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}
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.
And this is why math teacher should NOT be teaching spelling. 
mmm
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.
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.)
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.
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.
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).
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.
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).
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.
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.
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.
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.
I was in elementary school in the 80’s. (American)
Unless, of course, it does, which is true for many cases.
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.
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…