If a student thought that, they wouldn’t be following the proper OOO OR BEDMAS/PEMDAS, I don’t know if you understand my point.
In this context, the proper OOO is simple: Whatever gets a result different from what you got, because you are wrong, because the whole point of the exercise is to prove that the person posing the question understands basic arithmetic and you do not.
It’s ambiguous. It’s inherently, deliberately ambiguous, and to claim otherwise is dishonest. It’s impossible to just stumble upon such an equation because anyone who is actually writing math would have enough basic sense to make the equation unambiguous, by re-writing it, using more parentheses, or both.
From the wikipedia link earlier (thanks GamerHat):
My bolding added. Aside from using the difference word for the acronym the bolded parts highlight a key difference - multiplication vs division being first varies.
Basically, this. These kinds of problems are designed, by intention, to screw you up and cause you to get the wrong answer. So if you got the wrong answer, that was the answer you were supposed to get. So the wrong answer is the right answer. Except it’s still wrong. But if you get the right answer instead, that’s also wrong. It’s a trap!
Personally I’d limit that to implied multiplication. I want 1/2a to be 1 over 2a, but I want 1/2a to be 0.5a. That is the most convenient for writing the kinds of fractions I’m most likely to deal with in the limited form of plain one line text.
I’d avoid 20/5(22) entirely. If you mean 20/(522) you’re saving one symbol that way, if you mean 20/52*2 you’re using one extra just to make things ambiguous.
I believe in an earlier thread of this type I screwed with Wolfram Alpha to see what its parser said and it tended to agree with this. Trying it out now, it looks like WA normalized itself to treat multiplication by juxtaposition the same as explicit multiplication, presumably for predictability’s sake.
I believe someone also presented a cite from at least one well-known math publication that when submitting papers to the journal, to save on space it recommended reducing expressions such as
To 1/2x for brevity. I think for variables and perhaps function invocations, most people bind the “juxtaposition operator” higher than division, but below exponentiation.
Of course, in cases in the sciences, you can often use dimensional analysis to disambiguate, and for things like vector math nobody is going to mistake 1/x.x for (1/x).x when x is a vector because such an expression is ill formed.
The wikipedia article mentions this change explicitly.
The problem with your interpretation is that in all variations, m and d are given the same priority, it is incorrect to say that bedmas gives priority to division or pemdas gives priority to multiplication. It is explicitly explained that they are treated as equals. The acronym is just a simple way of remembering the order for children, they must still remember that multiplication and division are done left to right, rather than division before multiplication, same deal for addition and subtraction…
If we’re really talking about children, then adjacency as multiplication isn’t allowed. If there were a “*” or “/” there, the discussion would be much less contentious.
If we are talking about a broader “truth”, then you have to fold in other conventions beyond PEDMAS that hold. One convention is that you simply don’t write things this way. Someone who has written “20/5(2*2)” has violated one convention, so it would be odd to assume they have obeyed any other particular one.
And the reason there is a convention of “don’t write things this way” is exactly because it is, in the fullness of mathematical notation, a bit ambiguous to do so. This discussion is good evidence of that. It’s the same reason I would not choose to say “I fed her dog food.” I could say that to mean that I gave her dog some food, but it’s ambiguous, so I shouldn’t even try, because the reader won’t be quite sure what I mean.
[quote=“Senegoid, post:8, topic:705387”]
I larnt me my basic arithmetic in grammar school, circa 1958-1963, in the United States. The rule was:
[list=a][li] Evaluate any parenthesized group before combining that result with anything outside those parentheses.[/li][li] Perform all multiplications and divisions in order from left to right. Note that multiplications and division have equal precedence, neither of them having priority over the other![/li][li] Finally, perform all additions and subtractions in order from left to right. Again, note that addition and subtraction have equal precedence, neither having priority over the other.[/list][/li][/QUOTE]
Yes, you’ll find plenty of educators who advocate not teaching PEMDAS or whatever because it can wrongly be interpreted to mean that multiplication comes before division, or addition before subtraction, when, as you correctly stated, they have the same level of precendence.
For the record, my TI-83 calculator evaluates the expression in the OP as 16, which is correct according to that standard rule of the order of operations.
Evidenced a few posts above.
DaveBfd, do you agree that there are notation conventions that the expression violates, independent of order-of-operations considerations?
It’s as close to universal in math as anything is. The BEDMAS/PEDMAS/BODMAS/etc mnemonic is relatively new (about a century - the history is somewhat unclear whether it was the end of the 1800s or the beginning of the 1900s).
Some notion of an operator precedence is, of course, centuries old implicitly and explicitly around 200 years old.
That said, it’s simple pedantry to see a rule and blindly apply it without explanation.
I’ll quote myself from a couple of the threads linked above:
(bolding newly mine)
(bolding newly mine)
Well yeah, but I was specifically referring to BEDMAS. I was raised on my dear Aunt Sally’s knee. Saying ‘brackets’ instead of ‘parentheses’ and putting division ahead of multiplication sound as wrong to my Yankee ears as “clever chap.”
It depends. Obviously, bracketing 22 is useless, so you could say that is enough. My view is that the formal order of operations is meant to remove all ambiguity in formatting and perception to ensure a constant result. With this understanding, it is expected that if a person wanted to achieve any result, such as including 5 with 22, they would have had to intentionally made that effort in some way in their notation.
If you want to reduce the problem to the person making the equation having no clue how conventions work, I think the conversation becomes pointless as the whole formula is useless. If we want to solve any equation, we must apply OOO as intended, under the assumption that it was meant to be interpreted as written. I don’t think working outside of the OOO is acceptable simply because of any implied meaning from the writer when there is no precedent for it to be interpreted otherwise. The physical sciences take is a separate issue which I understand, but it is unique only to those sciences (and I still want to know why). Under no other circumstance are implied brackets accepted.
If you want to interpret 522 as being the denominator, you must change the formula as it has been written.
First, you mean “evaluate any expression,” not “solve any equation.” Second, OOO is purely a matter of convention, and doesn’t even represent actual usage. Similarly to deciding what constitutes Standard English, proper mathematical notation should be judged by what mathematicians actually write. And no mathematician assumes that division and multiplication associate strictly left-to-right. No mathematician would write the expression under discussion. If I saw that expression written by a colleague, I’d knock him upside the head and ask him what he meant to write. If I saw it on a student’s homework I’d mark it as a mistake.
To illustrate that this is really a matter of convention, mathematicians are quite comfortable assuming that addition and subtraction associate left-to-right. We regularly write things like 1 - x + x[sup]2[/sup], which is unambiguously (1 - x) + x[sup]2[/sup], not 1 - (x + x[sup]2[/sup]). For multiplication and division, this convention simply doesn’t exist among mathematicians.
I agree that the whole formula is (nearly) useless. This doesn’t make the conversation pointless, though. It makes it more interesting, since the dissonance comes about entirely because the formula is written in an odd way. And a lot of interesting observations fall out of that.
(Bolding mine.) There is no requirement that we be able to solve any equation or evaluate any expression. Some oddly written equations you would reject outright, like “4 + elephant = 1 mile”. The one in the OP is broken in a more subtle way (violating other conventions) that happens not to prevent the application of PEDMAS.
Notational rules needn’t allow all expressions to be valid. Some things are strictly forbidden by the rules, like “4+*8”. Others are merely forbidden by accepted convention, like “4++8”. Had that been “4±8” you might be okay with it as meaning “4 + (-8)”, but the double-plus version is outside of the accepted conventions of the language. It should normally be written “4+8”.
You could write “a=2x” or you could write “a=x2”. No one writes the second thing to mean “a=2*x”. If someone did, you could safely assume they were not fluent in the language. Forcing an interpretation onto “a=x2” by rigidly sticking to PEDMAS is to accept one convention while arbitrarily ignoring another. And the very fact that people don’t write “a=x2” holds the key to the discussion.
Everyone agrees that you can constrain yourself to a set of parsing rules and parse these expressions. But that’s the pointless discussion. The interesting discussion, and the source of the controversy in the first place, is that those parsing rules come tightly bound to a rich suite of other notational conventions.
The posters who are arguing that PEDMAS sometimes doesn’t hold are saying something different from me. I’m saying that even if PEDMAS holds, so do other rules, and this expression violates some of those other rules. Asking how you would apply PEDMAS anyway misses the heart of the matter.
I meant to say something about this as well. Physicists and mathematicians outwardly speak the same language, but it would be more accurate to say that they speak two dialects of the same language, each with their own conventions, including notational conventions. Something like American English and British English, where (among many other differences) the same words are sometimes spelled differently. Physicists use mathematics as a tool, not an end in itself, and feel free to modify that tool for their own convenience. Each side often wonders what the other is on about.
Not directly related to the topic, but one of my favorite incidents along these lines: When I was a postdoc and string theory was brand new, barely more than a gleam in Ed Witten’s eyes, he gave a series of lectures at Harvard on the topic, attended by both mathematicians and physicists. Almost more interesting to me than what he was talking about, was to watch the reactions of the eminences of both fields to what he was saying. The physicists didn’t really understand the math he was using, as it was a tool most of them hadn’t yet taken up; the mathematicians were familiar with the math (mostly algebraic topology) but found his approach strange. When he would do a calculation, he wouldn’t prove the result, he would simply argue that what he had written down was what it had to be because of the physics. Two groups of academics separated by a common language.
Though sometimes mathematicians work very hard to prove why some of the weirdness engineers and physical scientists do is technically correct or feasible.
The proofs apparently tend to get complicated.
Oh, absolutely. That’s part of why the exchange has been so fruitful in both directions.