. 
Dear ..., you probably refer to http://www.math.harvard.edu/~knill/pedagogy/ambiguity/ It is not about whether the division is the inverse of multiplication. That is the definition. This is not where the ambiguity is. It is that the notation is ambiguous (and experience shows that that it is a source for errors and misunderstandings). For example, if we write x/3x, then many humans understand the result as x/(3x) which is 1/3. If you give it to a machine, then it gives the result x/3x =x^2/3. Now, the question is whether there is a definite rule which tells, what is right. The PEMDAS rule, clearly puts multiplication before division so that x/3x = x/(3x) = 1/3. Most humans follow the PEMDAS rule. Because they have been taught so. There is also the BEDMAS rule in which the division comes before multiplication. In that case the result would be x^2/3. Now, if you look at the literature and history, then it turns out that there is no definite answer what is right. And if this is the case, we call it ambiguous. There is a camp which advocates PE(MD)AS where MD are on equal footing and where the order matters if multiplication is used together. But this makes things only more complicated as we have now three different interpretations. So if one writes such an expression like x/3x, one has to be careful and put the brackets. Everything else can produce misunderstandings. You are not the only one who feels very strong and become emotional about it. 
I was told that when you multiply and divide (since the order of operation does not matter) you never need to use parentheses, is this right? Because 2 * 3 / 4 * 6 on my calculator give me 9, and I was expecting 0.25! To me this should be the equivalent to 2*3/(4*6) because, since we don't need parentheses, it's the only way to type it without them. If I want to calculate 2*3/4*6, like my calculator does, I should type 2 * 3 * 6 / 4 is this correct?My answer:
The order of operations matter. You need to put parenthesis. I like the example you give. It already illustrates it well. Most humans would get 6/24=1/4 as you did. Most programming languages (computers) give 9. The computer follows PEDMAS (division before multiplication) 2 (3/4) 6 = 9 or do with the rule (MD) which means ``whatever comes first" ((2*3)/4)*6 = 9 Humans (and most recommendations, like professional societies like AMS follow PEMDAS, which means you first do multiplication and then division (2*3)/(4*6) = 1/4 But it does not make sense to follow a recommendation if different interpretations exist and computers do it different. There are lots of different opinions about it on the web. There is only one way out: just write the parenthesis, also when using a computer. Here is an other nice example which only uses division: ((2/2)/2)/2 = 1/4 (2/(2/2))/2 = 1 (2/2)/(2/2) = 1 2/(2/(2/2)) = 1 2/((2/2)/2) = 4 The computer follows here the rule (left to right) and gives 1/4. But in this example: (3^3)^3 = 19683 (3^(3^3)) = 7625597484987 the computer goes from right to left. Also here, brackets are required.November 4, 2018: S. A. added an interesting angle to the story: It is the recommendation to simplify first then remove parentheses.
I read your blog on programming issues as to MD or DM. The problem is they all conflict with the first law of Algebra. Simplify then REMOVE Parentheses. These conventions all violate that by saying only simplify brackets INSIDE. So at first I like you said AMBIGUOUS to 6/2(1+2) 1 or 9 However, I took a nap, astral surfed over to old Euclid and he laughed. Proof 1 and 9. 6/x=1 or 6/x=9 When x=2(1+2) 2(1+2)=2(3)=6 6/6=1 So not teaching students to remove parentheses in new math is conflicting with first law of algebra. All these acronym conventions need to be corrected to reconcile with 1st law of algebra. So do you agree the new math conventions need to agree with Euclids first rule of Algebra? I think it does.Here was my answer:
It is an interesting angle. But note that the recommendation to ``simplify" is where the problem of the ambiguity is located:December 3, 2018: Atmos added an other interesting angle
Yes, one can simplify 6/2(1+2) by introducing x=2(1+2) = 6, then get 6/6=1 But one can also simplify by defining x=6/2, then get x(1+2) = 9. Actually, this is also historically interesting. You mention Euclid. Euclid did not use any algebra as we know it. Symbolic algebra came only with Viete in the 16th century. As far as we know, one only has realized in the 20th century that there is really an ambiguity. It is clearly stated in Cajori's book on mathematical notation, which is the authority on the matter.
It has also become a pedagogical issue: students today get mostly taught the PEMDAS rule which formally puts multiplication before division and would recommend the result 6/2(1+2) =1. If you give the expression to a computer algebra system they all give 6/2(1+2) = 9. It was examples like that which produced all these discussions. The first rule of algebra is still a nice rule. It is good advise. Unfortunately it does not resolve the ambiguity. But I agree that it helps a writer to avoid the ambiguity. But you know, the issue came up mostly in educational settings. If a teacher asks a student what is 6/2(1+2) , the teacher does not want to simplify that, as it would already solve the problem. If today, a teacher asks students what is 6/2(1+2), then this is just asking for trouble. The right thing to do is to clarify and either write either (6/2)(1+2) or 6/(2(1+2)). It was clear already to Cajori that avoiding the parenthesis does not produce well defined mathematical expressions. Oliver
A potential solution for this controversy could be that when you have coefficients and variables written together without operators between them, e.g. 5ab, we should be able to treat this as a nested operation. In other words, the lack of an operator symbol implies that the relationship between them takes precedence over any operations outside, i.e. 5ab represents (5 * a * b). So if you had a / bc, there is only one operator written (the division symbol), and the "bc" part would be implied to be nested because of the omission of the operator within. So it would still be "a over bc," exactly what it looks like and how many of us were taught. And then if we need to signify that the operation between a and b actually takes precedence over the relationship between b and c, then we would just write a / b * c instead. No fuss, no muss. Isn't that a much more efficient way of communicating with the mathematical language here? And isn't that the point of a mathematical language, to communicate concepts effectively? Otherwise this kind of confusion will never go away, and we will have to write a lot more parentheses in our equations (and nobody wants to do that). Some people like the "new math" of the super strict PEMDAS interpretation specifically because it is an easy way to trick people and make the math more convoluted than it needs to be. Yet that seems to defeat the entire point of why we do this stuff in the first place. I get both ways of doing it, but the strict PEMDAS method seems counterproductive because it causes so many problems, and it does things like making an ostensibly simple fraction like 2x / 3y into actually meaning 2xy / 3 instead, which seems completely insane. But if instead we just use PEMDAS when the operators are actually written, then all of these kinds of problems would literally vanish overnight. The "old math" and the "new math" would finally agree, and we could accomplish it all with one very simple rule. What do you think about that?I wrote back
Hi Atmos, leaving away the multiplication signs is already routinely done. Actually most of the time. There can be an additional problem however when using numbers rather than variables like 3/45 is not the same than 3/4 5 But you contribute an interesting point because there is now even more ambiguity: 3/45 = 3 over 45 = 1/15 3/(4*5) = 3 over 20 = 3/20 (3/4) 5 = 3/4 times 5 = 15/4 The PEMDAS problem is not a ``problem to be solved". It is a matter of fact that there are different interpretations and that a human for example reads x/yz with x=3,y=4 and z=5 as 3/20 while a machine (practically all programming languages) give a different result. There are authorities which have assigned rules (most pupils are taught PEMDAS) which is one reason why many humans asked about 3/4*5 give 3/20 which most machines asked give 15/4: I type this in Mathematica x=3; y=4; z=5; x/y z and get 15/4 It is a linguistic problem, not a mathematical problem. In case of a linguistic problem, one can not solve it by imposing a new rule. The only way to solve the problem is to avoid it. One can avoid it to put brackets. OliverUpdate of December 14, 2018:
Question:
I am a Maths teacher and recently came across a particular question on PEMDAS (kindly check the attachment) where the students got two different answers (6 and 66). The reason for the getting two different answers was the way the students solved the last part of the question: 96 ÷ 6 of 4 
My answer: There are two ambiguities in this problem and yes, all answers given
by students should be graded as correct. 1) The first expression: It appears that the students did interpret 57 ÷ 19*2 expression as 6 even so it could be 3/2 if PEMDAS is used (and official recommendations of AMS or physical society and used by most scientific papers, especially if the expressions are variables). What happens is that if the question would have been ask as 57/19*2, then many would have interpreted it as 57/38. 2) The third expression is a new thing as the "of" as a "multiplication" is unusual. Also here, there are no definite rules. Both answers 6 and 66 are correct in the total. I would even count 3/2 as correct as this is what one gets if one uses the PEMDAS rule and not the PEDMAS nor the PE(MD)AS rule. So, here are the four possible answers. On the left we have now always unambiguous expressions: 57/(19*2) 64*2/32 + 96/(6*4) = 3/2 (57/19)*2 64*2/32 + 96/(6*4) = 6 (57/19)*2 64*2/32 + (96/6)*4 = 66 57/(19*2) 64*2/32 + (96/6)*4 = 123/2The example again shows that one always needs to put brackets. But it also illustrates what can happen if "colloquial language" is used to describe arithmetic operations as this can can lead to other ambiguities. "What is two third of 9" should be clear as (2/3)*9, while 2 ÷ 3 of 9 could be interpreted as 2/(3*9) too. The example indicates again that humans might interpret the obelus sign ÷ differently than the division sign /. 
In the Swiss newspaper 20 Min, the problem 6/2(1+2) = ???
is mentioned too.
The article already has 1384 comments. Similarly as for years on social media, the fight
goes on there. The most interesting thing is how certain most are that they are right, on
all sides. Which points again to ambiguity. The title of the article is "Millions fail at this math equation!" As ``proof", there is a youtube video which gives the answer 9. The author of that video, Presh Talwalker, gives in his blog the reference Lennes, N. J. "Discussions: Relating to the Order of Operations in Algebra." The American Mathematical Monthly 24.2 (1917): 9395. . One should better read that article. This article from 1917 indeed claims that ``most text books" use the left to right rule if division and multiplication appear mixed. But it also states the "established rule"
So, we have it: it is simply nonsense that the 12 million people who do it differently were not ``unable to do the problem". We definitely deal with a situation which must be considered ambiguous. The 1917 article is a nice reference. It confirms that already. But since 1917, the PEMDAS rule has been taught to millions of people. It remains astounding only how many claim to know the right answer. Maybe that is just human nature. 
"When a mode of expression has become widespread, one may not change it at will. It is the business of the lexicographer and grammarian to record, not what he may think an expression should mean but what it is actually understood to mean by those who use it. The language of algebra contains certain idioms and in formulating the grammar of the language we must note them. For example that 9a^{2} ÷ 3a is understood to mean 3a and not 3a^{3} is such an idiom. The matter is not logical but historical. 
That Vexing Math Equation? Here's an Addition The confusion (likely intentional) boiled down to a discrepancy between the math rules used in grade school and in high school. 8 ÷ 2(2+2) = ? The issue was that it generated two different answers, 16 or 1, depending on the order in which the mathematical operations were carried out. As youngsters, math students are drilled in a particular convention for the "order of operations," which dictates the order thus: parentheses, exponents, multiplication and division (to be treated on equal footing, with ties broken by working from left to right), and addition and subtraction (likewise of equal priority, with ties similarly broken). Strict adherence to this elementary PEMDAS convention, I argued, leads to only one answer: 16. Nonetheless, many readers (including my editor), equally adherent to what they regarded as the standard order of operations, strenuously insisted the right answer was 1. What was going on? After reading through the many comments on the article, I realized most of these respondents were using a different (and more sophisticated) convention than the elementary PEMDAS convention I had described in the article. In this more sophisticated convention, which is often used in algebra, implicit multiplication is given higher priority than explicit multiplication or explicit division, in which those operations are written explicitly with symbols like x * / or ÷. Under this more sophisticated convention, the implicit multiplication in 2(2 + 2) is given higher priority than the explicit division in 8÷2(2 + 2). In other words, 2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1. By the same rule, many commenters argued that the expression 8 ÷ 2(4) was not synonymous with 8÷2x4, because the parentheses demanded immediate resolution, thus giving 8÷8 = 1 again. This convention is very reasonable, and I agree that the answer is 1 if we adhere to it. But it is not universally adopted. The calculators built into Google and WolframAlpha use the more elementary convention; they make no distinction between implicit and explicit multiplication when instructed to evaluate simple arithmetic expressions. Sign up for Science Times We'll bring you stories that capture the wonders of the human body, nature and the cosmos. Moreover, after Google and WolframAlpha evaluate whatever is inside a set of parentheses, they effectively delete the parentheses and no longer prioritize the contents. In particular, they interpret 8 ÷ 2(2 + 2) as 8 ÷ 2x(2 + 2) = 8 ÷ 2x(4), and treat this synonymously with 8 ÷ 2x4. Then, according to elementary PEMDAS, the division and multiplication have equal priority, so we work from left to right and obtain 8 ÷ 2x4 = 4x4 and arrive at an answer of 16. For my article, I chose to focus on this simpler convention. Other commenters objected to the original question itself. Look at how poorly posed it was, they noted. It could have been made so much clearer if only another set of parentheses had been inserted in the right place, by writing it as (8 ÷ 2)(2+2) or 8 ÷ (2(2+2)). True, but this misses the point: The question was not meant to ask anything clearly. Quite the contrary, its obscurity seems almost intentional. It is certainly artfully perverse, as if constructed to cause mischief. The expression 8 ÷ 2(2+2) uses parentheses  typically a tool for reducing confusion  in a jujitsu manner to exacerbate the murkiness. It does this by juxtaposing the numeral 2 and the expression (2+2), signifying implicitly that they are meant to be multiplied, but without placing an explicit multiplication sign between them. The viewer is left wondering whether to use the sophisticated convention for implicit multiplication from algebra class or to fall back on the elementary PEMDAS convention from middle school. Picks: "So the problem, as posed, mixes elementary school notation with high school notation in a way that doesn't make sense. People who remember their elementary school math well say the answer is 16. People who remember their algebra are more likely to answer 1." Much as we might prefer a clearcut answer to this question, there isn't one. You say tomato, I say tomahto. Some spreadsheets and software systems flatly refuse to answer the question  they balk at its garbled structure. That's my instinct, too, and that of most mathematicians I've spoken with. If you want a clearer answer, ask a clearer question.August 5, 2019. An other treasure trove by Jenni Gorham just appeared on Youtube:
Formulas. You can help us to reduce printing costs by avoiding excessive or unnecessary quotation of complicated formulas. We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division. Thus, instead of the (coded) display:August 17, 2019 An other question:$${1\over{2\pi i}}\int_\Gamma {f(t)\over (tz)}dt$$
we might use
$(1/2\pi i)\int_\Gamma f(t)(tz)^{1}dt$.
I am curious what you think the answers is to this equation. 8 ÷ 2(4). To me, the simplest order of operation next is to multiply 2 * 4 first because I still have the parentheses to deal with. I'm just a believer that we have to do some kind of math to get rid of the parentheses. Some people are just dropping them without doing any math. I mean, why have them at all if one can just drop them at any time without doing any math to clear them out. S.  My Answer: Yes, this is one of the latest PEMDAS riddles passed around. A similar one has been discussed also by Strogatz in the NYT. What you mention is that the 2+2 is already evaluated but it does not change the situation to to 8 ÷ 2(2+2) which is now passed around. But it is the same story. The reason to put a bracket around 4 is so that it is not read as 24. But it does not clarify the ambiguity. Yes, one is tempted to do first 2*4 and get the result 1. Most computer programs evaluate it to 16. Example: Mathematica 12.0.0 Kernel for Linux x86 (64bit) Copyright 19882019 Wolfram Research, Inc. In[1]:= 8/2(4) Out[1]= 16 Almost all humans would evaluate it to 1. Opinions do not matter here as it is now well documented that there is just no consensus (neither by any authority, nor by historical growth, it is a linguistic phenomenon that one has realized on so late the need to define this precisely). Things can be interpreted differently and that will remain so. On has to place brackets to make things clear.September 17, 2019. An answer to some fact checking questions from the NYT. I wrote back:
Summary:  The question 8/2(2+2) has different answers depending on the rule which is used. One can interpret it as (8/(2(2+2))) = 1 or (8/2) (2+2) = 16 depending on the rule. There is no universally accepted rule as there are several: PEMDAS, BEDMAS, PE(MD)AS. It is not possible to say what is correct and what is incorrect. There are different rules, leading to different results. The expression is not well defined. It appears that most humans naturally give the answer 1 and most computers and programming languages return the answer 16. In order to make the expression unambiguous, one has to put brackets. It was realized only relatively late (about 100 years ago) that there is an ambiguity. No consensus has been reached ever since, so that there is no alternative to clarify the expression. Literature to that is Florian Cajori, "A history of mathematical notation", London, 1928. N.J. Lennes, Relating to the order of operations in algebra, Amer. Math. Monthly, 24 1917  Here are answers to your questions:  Accurate to say that according to PEMDAS, "2+2" should unilaterally be the first operation performed? Yes! But this is not PEMDAS. The 2+2 operation is performed first because brackets have been placed around it.  According to lowerlevel math, the solution to this equation should be 16? No. There is a rule called PEMDAS which is wide spread (kids learn "Please Excuse My Dear Aunt Sally") which when followed asks to do multiplication before division. This gives 8/(2(2+2)) = 1. As today, most kids in lowerlevel math also have access to calculators or online tools, they might give the answer 16. The reason for the claim that it is a lowerlevel math part is probably that most teachers now use a calculator to test things and the computer tells them that the answer is 16. As students and their parents get different results (if they do not use a computer) there is a controversy.  Fair to say that in algebra or higher math, the 2(4) operation takes precedence? No. Also here it depends on the rule used. If one uses PEMDAS as intended, then the answer is 1 which is to compute 2(2+2) first as the M comes before the D. It is not a question which level or subject one uses. The answer depends on which rule is used.  According to highermath standards, is the solution to this equation 1? No. Also this depends on the rule used. When doing the test with students, most give the answer 1. Actually, most humans give the answer 1 if they do not use a computer. Most people read such expressions as 1. Computers however, who treat multiplication and division often on the same level and almost universally give the answer 16. I had asked in 2014 incoming freshmen (not yet highermath students) in an intro calculus course and all except one used the PEMDAS rule. Most humans answer 1.  The piece asserts that the way this mathematical expression is written is misleading. Would you agree with this assertion? No. It is not misleading, it is ambiguous. It would be misleading if there would be a correct answer and the expression would lead to a wrong answer. This is not the case. There is no correct or wrong answer. The expression is ambiguous and depends on the rule which is used.  Fair to say that as one moves into higher math (post elementary school level), division is generally denoted as a fraction? No. One uses all kind of expression in higher math. Dividing into lower level math and higher level math is less important (in my experience) than whether one computes how one reads math naturally (and most textbooks do) or if one computes the expression with a computer. It might be psychological, it might be linguistic, it might be by being taught so, it might be reading texts, but most humans do multiplication before division in situations like the one discussed.  How would one write this equation in fractionform? Nonambiguous expressions either use brackets like 8/(2(2+2)) An other possibility is to use the expression 8  2 (2+2) This gives 1. In that it is important that there is a clear long fraction division line making sure that the 2 (2+2) is done before the division. The other version would be (8/2) (2+2) which gives the answer 16, as most computers do.November 2, 2019: This website features an online calculator mentioning BODMAS BODMAS is also an acronym which is short for Bracket, Order/Of, Division, Multiplication, Addition and Subtraction. The website and calculator completely misses the point however of identifying expressions which can be understood differently. Especially, because the website is called PEMDAS calculator, it implies to use the PEMDAS rule which has multiplication before addition. So, the calculator evaluates 2*9/3*21 as 11, as most calculators or programming languages do, but which is different from what PEMDAS actually suggests (2*9)/(3*2)  1 = 2 and which most humans evaluate it to when asked. So, the calculator, rather than helping to clarify the ambiguity which is present, is just a calculator as many. There are modern calculators which warn a student that an expression is ambiguous and enters parenthesis so that the user can, if these parenthesis are not what was meant can change them.
Hope it is okay I email you. I came across your page while having a debate over the Facebook problem. I have been furloughed so I have a lot of time on my hands. This version was 6/2(1+2). To me the answer to the above is 1 and only 1. But, I didn't want to debate that. I wanted to add something I didn't see represented on your site that I think the debate over PEMDAS misses. Math is often called the universal language. It allows people across different cultures to communicate. When you read the first example 2x/3y  1 when x = 9 and y = 2, how do you read it? Do you read it two times x divided by three times y minus one? Do you read it two x over three y minus one? Much like the English language has rules, math also has rules. English isn't perfect and doesn't always follow the rules. Math is the same. Sometimes multiplication and groupings can be implied.My answer:
yes, math is a universal language which allows communication among different cultures. But languages also have been designed by people and are not always perfect. The PEMDAS ambiguity is actually just an omission of design. There was no authority which once and for all told that it is necessary to specify this. The reason is historically clear. One only realized the problem too late. You say, for you 6/2(1+2) is 1, yes, almost all humans assume this. If you give it to a computer, it gives you 9, almost all programming languages do. Yes, like English, Math has rules. But they are far from perfect. Languages change and get refined over time. In the PEMDAS discussion it turned out to be too late to adopt rules. Some have tried and many are fanatic and think that their way is right. In the 6/2(1+2) case for example, we can see advocates (i.e. on youtube), who argue, only 9 is the right answer. But as you say, reading it as 6/2(1+2) is entrenched and changing that would upset many. The last few slides in this presentation [PDF] show a bit what linguists say. It is not really a math problem, it is a linguistic problem. What is good about the discussion is that all teachers and students now know about the ambiguity and write down clarified expressions.
I am not a mathematician. STEM has always been my strong area, but I prefer application to theory, thus I am an equipment technician in the semiconductor industry (I fix the robots that make computer chips). I mention this to say I may not be an expert, but also am not a lay person. My first experience with this question came when trying to program the quadratic formula in to my TI83+ calculator almost 20 years ago. To make the equation work correctly additional parenthesis are required. This is because standard algebra is notated in an implicit way and computers never pick up on implied meanings, or only began to recently. This is kind of covered in the NYT article you cite, but it is stated in a way I think is erroneous. It states in "simple or elementary" math 8/2(4) is the same as 8/2*4, and only in algebra does 8/2(4) become 8/(2*4). Calculators are fallible, as you pointed out MS calculator solves 1+2*3 as 9 not 7 which is never correct. The "distributive law of multiplication and division" proves the notion of two right answers invalid. 8/2(2+2)=16 is in complete violation of the distributive law, thus is invalid. If the equation were written (8/2)*(2+2) then order of operations would dictate that the division happen first then the result would be distributed within the second brackets as is the case if it were written 8/2*(2+2). However in any equation where it is written 8/2(2+2) then it is implicit (not explicit) that the distributive law is to be applied as part of the parenthesis before the division occurs, thus written explicitly it would be 8/(2*(2+2)). The fact that computers (or calculators) do not understand what we are implying doesn't mean the implication is wrong, it means computers can only understand explicit instructions. Explicitly solving an equation with implied factors usually results in the wrong answer. There are not 2 right ways of doing the math, just some people running an implicit equation through a calculator without explicitly writing it out and getting the wrong answer. Claiming that 8/2(2=2) is ambiguously written is like saying quadratic formula is ambiguous because you programmed it explicitly in to your calculator and are getting the wrong answers. My keyboard doesn't have math symbols but in a calculator quad form is roughly x= ((b+ root(b^2(4ac))/(2a)). Written normally it is x= b + root b^24ac/2a or maybe x= (b + root b^24ac)/2a, but both of those give totally wrong answers from a calculator. Obviously quadratic formula is a constant and should not get differing results no matter what you were taught, this proves we must be smart enough to use our calculator correctly, not that calculators are infallible. Another example would be the equation "ab * cd". Solving that expression explicitly would have you do "a*b*c*d=x" when we know it is actually (a*b)*(c*d) because the () are implied, just as above. Your algebraic example using 2x/3y1 is silly. In algebra there is always an implicit parenthesis around the 2x or 3y which is why (9a^2)/3a = 3a not 3a^3 as pointed out by Lennes in his 1917 article. The reason 2x/3y1=11 on a calculator is because the calculator doesn't understand there is implicit grouping. Every parent understood the implicit grouping, that (2x)/(3y)1 is the implied equation though not explicitly written as such. Only when you put in a calculator 2*9/3*21 do you get 11. But 2*9/3*21 would be written 2*x/3*y1 which is very different from 2x/3y1, but a calculator sees them as the same. This was proven when you had your calculus class do it, 2 of your students punched it in to their calculators, the other 58 did the math correctly. What this means is you need to teach those 2 students how to correctly use their calculators, not that there is some universal ambiguity. When programming quad form you must explicitly bracket all implicit grouping or it wont work, this is a limitation of software, not a flaw in mathematics. If there was any actual ambiguity the results from your calculus class would have been split closer to 5050, statistically speaking its obvious there is strong consensus amongst calculus students as to the correct method. The math teacher who got 2x/3y1=11 should be in a different field.My answer
thanks for your note. Yes, it is a very interesting topic, especially in connection with computers. You are right that with calculators one has to be even more careful and needs to put more brackets. One of the reasons, why HP calculators had a lot of success is because they used reverse polish notation allowing to skip many brackets. I was never in that HP camp but like you used TI calculators. The example 2x/3y 1 is not that silly. It is actually very interesting. Yes, you are right that every human reads it as (2x)/(3y)  1 but a computer gives something different. I calculate it here with Mathematica, one of the most advanced computer algebra systems and it gives 1+ 2xy/3 not 1+ (2x)/(3y) (see attached screen shot). I also agree that the PEMDAS ambiguity is not a flaw of mathematics as it is often presented, it is just a matter of fact that some expressions need more clarity (meaning brackets) to be meaningful. What is a bit offsetting is that there are so many around who believe that there is a definite way and only their way is right. In the example 2x/3y 1 most people just naturally assume that this means (2x)/(3y)  1 but according to some fanatics, there is only one way to see it right and that is what computers give you like 1+ 2xy/3. It is also quite amazing how long this discussion is still going on. But that makes it even more interesting. There is not only a mathematical or linguistic side, there is also a social aspect to the story. And as you mention as an engineer, it can be crucial. If one writes a program guiding a robot and overlooks something like that, it just does not work. I learned early on that programming something is the ultimate test of understanding. It is harder than to read or write or teach a subject. If a procedure does not work, it is proof that there is something, that one does not understand yet.After an other question about distributivity leading away from the main point: I tried to make the point that there is no correct or wrong in an ambiguous situation like 8/2(2+2) There are answers 1 or 16, depending on what rule you use. This is the reason why people continue to argue about it. The answer 1 is the answer most people get. The answer 16 is what most computers get. I tried to make this point from the very beginning in http://www.math.harvard.edu/~knill/pedagogy/ambiguity In my last email, I pointed out that it is not a problem with distributivity. The 2(2+2) is always 8, there is no dispute there. Again: the question is whether (8/2) is computed first and then multiplied with (2+2) to get 16 or whether 2(2+2)=8 is computed first and 8/8 =1 is obtained as a result. This is a question on whether the division or multiplication is done first. It is a matter of fact (an observation when looking what people write on the web) that there are some who believe one of the answers is right. There is no right or wrong here. The question is also not misleading. There is an ambiguity. The conclusion is that one has to write things down more clearly using brackets.And a follow up email: I think I may have made a break through. I accepted that as of current multiplication through juxtaposition has been taught as a higher order of multiplication or equal to all orders of multiplication. Thus 5/2x has been taught to mean (5/2)*x or 5/(2x) depending on your teacher. Thus the only solution is a clear rule defining multiplication through juxtaposition. How do we determine if multiplication through juxtaposition is a higher order or not? I think (though it pains me) that language and grammar may hold the key. But first, I have to ask about x*2xy? I have never seen anyone write something like that as x2xy or 2xyx or anything else, would it be correct to do so? If not, why? When I see something like this to me shows grouping implies parenthesis, not just multiplication, thought the result wouldn't be effected by them.to which I answered Some ambiguity can be avoided by juxtaposition, some can be done by ordering, but it remains still ambiguous, especially, if typed. You appear still to think that the PEMDAS ambiguity is an open problem which needs to be solved. It is not an open problem. It has been solved a 100 years ago when realizing that one just has to write more brackets in general. It had been a linguistic oversight. When writing the grammar rules for arithmetic, one has initially not realized that there were parts which require more brackets than anticipated. When using only multiplication, then the problem is not present. The reason is that there is associativity and commutativity of the product operation. Already the division is not associative nor commutative (3/3)/3 = 1/3 is not 3/(3/3) = 3 3/5 is not 5/3 

Just finished reading over your webpage regarding ambiguous equations. Imagine my surprise to learn that math convention does not treat multiplication and division equally. I actually majored in math for 3+ years (before coming to my senses) and tutored at the college level. I learned and taught that the mnemonic PEMDAS was actually PE(MD)(AS). Indeed, I taught my tutees to think of it as PEMA, since the nth root of x = x^(1/n), x ÷ n = x * (1/n), and x  n = x + (n). Once they caught on, they were amazed at the pitfalls they avoided. Regardless, uncovering long held conventions, I wonder... couldn't the ambiguity be resolved such that a solution to the equation 8 ÷ 2(2+2) can be found? I should disclose that I was vehement in arguing that the answer was 16, although I ignored the ambiguity. After reading your webpage, the ambiguity is clear. However, my brother, a retired engineer with Honeywell, whose understanding of math supersedes even my own, also came up with the answer 16. Interestingly, his first, instinctual answer was 1. However, your reference to ISO 310 and the link to a YouTube video ("PEMDAS is wrong") sent me down the rabbit hole. Incidentally, I discovered ISO 800001 now supersedes ISO 310, fyi. Unfortunately, ISO 800001 isn't available for free so I had to resort to ISO 310 for specifics. A possible (the?) solution to this ambiguous equation requires knowing rules and conventions I suspect the majority of people are unaware. Specifically ... that the obelus (÷) violates ISO 310. Division is to written as "a over b" (using the fraction bar), "a/b", or "a * b^1" (alternatively, "a * 1/b"). ( a product is to be written as "a b", "ab", "a * b", or "a x b". ) that when applying the rules of the AMS and APS, multiplication precedes division. ( that "/xy" is conventionally understood by mathematicians to mean "/(xy)"_ Solution 1: 8 ÷ 2(2+2) = 8/2(2+2) (to conform to ISO 310) 8/2(4) (per order of operations) 8/8 (per AMS and APS rules that multiplication precedes division) 1 Solution 2: 8/2(2+2) can be rewritten as 8/[2(2+2)] [per mathematical convention that "/xy" = "/(xy)", as well as due to juxtaposition, whereby "2(2+2)" could be seen as "2y", using substitution, and "8/2y" = "8/(2y)"] 8/[2(4)] 8/8 1 Many people state the problem is one of linguistics and/or grammar. I see it as a problem of translation and that it is the purview of the mathematician to translate the problem into common language. It is analogous to an immigrant hearing a colloquialism for the first time. Which reminds me of the Russian comedian, Yakov Smirnoff, who had a bit where he talked about odd idioms. He heard someone say they quit smoking cold turkey. He said, "what do you smoke, now, ham?" And so I now believe that a mathematician, applying international standards as well as the conventions of the AMS, can translate the equation and resolve the ambiguity. What do you think? Is my logic sound?  My Answer: Your example is a typical example, where people and machines get different things. If you feed it into a computer, it gives 16 [knill@knill11:] math Mathematica 12.1.1 Kernel for Linux x86 (64bit) Copyright 19882020 Wolfram Research, Inc. In[1]:= 8/2(2+2) Out[1]= 16 I think most people get 1 due to training, PEMDAS and because in the math and physics literature, there is an implicit assumption to put the 2(2+2) together first especially with the obelus. The ambiguity can not be resolved by searching for rules, or AMS guidelines or ISO standards (especially if they are not available in the public domain) it is a matter of fact (and you confirm that), that we can get different answers. Many people on the web try to argue one or the other way. This is futile. What does it help if 99 of people argue one way and 99 of computers give answers the other way? Yes, and the obelus even confuses things a bit more as you indicate. There is no logical way to solve an ambiguity. That the definition of an ambiguity. There is no clear standard and if there would be one, it has not been adopted universally. The only way to resolve the problem is to put brackets.
I was reading your article on the order of operations, or as you call it "the PEMDAS rule". One thing that, if mentioned, is not immediately obvious is that when you say that the "pemdas rule has been taught to millions of students", you are specifically referring to students in the US. I was taught maths in Sweden, and there we were taught that there were three levels of precedence: Exponentiation Multiplication and division Addition and subtraction Parentheses aren't really mentioned, because there is no ambiguity there. Other than to explain that they can be used to override the default rules. I now live in Singapore, and my kids go to local school here. I just checked with them, and they have indeed learned the same way as I was taught. In all programming languages that I know of, as well as the mathematical software I have tested, I note that they follow precisely this rule. The "PEMDAS" rule seems to be a way to make something rather trivial into something complicated. In addition to this, it also ends up teaching it wrongly, since if one insists on using it, one would have to write something like: PE[M/D][A/S] to clarify the equivalence within the groups. Just reading it outright gives the student the impression that multiplication should somehow be performed before division in all cases, which is rarely true. I say "rarely" here, because I do agree with you that there are certainly cases it makes sense to do the multiplication first, such as your example 1/xy. That said, if you write it as 1/x y it's not so clear anymore which implies that spaces are significant. As a software engineer who is also studying physics in my spare time], I frequently get annoyed with the lax notation in maths. Not just the issues with horizontal writing (as is the topic of your article) but also proper typeset equations using LaTeX. Authors tend to use shorthand notation, reuse symbols, etc, which means you have to have context in mind when figuring out what is happening. This is a type of problem I would have hoped were limited to the field of linguistics rather than mathematics, the only scientific field where something can be proven with certainty. Finally, I'd like to add that the reason I have spent too much time thinking about this is because I've been working on a new interface for Maxima (a symbolic maths system) which represents the programmatic forms in mathematical notation, which means I have to make it completely nonambiguous. If you're curious, here is a demo video.My answer:
thanks for the insightful comments. For me it is similar, from Switzerland, where we were not taught the explicit PEMDAS rule. Still, I would still maintain that millions have been taught PEMDAS. I agree with everything. Especially when working also with mathematical software, the need for precision and nonambiguity has become more and more important. The topic is not only pedagogical, one can make serious mistakes. Not only by doing wrong computations but also by using results which have been stated in an ambiguous way. It would be interesting to know, how many students in Singapore or Sweden would read say 8/2(2+2) as 16 and how many would read it as 1 especially if the division is given as an obelus. I would predict that most would give the answer 1. This is because I teach often students from all parts of the world (This summer also 2 from Singapore by the way) and that questions about PEMDAS often pop up, if one is not super clear in writing (even when writing expressions like 1/x+3, there are still students who read it as 1/(x+3) or are confused and ask. I started to even clarify such things by writing (1/x)+ 3 or using a horizontal fraction notation \frac{1}{x} to make sure that one reads it correctly.
Good day! I hope you are doing well. Very recently, my colleagues and I had several heated debates about the problem 8 ÷ 2(2+2) or the likes. As you would expect, there are people who answer this as 16 and people who answer this as 1. As for us, we are in the ambiguous camp. The people who had a unique answer to the problem keep citing PEMDAS, or BEDMAS, or PE(MD)(AS). And we told them that the convention is not universal. So we tried looking for documents to support our side. And then we found your article. I've read everything. I am amazed that you are so patient in responding to people sending you emails, and giving them same answers, repeating your explanations over and over again. Anyway, I just emailed you to let you know (in case you haven't seen it yet) Section 7.1.3, p.23 of ISO 800001:2009"
"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 sign on the same line unless parentheses are inserted to avoid any ambiguity." 
Also, Example 1 following the paragraph should settle all this debate. Thank you for taking time to read my email.My Answer
Thanks so much for the feedback. The ISO reference is very valuable! It has been pointed out before but unfortunately, that part of ISO is not publicly available. One has to purchase the standard.I actually happened to have access to the relevant pages from Section 7.1.3 of ISO 800001:2009:
Following a heated online discussion I was forwarded to your website http://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html. From what I read there, you seem to blame the confusion on the used notation. While I agree that this may confuse people, I would like to present a more universal explanation for this problem based on using (or not using) simple axioms of algebraic structures. The notation then becomes more or less irrelevant. The given equation 6:2(2+1) seem to have multiple answers, which appears illogical on first glance. But I think some people just forget to decide about the span of definitions they're working in. With simple arithmetics, multiplication and division are equally ordered and have just to be executed before addition/subtraction but after calculating brackets and exponents. That's where PEMDAS or other mnemonics come from. This just works from left to right and is correct in it's own simple context. Sequential input calculators do it this way. But when assuming the context of formal algebraic structures, in this case a field of rational numbers, additional axioms apply. Now, "distributivity" and "associativity of addition and multiplication" (among others) are defined as basic axioms in the context of this field. With that, the order of operations now is far more restricted by those properties. In this context, the equation 6:2(2+1) can be transformed as following: 6:2(2+1) = 6:(2+1)2 // Associative property of multiplication (just to prove the point by using this field axiom): ab = ba; division is nonassociative = 6:(4+2) // Distributive property: ab+ac = a(b+c) = 6:6 // Order of operations (now trivial), "brackets before exponents before multiplication/division before addition/subtraction" = 1 Respecting all axioms of a field structure, the division now has to be solved by calculating first the dividend and divisor side separately, then the quotient. Any other way violates one of the basic field axioms and would be invalid in terms of this field. More sophisticated calculators are programmed to respect this. However, if not restricted to the context of those axioms, other results may be valid as well (i.e. 6:2(2+1)=9, with a sequential calculation on a basic arithmetic level). So, in my view, it's not a problem of notation, but in giving the context and thus the defining axioms. Knowing the used tools and their assumptions respective abilities/restrictions  as always  is important. By the way, students in Germany learn the mnemonic "KLAPPS" (KLAmmer, Potenz, Punkt, Strich) before higher algebra, which thankfully ignores any suggested preference of multiplication/division ("Punktrechnung" = "point calculations" from the "point" characters used in German calculation '.'=Unicode U+22C5 for multiplication and ':' for division) and addition/subtraction ("Strichrechnung"="line calculations", from the "line/crossed line" characters '+' and ''). This lack of sequence suggestion seems to prevent the confusion in higher classes, at least a bit. Thank you very much for your time reading this. I'd appreciate to hear from you, especially if you can't agree with my conclusions. Mathematics surely isn't my strong point, but I'm just learning this stuff again with my son's homeschooling activities (thanks to COVID19 lockdown) and still seem to remember some of my former computer science math lectures long ago. Best wishes from Bavaria/Germany! Bernhard Nowotny, M.Eng.My answer:
That is an interesting angle. I especially am grateful to hear about the German KLAPPS rule. I went to school in Switzerland and have not seen that. Rules also need to stick with students and KLAPPS is a good name. Interesting that it does not address the PAMDAS ambiguity. Your suggestion to solve the ambiguity might well work, but the problem is more sociolinguistic than mathematical. There are already different standards around. I don't know about introducing a new one XKCD said it best.February 22, 2021: Vic:
You wrote that PEMDAS and BODMAS are ambiguous. I disagree. I have a masters in math and have been teaching math for 30 years. You gave a question using a division symbol /. I find this symbol for division confusing and never use it. I find it interesting that even though all calculators gave the answer as 11 to your initial problem you seem to think that they should what  give an error message? The answer is 11. There are no brackets and I have no idea why you think you can insert brackets willy nilly. People get it wrong because they do math by patterns instead of thinking "What are the rules here".My answer:
thanks for the feedback. You can disagree of course. I have given many reasons on my page why there is an ambiguity. Many students give the answer 2 and not 11 in the initial example. If you read my text, you see that ambiguity does not mean that there is an error message. In most cases, the disagreement comes from the lack of standard. The professional references given in my page confirm that there is no standard. Putting brackets can not be placed "willynilly", they are placed to make things clear and remove the ambiguity. As you are teaching, why don't you make the experiment and ask students what they think when they have to evaluate 2x/3y 1 for x=9 and y=2. Most humans give the answer 2 as they bracket 3y as inseparable (quite a few teachers have confirmed that). This is also the standard many books follow (especially if they adher to earlier professional standards like the AMS or rules like PEMDAS which are often taught and put multiplication before division. Cojori, today still the most authorative figure on mathematics notation wrote already in 1928 "If an arithmetical or algebraical term contains / and x, there is at present no agreement as to which sign shall be used first." This is still true today Oliverand a follow up by Vic
PEMDAS does not put multiplication before division. They go in order left to right. Just because students put 3y together as inseparable does not mean the students are correct. They make an assumption. . What books agree with the students? I have never seen any of these books. / means division. There is no other interpretation. So you are saying that PEMDAS is not a universally accepted rule? So an expression like 2/3 + 5/8 x 7 can have many different answers depending where you put the brackets?My Answer:
One would read the second part as 5/56 and so get 127/168 while computers read it as 2/3+ (5/8) 7 = 121/24 (actually not all computers but most, some calculator generations gave other results). PEMDAS as understood for decades was always understood in the sense MD meaning Multiplication before Division. There was also PEDMAS. More recently, one sees things like PE(MD)AS with the footnote that MD can be interchanged but that one reads it left to right. There are quite a few fanatic prophets out there one on the web. Of course, one can establish like this a new standard but it flies into the face of hundreds of years of practice and most books use things like 3m/4s meaning (3m)/(4s) and not (3/4) m s or write Q/mc^2 for Q/(mc^2) and not (Q/m) c^2. You say you divide a cake to 2 people meaning cake/(2 people) and not (cake/2) people. It is often, especially for established formulas that we naturally take them together. A popular example is 1/2pi because 2pi often is a standard unit by itself. Most people read it as 1/(2pi) but a computer reads it as pi/2. The former 1/(2pi) agrees with writing guiding standards which have put forward by AMS or APS.Vic again:
If 2x/3y means 2x/(3y) how do we write the expression 2 times x then divide by 3 and then multiply by y?My Answer:
The example 2x/3y is a typical example which most people read as (2x)/(3y). A computer reads it as 2xy/3. One would write (2xy)/3 or (2/3) xy or simply 2xy/3 if one wanted it to write it as a computer reads it. This is nonambiguous then also in general and nobody can misinterpret it.February 23 2021, James:
I am thoroughly enjoying your article (I guess I would call it that). It caught my attention because I have started seeing work from students who are plugging entire expressions into their calculators that reflects the discrepancies discussed in your article. One thing that I will note is regarding the example of 18/3*2, to which you state that most humans will arrive at the solution of 3, while computers arrive at 12. Immediately after seeing the expression, I calculated the solution to be 12. My reasoning being that I read this as eighteenthirds times two. I chalk this up to my frequently working with fourths, thirds, and halves outside of work and, whenever I see "/" I immediately think of a fraction rather than an operation. More evidence to support your argument about ambiguity in mathematics and the need for clarity in expressions. Again, thank you so much for the enjoyable read. I'm looking forward to exploring the rest of your site.My answer:
thanks for the feedback. Yes, it is a very interesting story. It seems very much also to depend on how one writes it. Often, in the scientific literature (as well as some guides from the AMS or APS), it is pointed out that one should read things like a/bc as a/(bc) and not (a/b)*c. The reason is also that often, products are read by humans to belong together. If you take 18/2π then most people read it as 9/π and not as 9π (as a computer does). One only can guess, but I suspect that this was the reason to build PEMDAS (the most frequently used acronym) and not PEDMAS or PE(MD)AS (which must come with the footnote that MD are equivalent and read from the left). The later can hardly be taught. Already PEMDAS is a hard sell (many Harvard students even get the basic PEMDAS wrong, and read + before *. In pedagogical matters, one always over estimates what complexity people can learn. Setting now a new standard like PE(MD)AS (they appear now in schools already) also has a problem that many texts would be read wrongly. If an expression belongs together like RT in thermodynamics or mc^{2} in physics. Back to your example: 18/3*2, yes, I guess many would read it as 12, but if you write 18/ab with a=3,b=2, most would read it as 3. Funny, how human psychology, linguistic and history as well as math come together here. That makes the subject so interesting.
I am a middle school math teacher at a small Catholic school. I recently came across the attached problem (#12) in my 6th grade spiral review program. It is similar to the question you pose, however it uses a division sign instead of "/". I posted it on a math teacher group I'm a part of for middle and high school math and HOLY COW! It has started quite the debate/argument. There are those adamant that the answer is 6, those adamant that the answer is 24, and those that are adamant that the answer could be either. Of course, every group thinks they're right! Your article was posted multiple times on the thread. I have read the vast majority of it. I am curious if you see any difference in the ambiguity of the attached problem with the difference in use of symbols or if it remains as much of an enigma as the problem you shared. Thank you so much for your time! 
Thanks. This is very valuable to hear about because it confirms that the only way to avoid such discussions is to be super clear and add brackets. I added it to the collection. The good thing is that everybody in your teacher group are right. There are strong arguments to group 2c together because this is often used in the literature, there are strong arguments to do division first because this is what most computers do. And then there is a strong argument that both are right. But who ever wins the argument would still face the dilemma to grade this ... And then face the storm of students, parents and other teachers who are upset. I will just cover algebra in my math for teaching course and can mention this
I was wondering about this equation. 6/2(1+2)=? My answer is 1. It's because I believe that 2(1+2) is a whole number. Another example is 9/3(1+2)=1 also Where 9/3(3) which can also be written as 9/3^2. If so PEMDAS RULE SAYS Exponents is 2nd. So my question really is Exponents are also form of Multiplication so why is it now M is same as D on the equation where it is now changed to PE (M or D left to right) (A or S left to right)? In my math years Multiplication is third because Exponential and Multiplication is the same. Also when we simplify 6/9 simplifies as 2(3)/3(3) we can't simplify this as answer of 6 when it should be 2/3. Here is a new one on social media: 24/4(8/4) and my answer is= 3. Some says it's 12. After what I have explained above. What is your answer? Thanks in advance.My Answer:
The case 6/2(1+2) is now a classic. Most people get 1. It is not because it is a whole number but because 2(1+2) is seen as a unit. Most computers get 9 because (M and D) are evaluated on the same level. I have not seen the new 24/4(8/4). You have got 3, what most people get. Millions of them because it is entrenched in PEMDAS rules (M before D) and because it is often written in books as such. Most computers get 12. There is no right or wrong answer here. The question is just posed in an ambiguous way. This has now been known for 100 years. It is interesting that the social media still hum about it. But that makes it interesting and aware. I wrote a short summary with sources in this document for a course I'm teaching right nowObviously things are not clear on a much more basic level. The following email illustrates confusion which even stellar students can have. It appears to be difficult even to get through the fundamental properties of PEMDAS like exponentiation before multiplication and division and parenthesis before which are completely undisputed.
I always thought that when there's an X next to any number they are considered a whole number. Also, when 9/3^2=1; how is that different from 9/3(3) if they both are the same? Also, I ask somebody if the equation was 6/1(1+2). He answered 18. Then I ask what is 6/(1+2). He said 2. Aren't those equations the same?My answer:
You bring up various different things. First of all, the discussion does not depend on whether X is a hole number or not. The rules do not depend on whether we work with integers or real numbers or any other number. If you write 9/3^2=1, then this is unambiguous because exponentiation comes before the other operations. If you write 9/3(3), then this is ambiguous because it can be read as (9/3)(3) = 3*3=9 or then 9/(3(3)) = 9/9 = 1. The case 6/1(1+2) is an ambiguous case because it can be read as (6/1) (1+2) = 18 or 6/(1(1+2)) =2. It is the situation we were talking about. No, 6/(1+2) is not ambiguous. It is clearly 6/3=2. There is no discussion about this case because parenthesis are done before.
I'm a technical writer by profession, so concise, precise communication is something of a passion of mine. I also majored in mathematics as an undergraduate, so this discussion was right up my ally. Thanks for maintaining this page. It was an enjoyable journey. As mentioned by one emailer, where possible, I prefer typesetting engines, e.g. LaTeX. Why anyone wouldn't write $\frac{2x}{3y}  1$ (or something similarly robust) if they have the option is irksome to me, but that's part of the philosophy of my profession. I take on the time and cognitive burdens of clearly explaining something to minimize the effort readers have to expend in understanding it. A separate emailer mentioned the use of the division sign or obelus: ÷ (LaTeX: $\div$), which made me realize, I don't evaluate ÷ and / the same way. I assign different mental probabilities to the different interpretations. If I were handed a document that contained a÷b(c), I'd, first, curse the author and hunt for contextual clues. If I found no clues, I'd note that they took the time to use a character outside the basic ASCII set. That tells me they were possiblymaybe probablydeliberately avoiding the notoriously, obviously ambiguous slash. So I think "What would the most common, laziest, kneejerk interpretation of that equation be if the author had used a slash?" That's probably the interpretation they were trying to avoid. One piece of context sometimes available to me in deciding how much effort they put into avoiding the slash is what device they're using. Typing ÷ within macOS is trivial (Alt + /), but typing ÷ on an iPhone requires either copying and pasting or switching to a separate, thirdparty keyboard. So that's my input! It's more of a behaviordriven approach: the amount of effort it took for the author to type something might suggest their intended meaning. Yours geekily, NikMy answer:
Hi Nik, nice points. For me also, it is that when I write latex formulas in the text and not displayed, I'm tempted to use a/bc rather than \frac{a}{bc} because it fits better into the page. Since I'm aware of this PEMDAS wars, I myself do no more hesitate to sacrificing beauty if it can increase clarity. This is for me as a teacher especially important in exams. The use of division sign, obelus and slash is all over the place. I also learned, especially from teachers, that for them the obelus is used differently and that it is a stronger division. I myself try to avoid the obelus. I'm only aware having seen it early in primary school and have not touched it since. Yes, clarity needs some effort, sometimes also needs the sacrifice of some elegance. But we know that from language also. There are many expressions which become clear only in context. The answer definitely is redundance. Even the language of our genome makes use of redundancy to avoid communication errors. OliverApril 5, 2021: from Jeff:
Wouldn't these 3 counter examples end the debate? 1 = 6 ÷ 6 = 6 ÷ 2(3) = 6 ÷ 2 × 3 = 9 2(3) = 6 and (3)2 = 6 but 1 / 2(3) != 1 / 3(2) 1 ÷ 3! is now 2/3 ? Or 6 ÷ (2 + 4) 6 ÷ 2(1+2) 3 ÷ (1+2) So wrong LolMy Answer:
yes, these are all good examples illustrating the ambiguity. The 2(3) is a new twist. Usually, one illustrates the point by defining a=2, b=3 and then write 6/ab. I like also the second part because especially factors 2 are often more closely linked by humans to what comes beside. Terms like 2pi are often read as a unit and 1/2pi which is often done and where very few readers would interpret it as pi/2 like a computer does. Mathematica: In[1]:= 1/2Pi Out[1]= Pi/2June 17, Stephen,
After yet another "MindYourDecisions" video popped up on my YouTube feed (the 6/2(1+2) one), I went searching for when PEMDAS was taught as an acronym, or more succinctly, when it became PE (MD) AS as the author of that vlog insists. That author also implies that PEMDAS with multiplication before division was last advocated in 2017 and therefore is archaic. And I fount your web page referenced in my subject line. Very interesting, and I found your views align with my own. I don't know your age, but I am old enough that I think I know where all the ambiguity comes from. In a word (or three), calculators and computers. I started school in Hong Kong and then moved to the US after third grade in 1968. I was taught in accordance with what you quote from Cajori's book, but never memorized any acronyms at all about order of operations. There were never any "trick" questions of the sort presented in viral videos. And notations for problems, once we had all learned the ground rules, were always understood uniformly by all. We also did homework and tests entirely by hand, written in pencil or pen on paper. In high school, a computer was something which was programmed with punch cards, in Fortran or COBOL. BASIC wasn't out yet. The first calculators came out while I was in high school, and the first reverse Polish notation calculators arrived my senior year. It didn't really matter because they were banned for use in school on tests, both in high school and in college. We could use slide rules though. Later, while I was in postgraduate studies, personal computers were invented and developed rapidly, and calculators that could do more than asmd and memory became widely used. It was then that issues came up with differences between what happened when you punched calculator buttons in the order an equation or expression was written versus what happened when you did the problem in your head or by hand began to be noticed. We dealt with it by learning our way around the calculator so that we got the same answer we had always gotten by hand. I have a feeling that where this "ambiguity" really came to the fore was with the advent of computer programs and calculators which allowed you to enter expressions the way they were written, and the advent of a computer telling you the answer, along with the acceptance by schools of the use of calculators and computers to do your tests for you, that this ambiguity became something everyone began arguing about. Everyone now assumes that the computer is always right. Apparently the acronym GIGO has been long ago discarded. But the computer is only computing the way it was programmed to, and as you correctly say, it makes no inferences or exceptions to its hard and fast acronyms, thus destroying the ability for everyone to understand a mathematical expression the same wayunless we all discard the convention of the last 100 years and embrace the new convention of the way computers are programmed and the acronyms promulgated by the sycophants who subscribe to that way of thinking. What we are seeing now with these "gotcha" problems and vlog experts is that young people who have grown up with mature computer technology already existing have turned their acronyms into something they were never meant to be, because, "that's how the computer does it, so it has to be right." They've never known any world except one in which tests are taken on computers and limited by the keys on a standard keyboard. But except perhaps for the newest generation of young students who don't even go to the classroom any more, but instead play games on their phones while muting their camera on the Zoom classroom, or those young collegians and graduates who were taught by a modern PE(MD)AS or BEDMAS evangelist, the vast majority of people get the original correct answer, even to ambiguous expressions, as you have found from your experience. Anyway, those are my thoughts on the subject.My answer
Hi Stephen, thanks for your thoughts. Very much to the point. The difficulty indeed flared up due to the use of computers. I think I have seen this video log. It was a couple of years ago popular. It is indeed a good observation that trick questions somehow get a lot of attention and become viral. The G7 summit reminded me about the St. Ives Problem (As I was going to St. Ives, ....) which is an ancient trick or gotcha question: an uninvited intelligent reader starts to calculate but the ``correct" answer is different. The St Ives riddle is a great example because it is ambiguous. When I saw the riddle first, I imagined that everybody would walk to St Ives, also me and that I would just pass the man, wives, cats, sacks and kit. All because I would just walk faster. ``Meeting" does not mean that we have to walk in opposite direction. These days, the PEMDAS story can make for a great discussion lecture in an algebra course. I myself in an algebra workshop for teachers used it here. The subject actually produces interesting discussions. It makes aware that one should avoid such ambiguities at all cost in exams. The solution to the problem is extremely simple: ``Just be clear. Be even redundant when asking questions". Yes, the advent of computers made the problem surface. It is really a linguistic phenomenon in a society where intelligence is both human and artificial and we expect humans to be syntactically 100 percent. And as you state, the actual problem is that some prophets advocate that there is a ``correct answer" or that ``one has to fix this". The "garbage in garbage out" picture is a very good one and a nice analogy. Maybe that is the best way to think about: language is a tool to communicate clearly. Ambiguities produce garbage. If one feeds a garbage question, one can only expect a garbage output. For me, garbage is also syntactically correct stuff which by a large part of recipients is understood wrongly. The Ives riddle is a ``garbage riddle" and that it is so, is just the reason why it became so popular. One of the best way to convince somebody that things need to be ``clarified" with brackets rather than ``fixed" is the exponentiation case like 2^3^4 It is a situation where different programming languages can give different answers depending on whether one interprets it as (2^3)^4 and 2^(3^4). There can be good arguments for both and most computers start evaluating from the right which is the second version. It is interesting that for such cases, there is no viral phenomenon because it is too obvious that it is ambiguous. In the viral pemdas stories on social media, things have flared up exactly because of this ``gotcha" effect and the almost religious quest to ``fix" this. The fact that these Vlogs from 5 years ago have created so much hype is a proof that the mechanism works OliverAn additional interesting remark by Stephen:
Thanks for the thoughtful analysis and response. After I wrote to you, I continued searching for the origins of acronyms for order of operations. Although the evangelists of viral fame imply that they were in use back to 1907 or so, I found this which analyzed textbooks looking for PEMDAS as an acronym. They found that it was not until the mid 1980s that the acronym was found and taught as an acronym and the associated mnemonic device in textbooks. This apparently was shortly after the acronym was found in study courses for the standardized exams, which became popular in that decade. I would submit that this lends support for the theory that PEMDAS as an acronym only began being taught after computers began to enter common use. It was most likely the programmers who introduced it because they would have to program their calculators and computers to handle entered expressions. Blindly following operations left to right and giving hard and fast rules about handling multiplication and division equally left to right, and no association inferences, would make the programming much simpler to write, and require less memorywhich was in very short supply those days as you had to be able to run the whole computer on one 5.25" floppy holding 360kb of data and pretty much zero RAM. Order of Operations is a convention. Basically, it was standardized around the turn of the 20th century by the authors you have cited, and everyone pretty much operated by the same convention until computers and calculators came on the scene. Because of limited computer capacity at the time, programmers developed a muchsimplified order of operations for their calculators to be able to handle variable expressions and solve them. I believe this was what essentially tried to force a change of convention on the order of operations, to make everyone change their practices to conform to the computer's compromised ability to understand variable mathematical expressions. That we still find the majority of humans using the original convention tells us the machines haven't fully taken over yet. Or we're all resisting the Matrix.