# Math problem **URL:** \ **Category:** Factual Questions\ **Created:** [April 28, 2011, 9:54pm UTC](https://boards.straightdope.com/t/math-problem/580084 "2011-04-28T21:54:00Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 28, 2011, 11:28pm UTC](https://boards.straightdope.com/t/math-problem/580084/21 "2011-04-28T23:28:36Z")
> [@Saint\_Cad](#): > > Short answer: 9 > Long answer: This is why I tell my student to forget PEMDAS that was taught to them in elementary school. The problem is 6 / 2 x 3. Students that memorized PEMDAS invariably interpret it as saying multiply then divide. What you are supposed to do is if you have both multiplication and division, do whichever is to the left so the proper solution is > 6 / 2 x 3 > 3 x 3 > 9 > > Math life would be easier if instead of PEMDAS (mini rant: what about roots? at least teach PERMDAS or better yet stay away from the mnemonic), elementary teachers taught > () > ^ r > X / > > - > - UMM, that’s STILL PEMDAS!!! :rolleyes: ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [April 28, 2011, 11:30pm UTC](https://boards.straightdope.com/t/math-problem/580084/22 "2011-04-28T23:30:34Z")
> [@etv78](#): > > UMM, that’s STILL PEMDAS!!! :rolleyes: Not if you consider each line/tier a set, rather than an ordered list. {parens} \> {exp,root} \> {mul,div} \> {add,sub} Now where do mods fit in… I’ll go with by {mul,div} ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [April 28, 2011, 11:57pm UTC](https://boards.straightdope.com/t/math-problem/580084/23 "2011-04-28T23:57:08Z")
> [@Great\_Antibob](#): > > I understand this is a bit a snark, but I wanted to address this point directly. There’s no implicit multiplication in C++. You need to use an explicit ‘\*’ for multiplication. That’s the crux of the debate. I suppose we should have engineered spoken and written language to avoid ambiguous statements altogether. What exactly are you saying here? That 6÷2(1+2) is an ambiguous expression, and 6÷2×(1+2) is not ambiguous? They both seem ambiguous to me. A programmer will answer 9 for both, but if you follow the PEMDAS rules as outlined by Johnny L.A. in post 6, the answer will be 1 for both. ---
**Author:** ![Mean\_Mr.Mustard](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mean_mr.mustard/32/3865_2.png) [@Mean\_Mr.Mustard](https://boards.straightdope.com/u/Mean_Mr.Mustard)\ **Post date:** [April 29, 2011, 12:15am UTC](https://boards.straightdope.com/t/math-problem/580084/24 "2011-04-29T00:15:14Z")
> [@Saint\_Cad](#): > > Then it is a different problem. Absolute value and division bars act as grouping operators and so > NUMERATOR > DENOMENATOR > is (NUMERATOR)/(DENOMENATOR) > > And I will also point out that your thought process in an earlier post as to how PEMDAS work is exactly why it should NOT be taught. And this is why math teacher should NOT be teaching spelling. 😃 mmm ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 29, 2011, 12:25am UTC](https://boards.straightdope.com/t/math-problem/580084/25 "2011-04-29T00:25:58Z")
PEMDAS was never meant to mean “Multiplication before division” and “Addition before subtraction”. It was always supposed to be PE(M or D)(A or S); within multiplication/division, or addition/subtraction, go from left to right. [Thus, “1 - 2 + 3” should not be read as 1 - (2 + 3); it should be read as (1 - 2) + 3] That having been said, I agree with **Great Antibob** and the others who note that implicit multiplication by juxtaposition (as opposed to with an explicit times operator), like most other uses of juxtaposition as an operator in mathematics and computer science, has a strong tendency to be intended with the highest operator precedence there is. A strong but not universal tendency… thus, the ambiguity pointed out. And it should be re-emphasized, over and over and over, that this has nothing to do with mathematics, per se, just arbitrary notational conventions over how to write that mathematics. Ideally, we’d have long ago settled on a notation system that didn’t need operator precedence disambiguation (with the hack of using parentheses for overriding this); e.g., Polish/LISP-style consistently prefix notation. Alas, that did not happen, but familiarizing oneself with PE(M or D)(A or S) is not learning mathematics; it’s just learning orthography, to speak. ---
**Author:** ![Pasta](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@Pasta](https://boards.straightdope.com/u/Pasta)\ **Post date:** [April 29, 2011, 12:28am UTC](https://boards.straightdope.com/t/math-problem/580084/26 "2011-04-29T00:28:53Z")
If I say, “Jim gave Bob the book,” there’s no problem. If I channel Yoda and say, “The book Jim did Bob give,” you could figure out what I meant to a reasonable degree of confidence, but you’d think I was an asshole for being so obtuse. So, too, with mathematical expressions. There are conventions that go beyond the order of operations, and when those conventions are violated, one must infer whether the author was intentionally violating some conventions but not others, and which conventions should be taken to dominate. If someone writes “1/ab”, am I to treat that as 1/(ab) or (1/a)\*b? Ignoring all conventions except order-of-operations leads to an unambiguous answer: (1/a)\*b, but there are different, and rather strong, conventions that say you would have written this as simply b/a if b/a is what you meant. Given these conflicting signals, I would have to ask you what you meant if it wasn’t clear from context. (And, if someone wrote that, they would almost certainly mean 1/(ab), as the grammatical conventions that say “write (1/a)\*b as b/a” are strong.) ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [April 29, 2011, 12:31am UTC](https://boards.straightdope.com/t/math-problem/580084/27 "2011-04-29T00:31:02Z")
6 / 2 (1+2) = ? Remembering that division is the inverse of multiplication, this can be rewritten as 6 x 1/2 x 3 Either way you multiply this out, you get 9. 3 x 3 = 9, or 6 x 1.5 = 9. The answer is always and only 9. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [April 29, 2011, 12:31am UTC](https://boards.straightdope.com/t/math-problem/580084/28 "2011-04-29T00:31:22Z")
> [@Arnold\_Winkelried](#): > > What exactly are you saying here? > That 6÷2(1+2) is an ambiguous expression, and 6÷2×(1+2) is not ambiguous? They both seem ambiguous to me. That’s precisely what I am saying. Multiplication/division should be performed left to right as encountered in a problem. The first expression contains an ‘implicit’ multiplication. As noted above, there are teachers and even some numeric solvers that perform the implicit multiplication before explicit multiplication/division. A rule is only absolute if everybody agrees on it. There’s sufficient argument over the application of implicit multiplication that there is a _de facto_ ambiguity. You can militantly insist one particularly way is correct, but that’s not going to change anybody’s mind or force the “mathematical community” (whatever that is) to accept it. Or, what **Indistinguishable** just posted. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 29, 2011, 12:37am UTC](https://boards.straightdope.com/t/math-problem/580084/29 "2011-04-29T00:37:53Z")
> [@Indistinguishable](#): > > That having been said, I agree with **Great Antibob** and the others who note that implicit multiplication by juxtaposition (as opposed to with an explicit times operator), like most other uses of juxtaposition as an operator in mathematics and computer science, has a strong tendency to be intended with the highest operator precedence there is. Er, except, still not stronger than exponentiation, I suppose… but still stronger than explicit multiplication/division. Anyway, the point is, the juxtaposition operator is syntactically different from the \* operator, and carries different precedence rules for many people, even if the intended semantics of the two operators are the same (multiplication). ---
**Author:** ![Cunctator](https://avatars.discourse-cdn.com/v4/letter/c/43a26b/32.png) [@Cunctator](https://boards.straightdope.com/u/Cunctator)\ **Post date:** [April 29, 2011, 12:38am UTC](https://boards.straightdope.com/t/math-problem/580084/30 "2011-04-29T00:38:25Z")
> [@Rigamarole](#): > > It’s not ambiguous. The order of operations in math is absolute. The answer is 9. That’s my response too. Maths has a fixed order of operation, which in this case gives the answer 9. > [@Johnny L.A.](#): > > PEDMAS. > > P. Terms inside parentheses. > E. Exponents and roots. > M. Multiplication. > D. Division). > A. Addition. > S. Subtraction. Interesting. I’ve never heard of the acronym PEMDAS. We were certainly never taught at school that multiplication took precedence over division, or that addition took precedence over subtraction. The rules that we were taught were consistent with \*\*Indistinguishable’s \*\*interpretation. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [April 29, 2011, 12:41am UTC](https://boards.straightdope.com/t/math-problem/580084/31 "2011-04-29T00:41:53Z")
> [@Cunctator](#): > > That’s my response too. Maths has a fixed order of operation, which in this case gives the answer 9.Interesting. I’ve never heard of the acronym PEMDAS. We were certainly never taught at school that multiplication took precedence over division, or that addition took precedence over subtraction. The rules that we were taught were consistent with \*\*Indistinguishable’s \*\*interpretation. PEDMAS (Please Excuse My Dead Aunt Sally) is supposed to be P E D and M A and S Since division and multiplication are the same thing but inverse, they are the same step. Same with adding and subracting. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [April 29, 2011, 12:49am UTC](https://boards.straightdope.com/t/math-problem/580084/32 "2011-04-29T00:49:04Z")
From [The Language of Algebra - Order of operations - First Glance](http://www.math.com/school/subject2/lessons/S2U1L2GL.html) > [@](#): > > When expressions have more than one operation, we have to follow rules for the order of operations: > > [ol] > [li]First do all operations that lie inside parentheses.[/li][li]Next, do any work with exponents or radicals.[/li][li]Working from left to right, do all multiplication and division.[/li][li]Finally, working from left to right, do all addition and subtraction.[/li][/ol] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 29, 2011, 12:52am UTC](https://boards.straightdope.com/t/math-problem/580084/33 "2011-04-29T00:52:16Z")
It’s no use merely quoting PEMDAS or such things, unless the source quoted shows some awareness of the issue that (as an empirical fact about people, and even about mathematicians) some people will treat the juxtaposition operator slightly differently from the infix “\*” or “x” operators so far as the syntactic rules for parsing go. A source which shows no awareness of this fact is not presenting anything to the discussion which isn’t already known to everyone involved. The syntactic rules for how to parse mathematical expressions aren’t handed down from God, any more than the rules of English are. They arise organically from the way mathematicians, in practice, use notation. It happens to be the case that they’ve been partially codified and formalized in schools to a degree beyond ordinary language, but nonetheless, you will find in practice that mathematicians do not look at an expression like 6 / 2x and uniformly feel a stirring for a single, unambiguous parsing which no one could dare to question. And since the actual linguistic practices of mathematicians does not resolve this question, there is no genuine rule of the mathematical grammar which resolves the question either (for the genuine rules of grammar are simply the ones determined by usage). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 29, 2011, 1:33am UTC](https://boards.straightdope.com/t/math-problem/580084/34 "2011-04-29T01:33:40Z")
> [@](#): > > It’s not ambiguous. The order of operations in math is absolute. The answer is 9. > [@](#): > > That’s my response too. Maths has a fixed order of operation, which in this case gives the answer 9. Math does not have an order of operations. Conventions for describing math have order of operations. And what that order of operations is depends on what convention is being used. There is a convention under which the answer to the OP’s problem is 1, and there are conventions under which the answer is 9. Which one is correct depends on which convention is intended, which probably means asking the person who posed you the problem. As an aside, on the calculator thing, this is one of the two most common mistakes I see with students and calculators. They’ll do a calculation like, say, f = omega/(2pi), and leave off the parentheses. I’m guessing that the fact that so many students interpret it that way is the reason why Texas Instruments changed which convention they use, but I fear that it’s just going to make things even more confusing. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [April 29, 2011, 1:42am UTC](https://boards.straightdope.com/t/math-problem/580084/35 "2011-04-29T01:42:35Z")
> [@Chronos](#): > > Math does not have an order of operations. Conventions for describing math have order of operations. And what that order of operations is depends on what convention is being used. There is a convention under which the answer to the OP’s problem is 1, and there are conventions under which the answer is 9. Which one is correct depends on which convention is intended, which probably means asking the person who posed you the problem. > > As an aside, on the calculator thing, this is one of the two most common mistakes I see with students and calculators. They’ll do a calculation like, say, f = omega/(2pi), and leave off the parentheses. I’m guessing that the fact that so many students interpret it that way is the reason why Texas Instruments changed which convention they use, but I fear that it’s just going to make things even more confusing. My answer in #27 shows that there is an order of operations. If you change the “divided by” to “times” then it is clear. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [April 29, 2011, 1:58am UTC](https://boards.straightdope.com/t/math-problem/580084/36 "2011-04-29T01:58:51Z")
> [@Indistinguishable](#): > > PEMDAS was never meant to mean “Multiplication before division” and “Addition before subtraction”. It was always supposed to be PE(M or D)(A or S); within multiplication/division, or addition/subtraction, go from left to right. [Thus, “1 - 2 + 3” should not be read as 1 - (2 + 3); it should be read as (1 - 2) + 3] As a math teacher for 14 years, I can assure you that even if the teachers does not teach multiplication before division, addition before subtraction (and yes I have seen elementary teachers make that mistake many times), PEMDAS is certainly learned that way by many students. ---
**Author:** ![Rigamarole](https://avatars.discourse-cdn.com/v4/letter/r/77aa72/32.png) [@Rigamarole](https://boards.straightdope.com/u/Rigamarole)\ **Post date:** [April 29, 2011, 2:07am UTC](https://boards.straightdope.com/t/math-problem/580084/37 "2011-04-29T02:07:48Z")
> [@Chronos](#): > > Math does not have an order of operations. Conventions for describing math have order of operations. And what that order of operations is depends on what convention is being used. There is a convention under which the answer to the OP’s problem is 1, and there are conventions under which the answer is 9. Which one is correct depends on which convention is intended, which probably means asking the person who posed you the problem. Can you show me any respected mathematical sources which assume a “convention” other than the standard convention (in which the answer is 9)? I’ve never heard of any other convention besides the one that has been explained in this thread already. If you are a math teacher teaching your students otherwise, it’s doing them quite a disservice. ---
**Author:** ![Snnipe\_70E](https://avatars.discourse-cdn.com/v4/letter/s/b5a626/32.png) [@Snnipe\_70E](https://boards.straightdope.com/u/Snnipe_70E)\ **Post date:** [April 29, 2011, 2:12am UTC](https://boards.straightdope.com/t/math-problem/580084/38 "2011-04-29T02:12:46Z")
> [@Saint\_Cad](#): > > And you would be incorrect. > > It is not ambiguous (see my post) and to claim 6÷2(1+2) should be read as 6÷[2(1+2)] is simply erroneous. I am afraid I would agree with 9. But remembering back to Miss Poge (Who I would rather not remember) and high school algerbra start left to right. But if she saw this written on any of her students papers she would have thrown something at the offender. ---
**Author:** ![etv78](https://avatars.discourse-cdn.com/v4/letter/e/d78d45/32.png) [@etv78](https://boards.straightdope.com/u/etv78)\ **Post date:** [April 29, 2011, 2:26am UTC](https://boards.straightdope.com/t/math-problem/580084/39 "2011-04-29T02:26:42Z")
> [@Saint\_Cad](#): > > As a math teacher for 14 years, I can assure you that even if the teachers does not teach multiplication before division, addition before subtraction (and yes I have seen elementary teachers make that mistake many times), PEMDAS is certainly learned that way by many students. I was in elementary school in the 80’s. (American) ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [April 29, 2011, 2:52am UTC](https://boards.straightdope.com/t/math-problem/580084/40 "2011-04-29T02:52:17Z")
> [@md2000](#): > > The trick is to see that 2(3) means 2x3, but is not a 2x3 that takes precedence of order over 6÷2 Unless, of course, it does, which is true for many cases. > [@Rigamarole](#): > > The order of operations in math is absolute. The answer is 9. Then why do we have differing orders of operations? Why do some say that the implied multiplication goes before explicit multiplication? Do you fail to acknowledge that this is, indeed, the case? It seems you’re just being purposefully obtuse. > [@needscoffee](#): > > My answer in #27 shows that there is an order of operations. If you change the “divided by” to “times” then it is clear. Of course it’s 6 x 1/2 x3. Unless, of course, you parse it as 6 x 1/(2x3), as many people would. Folks, it’s simple. We all agree that the 6 is in the numerator. We all agree that the 2 is in the denominator. The disagreement is whether or not the (1+2) goes upstairs or downstairs. The only thing clear about this syntax is that it’s ambiguous. This has really nothing to do with any order of operations because you don’t actually have to process the problem or do it in steps. Just determine whether the author intended to put the quantity in the numerator or the denominator and the argument solves itself. Of course, we all know that the author did this intentionally just to screw with us… [Previous page](https://boards.straightdope.com/t/math-problem/580084.md?page=1) [Next page](https://boards.straightdope.com/t/math-problem/580084.md?page=3)