Ambiguous PEMDAS

What is 2x/3y-1 if x=9 and y=2 ?

A lovely story

x=9; y=2;

(2x/3)y-1            gives  11
2x/(3y)-1            gives   2

What is x/3x?

x/3x         

         2
        x
Out[1]= --
        3

x/(3x)         

        1
Out[2]= -
        3

3.333333333333333
3.3333333333333335 

What is the source of the problem?

3+4-3  = (3+4)-3 = 3+(4-3)

3-4+3  = (3-4)+3 = 3-(4+3)

12/(3*2)      = 2
(12/3)*2      = 8

12/3x

12/(3*x)      = 4/x
(12/3)*x      = 4x

3-4+3   
(3-4)+3 = 2
3-(4+3) = -4 

An other example

x/x/x     ?

(x/x)/x = 1/x
x/(x/x) = x

    x
  -----
    x            =  x
    -
    x

    x
    -  
    x            =  1/x
   ----
    x

2/100/2

2/(100/2)   = 1/25
(2/100)/2   = 1/100

An ambiguous example with exponents

3^3^3  ?

(3^3)^3 = 19683
3^(3^3) = 7625597484987

2^10^2

(2^10)^2   = 1048576
2^(10^2)   = 1267650600228229401496703205376

256 // Sqrt // Exp // Floor // PrimeQ

PrimeQ[Floor[Exp[Sqrt[256]]]]

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}]

• 4/9/2014
  
• 4/13/2014 change title, found BEDMAS Excel page and put question first
  
• 4/14/2014: add picture, motivated from here using background drop from here.
• 4/14/2014: references
  • A hpmuseum forum example
  • A physics forum example: 48/2(9+3)
  • advocates Divide and multiply rank equally and go left to right. and has other memnonics like "Please Eat Mom's Delicious Apple Strudels".
  • This Science blog mentions the article by Tara Haelle which pretty well tells already what is going on (if I had seen this article written on March 12, 2013 before, I would not have bothered to write this down, because that article makes very clear that the initial assessment that there is no agreement on the order of multiplication or division is correct). Still, the topic had prompted me to say something new about the order of operations of the same type, like the D or E in PEMDAS which goes beyond the BEDMAS controversy. Here is an interesting quote from that article

          "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 high-school 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 1928-29, "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 don't see any indications of an advise given in the cited High school textbooks like here, here, but the mentioned entry this book of Webster Wells is clear about it on page 18:

  ---

  Update of May 18, 2017: As of recently, riddles like below have appeared which overlook that the number of fries has changed or that one cherry is used) have surfaced These puzzles have managed to become viral not because of the PEMDAS thing but because people don't look at the variations (3 instead of 4 bananas, 2 o clock rather than 3 o clock). Almost everybody gets it wrong at first. But there is also the PEMDAS challenge. Some get 88. But for getting 88, one would have to write parenthesis (2+3+3)*11. (thanks to Abitha Sukumaran for sharing this).

  ---

  Update of August 2, 2017: Presh Talwalkar writes
  
  

  "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."
  

  ---

  Update of August 5, 2017: Jacob Poscholann Koefoed Christensen sends an other example and a remark on the obelus.
  

  "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 division-sign." 
  

  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. "
  

  ---

  Update of November 2017:

  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:

  The exam still went well. And you can see on this photo taken just before the exam that everybody was happy to go: The lesson is very clear: as a teacher, even if you know better you have to be extra clear, even redundant. Even if there is no ambiguity, it is better to be on the safe side.

  ---

  By the way, the Wikipedia article mentions the example

  1+2x3 = 9     Microsoft calculator in standard view
  1+2x3 = 7     Microsoft calculator in programmers view
  

  It 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 TI-82 calculator
  1/2x = (1/2)x    in TI-83 calculator
  

  Self-proclaimed rules like this are hardly helpful.

  ---

  Update of January 19, 2018: Timothy Musgrove kindly drew my attention to a silly debate on youtube in which the question of 6÷2(1+2) again comes up (see above). Also this story shows how almost theological the debate can become but already the fact that the fraction of viewers liking the video and not liking the video is about equal shows that answer to that problem must be ambiguous. Above, I have given, (following partly Tara Haelle, who had written that Slate article), historical pointers showing how ambiguous things are. Here are the camps:
  1. PEMDAS (The multiplication comes before division)
  2. PEDMAS (The division comes before multiplication)
  3. PE(MD)AS (Division and Multiplication have the same weight, it depends what is left)
  4. Ambiguous (There is no established rule)
  Computers mostly follow the second or third. Most people and especially students (by experiment) tend to follow the PEMDAS rule. The literature points to Ambiguity.

  	PEMDAS	BEDMAS	PE(MD)AS
  6/2*(1+2)	1	9	9
  (1+2)*6/2	9	9	9

  There is a reason why the "PE(MD)AS" camp feels so much superior. We have in both cases the same answer. Also computers often follow "PE(MD)AS" and taking the "left" to "right" point of view.
  Even worse is probably the debate when asking what 8÷2/2 is (some high school teacher confirmed me that the divisions (obelus and backslash signs) are treated differently in some textbooks, see the "obelus" remark above by Jacob Poscholann Koefoed Christensen. Some would tell that the answer is 8 because / goes before ÷. When going from left to right, we get 2.
  

  ---

  Update of September 4th, 2018:
  
  I got the following nice email
  
  

  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.

  
  
  The only thing which is a bit troubling as the writer actually seems to be a teacher. Independent of the argument, the writer should probably change to a profession, where as little human interaction as possible is required. I wrote back
  
  

  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.

  ---

  Update of October 2, 2018:
  
  I was sent the link to the following Youtube Video. It is so far one of the best contributions on youtube about it. It makes a good point that in the real world, the expressions are used in a different way: for example, in published articles mn/rs is usually in publications interpreted as (mn)/(rs) or the Feynman lectures, one sees that 1/2N^1/2 is interpreted as 1/(2 N^1/2). In Engineering, one can read W = PVMg/RT. An other excellent point done in that video is that one would write x/2 if 1/2x would be interpreted as (1/2) x. Nobody would write 1/2x, if they mean x/2. So, in practice, one interprets the expression as 1/(2x), which is PEMDAS, but different from BEDMAS or the interpretation that multiplication and division are put on the same footing. It also mentions that in the AMS guide, one has PEMDAS (multiplication comes before division). Also the guide from the American Physical Society follows PEMDAS. The video illustrates again that the only way avoid the ambiguity is to use brackets.
  

  ---

  October 24, 2018: I originally had planned to post a youtube version of some slides from April 28, 2018 at the Harvard Extension STEM Club, but there was no time to do that. Thanks to Ana Carolina Smith for the opportunity to speak. Here are the part of the slides:
  SLIDES PDF (76 pages)

  ---

  October 26, 2018: An other nice example from somebody: Here the email:

  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: 
  
  
  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
  

  December 3, 2018: Atmos added an other interesting angle

  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. 
  
  Oliver 
  

  Update of December 14, 2018:
  
  In the newest Cartoon guide of the Larry Gonnik series (which are fantastic), there is also something about order of operations. But it does not go very far. "If no parentheses are present, multiply and divide before adding and subtracting." This is a very rough rule but it has the advantage that it does not get into the PEMDAS wars.

  ---

  Update of January 18, 2019:
  
  A math teacher sent me the following example. Not only is here the PEMDAS ambiguity involved. Also, the question "96 divided by 6 of 4" appears which can both mean "96/6 times 4" or then "96/(6*4)". It is a specially interesting case because of that:

  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 Method 1: Few students solved it as follows : 96 ÷ 24 Method 2: And the rest solved it as : 16 x 4 I need your help to ascertain which is the correct method, or are both the methods acceptable.

  
  
  

  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 = 123/2 The 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 /.

  Update of May 2, 2019:
  
  

  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): 93-95. . 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" "All multiplications are to be performed first and the divisions next". 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.

  
  
  Read at the end of the article of Lennes who wrote in 1917 already:
  
  

  "When a mode of expression has become wide-spread, 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 9a2 ÷ 3a is understood to mean 3a and not 3a3 is such an idiom. The matter is not logical but historical.

  
  
  One can not say it better! So, not the 12 million people are the idiots. Those who claim so, are.
  Update of August 5, 2019: Steven Strogatz gives in a New York Times article a new twist and blames a discrepancy between grade and high school: the example taken is 8 ÷ 2(2+2), where every computer gives 16, while humans usually give the answer 1. The article again makes very clear that the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!

  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 clear-cut 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: August 8, 2019. Greg McCann kindly pointed out this link to an archived copy of the AMS guidelines. Local copy. To cite from there for example:

  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:
  
  $${1\over{2\pi i}}\int_\Gamma {f(t)\over (t-z)}dt$$
  
  we might use
  
  $(1/2\pi i)\int_\Gamma f(t)(t-z)^{-1}dt$.
  
  

  August 17, 2019 An other question:

  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 (64-bit)
  Copyright 1988-2019 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 lower-level 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 lower-level math also have access to calculators or
   online tools, they might give the answer 16. The reason for the claim that it is a
   lower-level 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 higher-math 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 higher-math 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 fraction-form?
  
   Non-ambiguous 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*2-1 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.
  
  April 6, 2020 An other question:

  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.
  

  
  
  April 9, 2020 From an other email:

  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 TI-83+ 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^2-4ac/2a or
  maybe x= (-b +- root b^2-4ac)/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/3y-1 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/3y-1=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*2-1 do you get 11. But
  2*9/3*2-1 would be written 2*x/3*y-1 which is very different from 2x/3y-1,
  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 50-50, statistically speaking its obvious there is
  strong consensus amongst calculus students as to the correct method. The
  math teacher who got 2x/3y-1=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

  
  
  July 13, 2020: A high school teacher pointed out to me the article Rethinking the order of operations in the Mathematics Teacher magazine from October 2017. She pointed out also a new acronym "GEMA" appearing in the classroom which stands for "Grouping, Exponents, Multiplication and Addition". Division by x is then seen as multiplication by (1/x). This approach has the effect that division is always done before. It does not address questions like 1/x/y or 2^a^b. One obviously wants to get away from the PEMDAS rule, but the ambiguities remain if no brackets are used. What is nice about GEMA is that it only provides rules which have never been disputed. Since mathematical notation has been invented, we always perform computations which are grouped before exponentiation and before multiplication and before addition. But it does not give a student a guideline what to do with 12/4/2 or 12^4^2 or expressions like 2x/3y-1 if x=9 and y=2. Also the above article "Rethinking the order of operations" makes things more confusing. It is an other opinion. There is no rethinking necessary if one remains clear. Pointing out that the order of multiplications, divisions or exponentiations need to be clarified in writing is the smartest more. Every student should be aware of the fact that expressions like 18/3*2 are often evaluated to 3 by humans (centuries of mathematical writing have entrenched this), while computers evaluate it as 12. The expression 12/4/2 is evaluated from the left to right while 12^4^2 is evaluated from right to left by computers. Also GEMA does not give any guide lines here. It simply remains ambiguous without additional clarification. No ``rethinking" rectifies this. Just use brackets and all the problems are solved.
  
  August 4, 2020:

  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 31-0 and the link to a YouTube video
  ("PEMDAS is wrong") sent me down the rabbit hole. Incidentally, I
  discovered ISO 80000-1 now supersedes ISO 31-0, fyi. Unfortunately,
  ISO 80000-1 isn't available for free so I had to resort to ISO 31-0
  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 31-0. 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 31-0) 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 (64-bit)
  Copyright 1988-2020 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. 
  

  
  
  August 8, 2020: Here is an insightful comment from Elias Martenson from Sweden:

  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
  non-ambiguous. 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 non-ambiguity 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.
  

  
  
  Added August 24, 2020:

  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 80000-1:2009"
  

  "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 80000-1:2009:
  
  Added January 16, 2021: An email:

  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 non-associative
  
  = 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 socio-linguistic 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 "willy-nilly", 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
  Oliver
  

  and 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 non-ambiguous 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 eighteen-thirds 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 mc2 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.
  

  ---

  February 28, 2021: a nice new example:

  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!

  
  
  My Answer:

  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
  

  ---

  March 4, 2021: the social media discussions still seem to go on: I got the following email:

   
  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 now
  

  Obviously 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.
  

  ---

  March 13, 2021:

  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 possibly-maybe probably-deliberately avoiding 
  the notoriously, obviously ambiguous slash. So I think "What would the most common, 
  laziest, knee-jerk 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, third-party keyboard.
  
  So that's my input! It's more of a behavior-driven approach: the amount of effort it 
  took for the author to type something might suggest their intended meaning.
  Yours geekily,
  Nik
  

  My 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. 
  
  Oliver 
  

  April 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
  Lol
  

  My 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/2 
  

  June 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
  way--unless 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
  
  Oliver
  

  An 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 memory--which 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 much-simplified 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.
  

  A remark from Kevin (September 13 2021)

  I'm a C programmer, so * and / have the same precedence, as do binary +
  and -, with left-to-right ordering, so effectively we have ( BO) > (M or
  D) > (A or S) i.e. neither multiplication nor division are "done first",
  you do them sequentially in the order they are written. And similarly
  for addition and subtraction. You've covered that before.
  However, my contrary thought is what is meant by "2x" and "3y" in your
  apocryphal sum "2x/3y-1", because there is NO OPERATOR between the numeral
  and the variable.  This conventionally means "multiply" but I would argue
  that this zero-length multiplication operator has HIGHER precedence than
  the * operator. Thus your problem can be re-written (2*x)/(3*y)-1 and
  then C, Java, Perl et al will give the answer 2 when x=9 and y=6.
  The acronym should be corrected to BOZMDMAS where ZM is the zero-length
  multiplier operator.
  The ZM operator intuitively has higher precedence than even the
  exponential operator, because it is obvious that
  2^3x*4y = (2^(3*x)) * (4 * y)
  or it would be obvious if "^3x" was written as "3x" in superscript,
  which after all is just a typographical short hand for the ZE (zero-length
  exponential) operator.
  I hope this contribution adds to your enjoyment of this timeless
  conundrum.
  

  My reply:

  Thanks for the original addition. I like the word BOZMDMAS.
  Would be fun teaching it!
  Apropos programming languages, most don't allow things like 2x.
  Computer algebra systems are smarter and allow both spaces 2 x or
  just 2x or 2*x. This is actually quiteinteresting because 23 or 2 3 are
  different. While I like your modification, good luck teaching it. Many students
  do not even honor the basic multiplication before addition rule which is undisputed.
  Oliver
  

  A remark from Nathan (September 26 2021)

  I read through your page on Ambiguous PEMDAS, and I found something that
  is wholly incorrect. No, it's not in mathematics; I'll recluse myself from
  that, thank you very much.  In the comments section, you refer to Stephen
  as commenting that PEMDAS wasn't around "back to 1907 or so", wherein he
  referenced 
  this article, dated Friday, March 6, 2009, entitled "PEMDAS:
  Terminate With Extreme Prejudice".  I've read that particular article,
  and I haven't found a reference to any analysis of "... textbooks looking
  for PEMDAS as an acronym."  Maybe he referenced the wrong article, maybe
  the comment was removed, or maybe Firefox's edit and search functions
  are missing an occurrence of "PEMDAS".  If that article claimed that
  their analysis' found that PEMDAS "was not until the mid 1980s that the
  acronym was found and taught as an acronym and the associated mnemonic
  device in textbooks", they need to use a larger sample of textbooks.
  Their date of "the mid 1980s" is contradictory to your comment that
  "But since 1917, the PEMDAS rule has been taught to millions of people."
  Alright, I'm being picky here.  Let me throw in my 2-cents worth: I
  graduated highschool in 1975.  We learned PEMDAS in school, long before
  we graduated.  I'd love to find that article; not to be contrary, but
  I believe that it would be humorous to read just how they went about
  their sampling.
  
  Sincerely, Nathan
  

  My reply:

  Hi Nathan,
  thanks for the valuable comments. I must say that I do not know when PEMDAS first emerged.
  Yes, there was one comment which claimed that it is since 1907 but you are right that
  this article from 2009 does not back that up. It is an interesting
  historical question, who first came up with the acronym PEMDAS or similar acronyms.
  I have passed high school a few later than you but also for me (in Switzerland), no
  acronym was taught along for the rules of arithmetic. We just learned
  "multiplication and division" before "addition and subtraction" and to
  avoid ambiguities by placing brackets.
  
  Oliver
  

  A remark from Louis (September 29, 2021) [PDF] My Answer:

  Hi Louis, 
  
  thanks for the contribution. I would not call things right or wrong
  which are ambiguous. Your initial answer of 6/2(1+2) = 1 was actually
  the most natural answer. It is not the answer a computer gives who uses
  the PEMDAS (left to right) rule as you describe it but what most people
  do, especially also if you were taught so. Actually one has tested it
  and the huge majority of humans interpret such expressions as you did.
  You try to rectify something which can not be rectified for any practical
  reasons. Even if you were a government and put a document making it
  the rule of the land, the majority of students would still interpret
  it differently. It does not matter how one wants to ``rectify this",
  it is a linguistic ambiguity which will be misunderstood by a large
  part of the users even with new rules. The solution to the problem
  has been given already 100 years ago by make it unambiguous by putting
  parenthesis. Math teachers know that even the most basic rule of putting
  multiplication/division before addition/subtraction is misunderstood. The
  right thing to do is to acknowledge that and to avoid expressions like
  (3/4+5) and make it clear with (3/4)+ 5 or use typesetting with horizontal
  division sign which can not be misinterpreted. There are lots of other
  things which are often understood differently. For example, 2 3/4 is
  often understood as 2+ 3/4 but mathematically, it means 6/4. In many
  school textbooks, expressions like 3 1/4 which you have on the bottom
  of the second last page would be interpreted as ``three and a quarter"
  meaning 3+1/4 = 13/4. I just added this as an other example on how
  mathematical notation has historically grown to be ambiguous.
  Where language could be misunderstood, it is important to be even
  redundant and over clarify. I have a few sources in my summary article
  https://people.math.harvard.edu/~knill/pedagogy/ambiguity/ambiguity.pdf
  which sites experts in the field and confirm that we deal with a problem
  in linguistic.  
  
  Oliver
  

  ---

  November 8, 2021: Nigel sends me this screenshot from a youtube comment. I have not seen the video. Nigel:

  Recently I've fallen into the rabbit hole of this ambiguity case where 
  there can be multiple answers to a specific question. Looking 
  through youtube comments, I discovered this gentlemen who explains 
  it in a way which is supposedly contradictory to your case study. 
  I've attached a image of his analysis on the topic and humbly 
  request you to explain into further detail.
  

  My Answer:

  Short explanation:
  
  The algebraic convention like 6/2y clash with other
  rules like the (6/2) y =3y (the result a computer gives) 
  is exactly the point. It often happens that 2y is read as a 
  unit. This is why the expression must be considered 
  ambiguous. 
  
  Let me explain things a bit more detailed as you ask for it. 
  
  1) You must first realize that the expression 6/(2y) is 
  different from the expression (6/2)y. In the first case
  you get 3/y. In the second case you get 3y. Ok? 
  
  2) Once you have realized that there are different expressions,
  you have to find out which one is the right, if one just writes
  6/2y. Right? How does one find this out? 
  One has to search for rule books written down by humans and to 
  see what has been decided.
  
  3) The problem is that there are different rules and that these
  different rules do not give the same answer. Lets look what has
  been said: 
  
  --------------------------------------------------------------
  A) There is the famous PEMDAS which treats multiplication M before 
  division D. This gives the first answer 6/(2y) = 3/y. PEMDAS is
  what has been taught for decades. 
  
  B) There is the BEDMAS rule which treats division before
  multiplication. This gives the second answer 3y. 
  
  C) There is the PE(MD)AS rule (followed by most computers) which 
  treats multiplication and division on the same footing but looks
  at the order. This gives the second answer 3y. 
  
  D) There are writer rules like one given out by the 
  American mathematical society which urges not to use too many
  brackets and read expressions like most humans do. This gives the
  first answer 3/y. One has tried it out (and I have tried it out
  with students): most humans read 6/2y as 6/(2y)=3/y. This is
  again the first answer.
  ----------------------------------------------------------------
  
  4) Ok, now we have looked at the rules or conventions and seen that 
  they disagree, we want to look what textbooks, 
  linguists, experts in mathematical notation say.
  (see "Ambiguous Pemdas" [PDF])
  is that the expression is AMBIGUOUS. It can be read in different
  ways. There is no right or wrong because there is no accepted rule.
  
  5) Now, if you look at the discussions in social networks and 
  youtube videos, it is quite common that somebody comes up with a new
  rule or new suggestion to ``solve" the ambiguity. This does not help
  as it just adds more options to A),B),C),D). One picture explains
  this well in this cartoon. It is
  a case for "competing standards". If there are competing standards,
  the solution is not to come up with a new standard. The solution is to 
  AVOID the ambiguity. In the mathematical expression under consideration
  this is done BY PUTTING BRACKETS. This solves the problem once and for
  all. It is funny that this is the recommendation was put forward already 
  100 years ago like by Cajori, the one who wrote the first serious textbook
  on mathematical notation. 
  

  I just stumbled upon a intelligent remark by George Bergman from Berkeley. By the way, Gorge Bergman has made Many photos in the Oberwohlfach collection. I agree 100 percent with the following citation:

  "From correspondence with people on the 48/2(9+3)  problem, I have learned that in many schools today, 
  students are taught a mnemonic "PEMDAS" for order of operations: Parentheses, Exponents, Multiplication, 
  Division, Addition, Subtraction.  If this is taken to mean, say, that addition should be done 
  before subtraction, it will lead to the wrong answer for  a-b+c.  Presumably, teachers explain that 
  it means "Parentheses - then Exponents - then Multiplication and Division - then Addition and Subtraction", 
  with the proviso that in the "Addition and Subtraction" step, and likewise in the "Multiplication and Division" 
  step, one calculates from left to right.  This fits the standard convention for addition and subtraction, 
  and would provide an unambiguous interpretation for  a/bc,  namely,  (a/b)c.  But so far as I know, it is 
  a creation of some educator, who has taken conventions in real use, and extended them to cover cases where 
  there is no accepted convention. So it misleads students; and moreover, if students are taught PEMDAS 
  by rote without the proviso mentioned above, they will not even get the standard interpretation of a-b+c.  
  Should there be a standard convention for the relative order of multiplication and division in expressions 
  where division is expressed using a slant? My feeling is that rather than burdening our memories with a 
  mass of conventions, and setting things up for misinterpretations by people who have not learned them all, 
  we should learn how to be unambiguous, i.e., we should use parentheses except where firmly established 
  conventions exist."
  

  ---

  From December 3rd 2021: (only the first part of the email is displayed)

  I will state up front, I am in the MD, or AS, being equal in priority
  and done left to right however.  I think implicit multiplication should 
  be a special case Implicit multiplication includes both coefficients and 
  their follower(s) Coefficients first use is circa 1715 or so I read, a FYI.
  
  Implicit multiplication also includes an integer or variable in front
  of a parenthesis, ie distribution.
  It could also be assessed as a coefficient and a follower
  I believe IM logically has a higher priority than MD, but less than E
  
  Why?            E=mc^2 = m(c^2) But does not equal    (mc)^2
  In PEMDAS, it would be placed here               P E IM MD AS
  
  I think most globally accept that MD and AS are equal in priority
  Although you have shown there are exceptions of those who haven't
  
  I have seen on social media that some currently believe M must be done
  first, but it's very rare.
  
  Factorials are also not mentioned in the mnemonics but do have a place
  in order of operations
  
  Do you know what it is?  Is it taught?
  Is it shorthand for implicit multiplication? Or shorthand for a different
  function? IOW a linked set of factors that produce a standalone product, similar
  to implicit multiplication but different?
  

  My reply:

  Hi Charles,
  it is an interesting new angle  P E IM MD AS
  One of the main difficulties with that is how
  to recognize implicit multiplication. We use
  shortcuts like 1/2pi rather than writing 1/(2pi)
  but somebody might read it as (1/2) pi.
  The implicit multiplication is sometimes used
  and was encouraged at a time when typesetting was
  more expensive to do. It can be found in many
  places. It is an ambiguous notion unless one specifies
  in each case, whether one deals with implicit multiplication
  or not. One would have to use a new notation for implicit
  multiplication.
  About factorial: this is universally done before anything
  else. Again, one could be tempted to make a rule out
  of it and say  F P E IM MD AS
  I can only reiterate that producing more complicated rule sets
  or adding an other conventions does not help, as it just
  produces a new standard    https://xkcd.com/927/
  Oliver
  

  ---

  From January 3rd 2022:

  I came across your page
  discussing expressions of the form 8/2(2+2) and questions like "what is
  2x/3y-1 with x=9 and y=2?"
  I have read much of it and scanned the rest. Since the most recent
  entry is only about a month old I feel comfortable taking the liberty
  of sending this email.
  I believe that my own thinking on the topic has been much clarified by
  explicitly introducing the concept of grouping. That is, in PEMDAS or
  BODMAS or any of the variations thereof, the P or B should be generalized
  to G for grouping. Notations that indicate grouping include parentheses
  (aka brackets), the absolute value sign, the radical, and the horizontal
  bar that indicates fractions when mathematical expressions are properly
  typeset.
  
  With that in mind, the different interpretations of expressions like
  2x/3y can be resolved by agreeing (or not!) that an expression like
  "3y" is considered to be "multiplication by juxtaposition", and
  that multiplication by juxtaposition is a form of notation that
  indicates grouping. This would make expressions like W = PVMg/RT
  unambiguous, because it is then understood that the whole of RT is
  in the denominator. It's also my impression from watching many "viral"
  youtube videos and reading the comments, that there actually is widespread
  agreement that expressions of the form 2x or RT are considered grouped
  (though the term "grouped" is rarely used).
  
  This could also resolve the ambiguity of 8/2(4), if there was agreement
  about whether or not the form 2(4) indicates grouping. But there clearly
  is not agreement. (I do not consider it grouped, because I believe the
  absence of a multiplication operator is based on a different convention
  than its absence from expressions like 2x. But that's just me.)
  
  To summarize:
  
   
   I think there is already widespread agreement on the concept of
   grouping, even if it isn't always recognized as such (expressions like
   "more tightly bound" are sometimes used).
  
   
   I think there is already widespread agreement that expressions of
   the form 2x  or NkT (from PV=NkT) are grouped.
  
   
   There is widespread dispute whether expressions of the form 2(4)
   should be considered grouped (hence 8/2(2+2) is ambiguous)
  
   
   Evaluating 8/2(2+2) is not about order of operations. It's about
   interpreting the notation.
  
  
  Oh, and I take it for granted that the "MD" or "DM" in the acronyms
  should be understood using the left to right rule; this just makes the
  acronyms less than ideal as teaching tools.
  
  Thanks, and I hope you don't mind receiving yet another email about this.
  Don
  

  My answer:

  Hi Don, 
  
  I agree that clarifying the concept of grouping is a central part of
  the controversy and clarifying this would help. However,
  there is no consensus what is acceptable as grouping and as long as
  it is not clear, it will be misunderstood. Terms like 2pi are seen
  by most readers as grouped because 2pi is a unit. What about 8pi
  or (1+1) pi or 4 pi^2.  There is already a very nice solution to the
  grouping problem: just place brackets.
  
  
  Oliver 
  

  ---

  From March 21, 2022:

  Many thanks for your web page on order of operations. As an engineer I
  would get nowhere writing equations in an ambiguous way as seems to be
  the case in many schools.
  I have talked with teachers on Facebook and as you point out they apply
  rules almost religiously or dogmatically and they cannot be convinced
  of anything alternative.
  Why on earth if y=1/ab, if the person means b/a, would he not write
  it that way? Of course we all write using fractions anyway and using
  brackets, but there must be a lack of things to understand at school if
  they dwell so long on such problems when brackets can always be used. As
  you point out, when brackets are used there is no problem in the first
  place, so teachers seem to not understand that mathematics is a language
  to convey information to others, and is not a quiz. The importance of
  good notation is of prime importance.  Tom
  

  My Answer

  yes, exactly my sentiments. I made this semester the test again (among
  them many teachers who see it like you) and almost nobody would read
  1/ab as b/a. Yes, it has become a ``gotcha" issue overriding the basic
  principle that language is a social construct. It probably has emerged
  from the fact that 1/ab is interpreted by b/a by computers, and then one
  tries to pop a rule onto students, many of which (also very smart ones)
  would read 1/a+b as 1/(a+b). A little bit of clarity and redundancy does
  never hurt. Even our genetic information is stored in a highly redundant
  way so that a small mistake does not immediately lead to a failure.
  Glad to hear the voice of an engineer. Oliver
  

  ---

  From June 15, 2022:

  Hello,
  
  I just ran across this article
  https://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html, and
  was glad to read it, as I have seen this confusion many times. One
  great example I don't think you have, and just wanted to throw in
  is computer language design. In computing, we have a large amount
  of additional operators compared to layperson arithmetic, such as
  increment operators, assignment operators, etc. The standard solution
  is to essentially define a more elaborate PEMDAS rule and include
  it in the documentation, for example python's can be found here:
  https://docs.python.org/3/reference/expressions.html#operator-precedence,
  which has 18 bands!
  
  As you can see, much like the PE(MD)(AS) rule, it is quite common
  in programming to group left to right within a band, treating them
  as equal (with special rules for exponentiation, similar to how you
  identify it is treated specially in the article). This is also a
  clean extension of what I was taught in the 90s in Canada with
  BE(DM)(AS). As you might imagine, having an even longer version of
  the rule doesn't fix the ambiguity. The best approach is once again,
  to just add brackets to clarify the order, or to break into multiple
  explicit equation steps e.g. " top = 2 * x", "bottom = 3 * y",
  "result = top / bottom - 1". Quite often it is considered best
  practice to have maintainable code to remove the ambiguity, especially
  since adding a few more characters in the source file of a program
  is, essentially, free with the storage spaces now available.
  
  If for some reason, there is still ambiguity after this, languages
  may provide a full "grammar", often in BNF notation. Most programmers
  never have to go into that level, but it is good to know it is there
  if a reader ever ends up with one of these problems.
  
  Apologies if you did have this already, it is a very long article
  at this point.
  
  Thanks for the article! 
  Glen
  

  Hi Glen,
  thanks for the insightful note. Indeed, a good programming language
  should give guidance about the order of operations. Most of the time,
  this is not done or then hidden away. 
  Yes, there are layers and in the case of operations
  like e^x, the behavior is often not seen in the language specifications. 
  I agree that additional clarity is even good if the specifications are
  clearly given. Not everybody knows or looks at the exact specifications when 
  casually programming. This is a situation where one could even in a 
  program make a note that the parenthesis were put for redundancy reasons.
  A future maintainer of the program might remove some brackets to 
  ``make things more elegant". It happened to me several times that I was
  trying to optimize a code and making it shorter, not noticing that some
  additional bracket actually was needed in that language. 
  By the way, here is a joke about what happens if one includes also the 
  equal sign into the mix: 
     x = 12; x += 3              gives 15
     x = 12; x =+ 3              gives 3 
  Plus and Equal can be combined in both ways but do not commute. 
  Oliver 
  

  ---

  From September 9, 2022:

  Fascinating page. The thing I find interesting is how militant people are 
  in their beliefs over the right answer. It is almost religious. - Marcus
  

  Hi Marcus
  yes, this is maybe the most interesting part of the story.
  An allegory for many other things non-mathematical.
  The surprise is that this happens in such a relatively well
  defined narrow part of a very exact science.
  Oliver
  

  ---

  From October 9, 2022: a funny contribution (goes with the picture)
  
  

  I came across the 6/2(2+1) thing a couple weeks ago, and in trying to search for the rule I was taught about treating things like 2(2+1) as a unit I came across your chain. I was taught that it was important to treat it as a whole because of formulas I would eventually learn and have to use (and when I started taking higher level math, I saw what my teacher meant). I've been stewing over this for a couple of days, trying to figure out what exactly it was about this particular equation that was bothering me, and I believe I've figured it out. ... This bothered me so much because there were so many people stuck on 9 as the answer saying that this was how they learned it, and this is why kids don't like math. If you dumb down the basics incorrectly, without explaining the simplification (or worse, that the simplification even exists), when it gets more complex later it won't make any sense and they won't be able to understand why. My brother's kindergarten teacher tried to tell his class the earth moves very slowly, and that's how we all managed to stay on. Imagine how confusing the rest of his life would be if our mom hadn't corrected that nonsense! But, as the teacher explained, it was easier for the kids to understand. Regardless, there IS a correct answer to 6÷2(2+1), and that answer IS 1, because it is a circle. Don't disturb my circles. Maggie

  
  
  

  Hi Maggie,
  thanks. I like that proof using the circle ...But yes, you hit the point
  exactly. What humans do is that they tend to group things like 2(2+1)
  as a unit. It become more prevalent in expressions like 6/2pi
  which most people read as 3/pi and not as 3pi as some fanatics or
  computers want to teach us. And yes, 2pi is a round thing.
  Oliver
  

  ---

  From October 26, 2022: a nice video of Clemens Winklmaier (MOV file) about CASIO calculators. One can not illustrate the ambiguity better. As we can see, even some calculators have become ``human" (Humans evaluate 8÷2(3+1) usually as 1 because they see 2(3+1) as a unit together.
  
  

  Today was one of those days I stumbled upon one of the "ambiguous" maths problems while scrolling down on Facebook ... yet again. Searching for an older answer from some professor I sadly can't remember, I did come across your page "Ambiguous PEMDAS" which I shall bookmark right away. Reading most of you examples and comments I was thinking you might be interested in this little video I recorded. The calculators used are all the Casio 991, which is used in most German high schools today. Interestingly each iteration of the model handles the 8 \div 2 (3+1) different. Feel free to add it to your collection. Kind regards Clemens Winklmaier Dear Clemens, wow! That is fantastic. I will add it to the conversation. Das Video ist ein Hammer. Oliver

  ---

  From November 5, 2022: Joerg sent me a link to a youtube video which suggests GEMS instead of PEMDAS. The name GEMS stands for Grouping, Exponents, Multiplicative structures and Subtracting. My Answer:

  Hi Joerg, 
  thanks. I did not know that one. I like the name, 
  it is easier to remember. A bit a hack, that addition
  is replaced by subtraction to make the word GEMA sound
  better. It of course, like any other approach, will not
  solve the major problem that we often group things not
  from left to right. It does not replace the 100 year old
  advise to write extra clearly if a misunderstanding is
  possible, like with in an expression   1/2r   which 
  by most is read as 1/(2r), while one can understand it also
  as (1/2) r. I have seen so many times also even professional
  Mathematicians write things like 1/a+b and understand it as
  1/(a+b) or very smart kids read 1/a+b  as 1/(a+b). 
  All teaching or smart acronyms does not change the fact that 
  some assumptions are wired in and in a time when things had
  to be typeset with a typewriter, authors tried to avoid brackets
  as much as possible and the AMS even gave this as a rule like
  for 1/2pi which is often written without a bracket and everybody 
  reads it as 1/(2pi) while a computer reads it as (1/2) pi. 
  
  The youtube video asks "Better than PEMDAS"? The standard answer
  to such questions is    https://imgs.xkcd.com/comics/standards.png
  It again just provides a false sense of security pretending we have a
  rule which solves the problem. The problem is that even by knowing
  the problem, the majority of people read things differently than intended
  in general. Independent of education level.  A teacher writing a text
  following this rule and having all students (experiments with real
  people show that almost none really follow such rules) read things
  differently will have a big problem and get torn apart, especially by the
  parents. (such a story has initially brought me to this fascinating area).
  
  Oliver 
  

  Joerg again:

  Thx for swift reply. Yeah, you`re right: rules are not everything. I just find it strange that in the English speaking countries 4 precedence levels are taught with 6 letters. This is the reason why I like GEMS so much = four letters.
  
  I was raised in a German speaking country and we don`t know nor 
  use PEMDAS or BODMAS or variations thereof.  We were taught 
  "Punkt vor Strich" = first points (# :) then lines (- +) 
  2 precedence levels, 2 words.
  

  This is actually a nice point. Simple rules are better to remember and easier to follow. I remember now that this was also what my teachers have told me "Punkt bevor Strich". Maybe also something to consider "Point before Line" !

  ---

  From November 8th, 2022: Alex Staub has some excellent points which I have never seen addressed. They illustrate very strongly why it does not work without clarification.

  I have recently been embroiled in one of those "ambiguous" math
  problems using the obelus, and ran across your article. Some of what I
  say here was touched upon by other people who have been in communication
  with you, but I wanted to add my input that implicit multiplication
  should take priority, and here is why:
  
  A monomial represents one term. So 2x is not quite the same thing as
  2*x. 2x is the single value which is the product of 2 and x. So the
  parentheses around 2x are actually implied.
  
  Further, what of x÷x? The variables have a coefficient of 1, yet few
  people would argue that this expression equals x^2, as strict PEMDAS
  would seem to indicate...indeed, even algebra calculators which equate
  2x÷2x to x^2, evaluate x÷x=1.
  
  One may argue that "well, the equation doesn't have the 1 written, so it
  can be ignored" but this is one of the keys to why implicit multiplication
  must have a higher priority. WHY is the coefficient of 1 virtually
  always omitted? Because it is done automatically. The multiplication
  is implied to have already taken place. We aren't dividing by just x,
  but the product of x and 1. Other coefficients should be the same.
  
  Finally, one more example:
  
  How are strict PEMDAS proponents to perform: x÷x^2?
  
  Exponents first, so x÷xx, division and multiplication left to right,
  so 1*x=x
  
  However, if x=3, then the result is quite clearly 1/3 or 1/x. Because we
  are dividing by a monomial. The product of x multiplied against itself.
  We are not dividing by its factors.
  
  It just seems like such an easy fix to prioritize implicit multiplication.
  
  Thanks
  Alex
  

  Here was my answer:

  Hi Alex, 
  fantastic points. 
  
   1. I believe it would matter in 1/2x or 1/2*x when doing experiments.
      Most people probably would think in the first case as 1/(2x) and 
      in the second case argue (1/2)*x. Some might still think of 1/(2*x).
  
   2. I love x ÷ x = x ÷ 1*x = (x ÷1)*x = x2. Indeed, nobody would
      understand it as such. I agree. Everybody would understand
      x ÷ x as being 1. 
  
   3. I also like your example x ÷ x^2 = x ÷ x*x. By PE(MD)(AS), with exponentiation 
      before multiplication and division, everybody would think 1/x. In the 
      second case, we could argue the answer to be x if division and multiplication
      are done on the same level and done left to right. 
  
  All these three examples perfectly well illustrate that there is no alternative
  than just to clarify the expressions. I have never seen these examples mentioned
  in discussions. Very nice. 
  
  Oliver 
  

  ---

  From November 28, 2022: Lucas writes:

  Hello, I have read your write up on the ambiguity of the order of
  operations. I was a little disappointed because it seemed you preferred
  anecdotal evidence to make your point. I have primarily two questions
  with some preface attached.
  
  The ambiguity as I understand it mainly comes into play when reading or
  writing math problems on one line. The classic example is 8/2(2+2) which
  could be read as either (8/2)(2+2) or 8/(2(2+2)). I see people reference
  your write up to support that this problem specifically is ambiguous.
  You go on to offer a few more examples such as 2^2^2 as ambiguous.
  My concern is that you seem to be intentionally avoiding an international
  standard which clears up that ambiguity.  Namely ISO 80000-1 section
  7.1.3 covers much of it.
  
  Your suggestion of improving clarity through the use of parentheses is
  as much a learned skill as is the actual standard.
  
  Would you agree that the real problem is a failure to teach the standard
  rather than the standard being flawed?
  
  Personally I don't see how you could improve the standard any
  further as it reads to me as essentially "work left to right unless
  it violates the fundamentals of math". It seems that any ambiguity
  is intentionally resolved by this. The standard gives no priority to
  coefficient multiplication or multiplication by juxtaposition unless I
  am misinterpreting.
  
  I have heard the argument that standards should follow intuition but
  it seems to me that the standard is as intuitive as could be reasonably
  expected. There is an inherent ambiguity when writing expressions on a
  single line but the standard exists to resolve that.
  
  What would the purpose be to ignore that standard and claim that the
  problems are ambiguous in the face of it?
  
  Thank you in advance for your time.
  

  My answer

  Hi Lucas,
  
  1. Used language is mostly about anecdotal evidence. Look at the Wolfram
  Citation in my article. It does not make sense to have rules which nobody 
  observes.
  
  2. I have mentioned the ISO standard and have a copy on my website. The 
  There are various problems with that standard. 7.1.3 does not clear up 
  all the controversy which has been discussed! Please read that standard
  and read the examples which were under dispute in the last 10 years. 
  
  3. I agree that what I say here is not very original. Clarity should always
  have priority. Clarity means also to be redundant if there is a chance
  of having it read wrong. Even if it would conform to a standard. Everybody
  agrees that 1/3+4 is clear enough. Every standard agrees. Still, lots of
  people read it as 1/7. It does not hurt here even to be more clear. In cases
  like 4/2c   (not addressed at all in the ISO standard), there are
  standards which claim (4/2)*c is the right answer. But as a matter of fact, 
  most humans read it as 4/(2c). 
  
  4. Intuition is great. But all the discussion is actually about cases where
  intuition for humans is different than intuition for computers. Changing the 
  standards to address intuition would still also be wrong. It would disappoint
  all computer languages which use an other standard. 
  
  5. It is not about ignoring the standard. It is to anticipate where 
  misunderstanding could occur. For us this is actually a very reasonable 
  rule. Whenever you talk and the danger is that you are misunderstood, then 
  it is better to clarify what you say.  It is a meta rule for all communication. 
  
  6. The interesting thing about all this PEMDAS thing is that there are a few
  folks who want to push some sort of standard against all common use. This
  produces conflict. One can say this does not matter. But it matters say for a
  teacher who writes an exam in which such an ambiguity occurs. It drives everybody
  mad. 
  
  Thanks for your email
  Oliver 
  

  Thank you for your response and I will look into the standard further
  especially regarding disputed cases.  I appreciate your take on clarity
  but I was worried that teaching these problems as ambiguous will ensure
  that they remain ambiguous. Whereas teaching the standard will eventually
  put everyone on the same page.  At risk of sounding pedantic the standard
  does actually cover 1/2c or in the language of the standard a/bc.This is
  towards the bottom of page 23 in example 1 and it continues to clarify
  on the next page explicitly stating that a / bc is a/(bc) and not a/bc.
  

  
  Hi Lucas,
  
  Thanks! Note however that this is written on the top of page 24 as
  a
  ---
  bc
  
  Which is non-disputed. This is how we write it. The dispute comes when
  writing  a/bc. If you feed that in a computer (or one of the fanatics on the web)
  you read a/bc as (a/b)*c which is ac/b. No normal human does that.
  
  I'm glad you mentioned the ISO. This is interesting.
  
  Oliver
  Oliver
  

  ---

  From February 2, 2023: Brent writes

  Good morning,
  
  The internet seems to be littered with Ambiguous PEMDAS and while looking to understand 
  more, I stumbled across this page of yours:
  https://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html
  Over my career (IT, Software, and Business), I've come across countless examples of 
  improper gathering and combining of numbers.  As the story goes... a client doesn't 
  understand why they are losing money.  Soon after we discover fuzzy math in their 
  golden quoting formulas. Such experiences have left me as a potential over-user of 
  brackets.  It's ok, I am no math wizard and brackets help me sleep at night. 
  Aside from sending a thank you note... I do have a question about one part. 
  Under the section " What is the source of the problem?" and toward the end of the section.
  3-(4+3) = -5
  3-(7) = -4 ???
  Maybe a typo, maybe I am totally overlooking something.
  Time to get back to work.  At least until I follow up on the ÷ vs / stuff.
  Thank you for reading.  Have a great week!
  Brent
  

  Hi Brent,
  thanks. Yes, that had been a typo. 3-(4+3)=-4. 
  Glad you point that out. It is corrected in the above. 
  I can imagine that notation problems can be a
  source of problems in many spreadsheets and errors in 
  business computations. In general, it is the lack of 
  brackets but of course, it can also be the wrong use
  of brackets or simple computation errors or typos. 
  One of the good things of the attention this matter has
  obtained is that one is maybe a bit more sensitive to 
  these issues now. Especially quality control or auditing
  of finances can improve when being aware that ambiguity
  in formulas can be a problem. 
  Oliver 
  

  ---

  April 4, 2023:

  Just found on twitter: a teacher asked a small class about 8/2(2+2) and got 3 answers 1 and 3 answers 16. One wrote that it is intentionally ambiguous and that the notation sucks. It seems that historically the problem occurred not "intentionally". It just happened to occur and then one was surprised how ambiguous it is. What surprises me is that we are still debating it, it is now almost 10 years into "heavy exposure" to the phenomenon in social media (I learned about it in 2014 but it had been discussed much before on facebook for example). The sample size is very small (7 answers) but in my experiments in 2014 with a 60 student calculus class, where almost all except a handful of students would pick 1.

  ---

  From April 16, 2023 Stuart writes:

  Hi Oliver
  I've been following the PEMDAS ambiguity for some time now, including
  the insightful discussion on your page.
  
  My background is in engineering and physics, where implicit multiplication
  taking precedence is fairly standard, particularly where the precedence
  is clear from context - for instance 1/jω and E/kT are very common
  terms that are often used without parentheses. It's not just a case of
  "why would you write it like that?" if you meant ω/j and ET/k, but the
  expressions often end up being meaningless from a dimensional analysis
  perspective (-E/kT is a common exponent, but ET/k is not dimensionless).
  
  But it is all fundamentally a semantic argument, from a prescriptivist
  perspective it's somewhat unfortunate that there isn't much in the
  way of freely-available authoritative answers - even the  (non-free)
  ISO standard says "shall not be followed by a multiplication sign" and
  does not mention implicit multiplication, even though it is apparent
  that implicit multiplication only worsens the ambiguity.
  
  The AMS style guide
  (https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf) seems
  to be even worse - in section 13.10 it states: "In slashed fractions,
  stet parentheses or lack of parentheses-1/2a + b or 1/2(a + b)-unless the
  same construction is inconsistent within the paper, in which case query
  the author."  So they give 1/2(a+b) as an example to stet (i.e. accept)
  but don't actually explain what it should be interpreted as, because it
  is seemingly so obvious!
  
  From a descriptivist perspective, though, it's clearly ambiguous and the
  fact that it's so easily misunderstood should be reason enough not to
  write something like this. But I think I'm preaching to the choir here...
  
  Regards,
  Stuart
  

  My answer:

  Hi Stuart,
  
  I agree with everything. Having to "query the author" is indeed not a
  good option. Yes, if somebody writes E/kT, then one knows in the context
  what it means. I like the example 1/j&omega. One reason why authors like
  such notation is because it is more elegant. E/(kT) looks more ugly and a horizontal
  division sign needs more time to write. Fortunately, modern writing techniques allow
  us easily to express things today, like \frac{E}{kT} in LaTeX.
  I recently looked through a "Formula collection (1990)" of Urs Kneubuehl
  in physics which I had used extensively throughout college and saw that he has almost
  fanatically avoided any terms a/b and everywhere already used the horizontal division
  sign even in non-ambiguous examples like  G m r/r^3 ,  even in cases
  like m v^2/2. He must have felt that especially in a formula collection which
  is used widely, any smallest ambiguity would be devastating and be a source
  of errors, not only in schools but in research papers.
  AMS should probably in a style guide simplify the message too and simply
  advice for example to avoid A/BC and use \frac{A}{BC} or A/(BC)
  
  Oliver
  

  ---

  From May First, 2023 Julian writes

  Dear Oliver:
  
  Perhaps you are sick of the "Ambiguous PEMDAS" debate by now, but I just
  wanted to ask a question whether there is an ISO 80000 standard or any
  mathematical rule regarding coefficients with variables.
  
  QUESTION: Is 2x considered one term?
  
  Is it a single term with 2 being the coefficient of the variable x,
  or is this merely 2 times the variable x? I think this would resolve
  many of the ambiguities. If a coefficient next to a variable is always a
  single entity, then 1/2x would always mean 1/(2x) without the need of a
  parenthesis. To represent 1/2 of an x, it would need to be written x/2 or
  (1/2)x. And (1/2)x would always mean ((1/2)x).
  
  In general, I agree with the email from Charles dated December 3rd 2021: P
  E IM MD AS. Therefore: 2x = (2 * x) 3(4) = (3 * 4) 2(1+2) = (2 * (1 + 2))
  
  Anyway, I'd appreciate an answer to the question above. Thank you.
  
  Julian 
  

  My Answer:

  Dear Julian,
  no, the topic of mathematical language is extremely interesting. It is
  also important because if mathematical communication does not work, we
  are in trouble. The question about coefficients with variables
  is an interesting topic. In most computer algebra languages for example
  you can write for example 2a or 2Pi and the computer understands this as
  2*a or 2*Pi. However, our computer friend does not understand 1/2a,  as we
  humans do. The computer interprets 1/2a as a/2   or 1/2Pi   as Pi/2  .  .
  There is some comments in the ISO 80000 standard like 7.2.2 or 7.1.3
  or A.3.      The difficulty is to see in an expression whether the
  author saw it as a coefficient of a variable. A silly example is
  when writing 23.141 . We all understand it as the number twenty three point
  141. An AI entity who has learned that 2Pi is interpreted as 2*Pi might infer that
  the author means 23.141 = 2*3.141. Why should 3.141 be treated differently
  than Pi? Actually, my initial experiments with Chat GPT when it became viral last
  November showed that it shows basic lack of arithmetic skills. GPT 4 is better.
  And now bots are bonding with computer algebra systems (something which had
  been done long before. Siri had been consulting with Wolfram alpha for decades. We had
  combined chat bots with computer algebra systems 20 years ago and it had already
  then not been considered original
  see our report
  In any way, especially with the upcoming of AI, we might need to be more precise
  in writing mathematics. Computers understand things differently in general than
  humans (that was one of the main points in this entire topic). And also with
  the inclusion of AI, the general rule to communicate unambiguously and if necessary
  in a redundant way, stays important.
  
  Oliver
  

  ---

  From May 14th, 2023:

  On May 1 2023, Julian asks whether ISO 80000 says anything about whether
  "2x" can be considered "one term".
  
  I was able to find a copy of ISO 80000-2 online, and in it I find the
  following:
  
  "The symbol for multiplication is a half-high dot or a cross.
  Either symbol may be omitted if no confusion is possible."
  
  This is as close as it seems to come to telling us how to interpret
  "2x". It does not define "term", not that I can find.
  
  Julian may be thinking that if we consider "2x" to be one term, it
  follows that the multiplication would be done first in the expression
  1/2x. My own view would be that expressions like 2x are widely (perhaps
  universally) considered to be grouped. This causes the multiplication
  to be done first in 1/2x. In this view it doesn't really matter whether
  we consider it to be one term. It's the grouping or not that counts.
  
  As to how computer algebra languages interpret 1/2a, or 1/2(3), we are
  at the mercy of how the language's designers chose to interpret it. As
  such, and especially for controversial expressions like 1/2(3), they
  represent nothing more than another opinion, one more among many. What
  these languages choose to do gives us no evidence one way or the other
  about how to interpret the likes of 1/2(3). (Unless the designers are
  internationally recognized experts of such stature that they should be
  considered authoritative, Which seems unlikely.)
  
  Ordinary computer programming languages (all the ones that I am familiar
  with) won't accept  1/2x as a valid expression. Omitting operators is
  not allowed. The programmer has to rewrite it to mean whatever it is
  they intend it to mean. This puts the responsibility where it should be,
  on the user/programmer, to give the computer explicit instructions.
  
  Thanks for your time, -Don
  

  My answer:

  Hi Don,
  thanks for the comments. As you confirm, the expressions can be
  ambiguous. Two comments
  
  -  Yes, the notion of "term" is ambiguous. There are places, where
  one can find definitions like
  "A term is either a single number or variable, or numbers and
  variables multiplied together"
  
  -  About computer interpretations of 1/2x, advanced
  computer algebra systems allow it
  
  ------------------------------------------------------
  Mathematica 13.1.0 Kernel for Linux x86 (64-bit)
  Copyright 1988-2022 Wolfram Research, Inc.
  
  In[1]:= 1/2x
  
          x
  Out[1]= -
          2
  -----------------------------------------------------
  A computer reads it as x/2, while a human would read it
  as 1/(2x).
  
  This example  confirms that one needs to clarify.
  If a teacher writes 1/2 x , then some of the students
  will interpret it as (1/2) x while others will interpret
  it as 1/2x.
  
  Oliver
  

  Hi Oliver,
  Thanks for the response.
  
  And Wolfram Alpha has two modes, one called "Natural Language", the other called "Math Input". 
  I typed "1/2x" (those characters in that order)  in both modes and the results were different. 
  See the attached screenshot for the "Math Input" mode result.
  Turns out it puts the expression in the URL, so I can show you
  
      Here and
     here
  
  For further exploration, try typing "1/1+2" in each mode; and watch carefully what it does immediately after each character is typed.
  
  I'd expect an advanced computer algebra system  (which I have called systems that interpret symbolic math) to accept 1/2x as input. 
  The question, of course, is what do they do with them. I'm a little surprised that Mathematica would do what you showed; 
  I thought one of its main things was to interpret symbolic math the way people do#but my expectations can easily be incorrect.
  
  -Don
  

  Update of May 26, 2023:

  I'm thoroughly enjoying browsing through your Ambiguous PEMDAS page. It
  seems like we might be on the same page. I don't think it's an issue of
  PEMDAS vs. BEDMAS, though. I think other things need to be considered
  like applying the distributive property as part of the "parentheses"
  step and recognizing that coefficients (cofactors) shouldn't be separated
  such that one becomes a divisor. I've written a blog post on it, and if
  you find my comments worthy, you can reference it on your page. Thank
  you for considering this. I'm not a mathematician, but I did make it
  through Differential Equations in college before switching majors,
  so I'm no slouch when it comes to math.  8 ÷ 2(2 + 2) = 1: 
  Why PEMDAS Alone Is Not Enough.
  

  My answer

  Hi Scott,
  thanks. 
  Just one thing about using spaces. In Mathematica or wolfram alpha,
  x/ab keeps ab as a variable and when writing
  
      x/a b 
  
  then the computer interprets it as bx/a. Good collection of examples
  on your blog. 
  Oliver 
  

  Update of May 27, 2023:

  I don't mean to bother you too much on this question, but I have a question about your October 26, 2018, e-mail exchange on your page. I appreciate that you have some material on your page that addresses the language aspect of the problem. My formal educational background is actually linguistics (biblical Greek and Hebrew), so part of my attempted solution is to address the linguistics or the "syntax" of the ambiguous expressions.
  
  It occurs to me that there is another possible process at work in the computers on the 2 * 3 / 4 * 6 expression, and you sort of hint at it, but it doesn't have anything to do with whether the computer is using PEDMAS or PEMDAS. I don't know enough about the inner workings of programming languages to understand the details of how they work, but is it possible that the computer is treating the division slash (or the obelus if used elsewhere) as a cue for the computer to multiply by the inverse of the term that follows? If so, then in the computer's "mind," that removes any ambiguity, and order at that point doesn't matter, because it's all multiplication and the associative property applies. So the computer looks at it like this, with the inverse of 4 in brackets to clarify it's taken as a single term:
  
  2 * 3 * (1/4) * 6 = 9.
  
  But when I hear the problem spoken, there can be verbal cues that are difficult to express in the written language. If I were to see the original problem and say it out loud, I might emphasize the connection between the multiplied terms. I'm using the ~ symbol to indicate where I would have a slightly longer word-space pause for effect; the hyphen indicates I'm trying to say it as a single term:
  
  "Two-times-three ~ divided by [or "over"] ~ four-times-six."
  
  Saying it that way, at least in my mind, emphasizes the multiplication first, then the division last. My answer would be 6/24, or 1/4
  
  However, if I were to say it as follows, I'd be emphasizing the LTR PE(MD)AS, processing each sign as it comes:
  
  "Two times ~ three-divided-by-four (or "three-over-four") ~ times six," or
  "Two times three-fourths times six," to make it clear I'm dealing with three terms.
  My intended answer then would be 9 for either of those statements.
  
  I do agree with you though, that using parentheses or brackets is the best solution.
  
  What would it take pedagogically to move toward a more robust application of PEMDAS? Maybe call it "Active PEMDAS"? Here are some suggestions:
  
  1. Insist (maybe too strong a word, but looking for some uniformity across the board) that mathematical Properties (esp. Distributive) be considered as part of the P in the PEMDAS (or PE(MD)AS) step (i.e., P = Properties & Parentheses). When such properties are properly applied, require the use of parentheses/brackets, etc. to indicate so.
  2. Similar to 1, for expressions/equations presented ambiguously, require students to indicate by the use of parentheses etc. how they went about solving the expression. In this case, the right answer depends on how the student organizes the equation.
  3. Stronger encouragement of the "division is multiplication by inverse" concept so as to eliminate the ambiguity of multiplication or division first vs. left-to-right application.
  
  In other words, actively "force" the use of PEMDAS through intentional parentheses/bracketing rather than passively assume intention based on signs or symbols used. From a linguistic perspective, this aligns with using active voice to clearly indicate the subject (actor) who is doing the action of the main verb ("I washed the car."). Passive voice often hides the true actor of the verb either by omitting the actor altogether or by moving it into a prepositional phrase at the end of the sentence ("The car was washed [by me].").
  
  Thank you for considering this. If you want to continue the conversation, I'm open to that. However, if your schedule is such that you can't devote a lot of time to this topic, I'll completely understand.
  Scott 
  PS: Thank you for the tip on spaces in Wolfram.
  

  My Answer

  Hi Scott,
  thanks. I can put your comment on the collection page. 
  Yes, it is a linguistic phenomenon. I'm not a linguist
  although. Language is a communication tool. As wolfram 
  has pointed out once, it grows and depends on social
  cultural and other factors. But language also has to be 
  so clear (maybe even redundant) that there can not be 
  any misunderstanding. Especially in classrooms, where 
  students have to take tests and exams, it is pivotal 
  that the problems are clearly stated, even if there should
  be established rules, one should be so clear that everybody 
  works with the same math problem. 
  Oliver