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DaveBfd
11-22-2014, 10:33 PM
I have been reading arguments about a simple math problem recently, and it is seriously frustrating me.

20/5(2*2)=

There is, for some reason, a large backing of people who state the solution is 1. BEDMAS would dictate that:

2*2=4 -----20/5*4
20/5=4 -----4*4
4*4=16

How does the argument of assumed parenthesis after the division hold so much support? Why is BEDMAS so contested in this situation? What is the TRUE answer? (as much as I know the answer, I want affirmation damnit)

JKilez
11-22-2014, 10:44 PM
Because for many places multiplication is performed before division. Are you outside the US?

GameHat
11-22-2014, 10:49 PM
This isn't a math question, it's a language question.

It all depends on how you learned to interpret written math into Order of Operations (http://en.wikipedia.org/wiki/Order_of_operations)

Chessic Sense
11-22-2014, 10:58 PM
For me, the answer is 1. Implied multiplication, the type of which appears in "xy+4" or "5(x^2)" acts as parenthesis. If the author intended the latter half to be in the numerator, he'da written 20/5*(2*2).

You must be outside the US, or you'd be writing PEMDAS, not BEDMAS. Regardless, notice that radicals and fraction bars aren't included in the acronym. Neither are implied functions. It's just the way it is.

DaveBfd
11-22-2014, 11:00 PM
Because for many places multiplication is performed before division. Are you outside the US?

Where would this be? That goes against PEMDAS/BEDMAS.

The OOO page on Wikipedia says that the only time where things would be interpreted outside of BEDMAS is in the physical sciences, which would support my interpretation (since this is not a physical science issue).

RE: the above post, 20/5(4*4) = 20/5*(2*2). It is the exact same thing.

I'd like to know why the physical sciences use a different approach though, if anyone knows.

Chessic Sense
11-22-2014, 11:09 PM
RE: the above post, 20/5(4*4) = 20/5*(2*2). It is the exact same thing.

I'm assuming you meant "20/5(2*2) = 20/5*(2*2)"

They aren't the same thing. The former is 20 divided by 5*4, which is 1. The latter is 20 divided by 5, times 4, which is 16. The difference is that the 4 is in the numerator in one and denominator in the other.

DaveBfd
11-22-2014, 11:21 PM
I don't think that your argument of the use of a * defining numerator or denominator is useful. No matter how you understand that interaction, you are multiplying 5 by 4.

How can you justify computing 5(4) first? If you do that first you are explicitly not using BEDMAS.

Senegoid
11-22-2014, 11:32 PM
I larnt me my basic arithmetic in grammar school, circa 1958-1963, in the United States. The rule was:
Evaluate any parenthesized group before combining that result with anything outside those parentheses.
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!
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.
(Exponents were introduced later, having greater precedence than multiplication and division, and associating from right-to-left, unlike + which associate from left-to-right.)

I have an Associate of Science degree majoring in Math. Nowhere in all that education did I ever learn otherwise. And in particular, nowhere have I ever heard that "Implied multiplication, the type of which appears in "xy+4" or "5(x^2)" acts as parenthesis"; these two particular examples given by Chessic Sense don't rely on any such rule anyway.

The original problem given by OP, to-wit:
20/5(2*2)=
is a weird way to write it. As given, I agree with OP's analysis, that the answer is 47 16. But, being weirdly written, it would be most clearly written as either:
(20/5)(2*2)
or
20/[5(2*2)]
according to the author's intention.

ETA: I'm getting some Spidey-tingly-like sense of deja vu here. Didn't we discuss exactly this same arithmetic problem in a thread once before?

DrCube
11-22-2014, 11:41 PM
This is another facebook-style notation question, in which the main confusion comes from using ASCII text to approximate real math notation. On a chalkboard, the parenthetical would unambiguously be either in the denominator or next to the fraction.

In plaintext, I would assume this equates to 1, and would have written it (20/5)(2*2) had I meant for it to simplify to 16. (I would use parentheses in either case to make it clear. The "5(2*2)" to me necessarily implies 5*4, NOT necessarily (20/5)*4, so I would definitely clarify.)

You should use parentheses when writing expressions in such a way that you cannot be misunderstood. That's what they are for. The fact that it is ambiguous in this case means the writer failed.

Senegoid
11-22-2014, 11:47 PM
Dr3 and I may disagree about how the given expression should be evaluated, but we do seem to agree that it isn't perfectly clear, and we agree on how it could be more clearly written.

Okay, I found some prior threads on the same topic (although the specific given problem is different in each):

62(1+2)=? http://boards.straightdope.com/sdmb/showthread.php?t=606537

6-1X0+2/2=? http://boards.straightdope.com/sdmb/showthread.php?t=664154

Pasta
11-22-2014, 11:49 PM
One can at best guess what the original writer of those symbols meant. Even though one can invoke a set of rules to come up with a definite answer, the elephant in the room is that the expression isn't written in a very natural way. There are lots of ways to write any expression, but fluency with the language of math (like with any language) carries with it some expectations. This expression is out of the norm enough that the only right answer is to take pause and ask for clarification from whoever is trying to communicate with that jumble of symbols.

An English analogy: If someone had a son and a daughter and no other children, they could say to you "Can you pick my son's sister up from soccer practice?" There's nothing wrong with that sentence, and grammar rules would suggest that he means his daughter, but it's such a non-standard way to say it that you would naturally ask for clarification.

There are multiple non-canonical aspects to the expression in the OP, and it would be odd to assume that the writer was strictly obeying a particular set of order-of-operations rules when he or she clearly ignored expectations for how that quantity should be written. (If it's to equal 16, then it should be something like "(20/5)*(2*2)" or maybe "(20/5)(2*2)". If it's to equal 1, then it should be something like "20/(5*2*2)" or "20/(5(2*2))". Just because you can make an interpretation of "20/5(2*2)" doesn't mean you should. It's just too far from a normal way to write it.)

DaveBfd
11-22-2014, 11:52 PM
I agree that in a perfect world this is a bad example of a problem. Thanks for finding older examples!

In plaintext, I would assume this equates to 1, and would have written it (20/5)(2*2) had I meant for it to simplify to 16. (I would use parentheses in either case to make it clear. The "5(2*2)" to me necessarily implies 5*4, NOT necessarily (20/5)*4, so I would definitely clarify.)

This seems odd to me. Why would you use parenthesis (20/5)(2*2) to denote the value of 16, when the parenthesis do not change the value if you were to follow BEDMAS? Of course 5(4) implies 5*4! BEDMAS still dictates that this results in 16 because 20/5 must be done before the 5*4. I would think that if you wanted to imply the answer was 1, you would add parenthesis to achieve that, rather than add parenthesis which do not change the equation as it is written.

You should use parentheses when writing expressions in such a way that you cannot be misunderstood. That's what they are for. The fact that it is ambiguous in this case means the writer failed.

Is the purpose of BEDMAS not to make it so that equations can not be misunderstood? BEDMAS applied properly ensures you always get the proper answer.

Senegoid
11-23-2014, 12:00 AM
On a related topic, there is sometimes some discussion about how to best write that expression that, in modern usage, is commonly written
sin2 x
In some textbook or other, I saw a quote from one of the major math sages (I forget who; maybe Leibniz?) that
sin x2
was a reasonable notation, on the grounds that the interpretation
sin (x2)
was unlikely ever to appear in any real-life application.

Seems like the mathematical purists won that argument, saying that sin x2 must be interpreted literally as sin (x2) whether that case ever really arises or not. Thus, the modern notation of sin2 x evolved to be the accepted notation, winning out over the equally correct but more cumbersome and thus less common (sin x)2

Senegoid
11-23-2014, 12:08 AM
I think we should all begin by thoroughly purging our minds, and the world's textbooks, of that abomination of a mnemonic PEDMAS, or BEDMAS, or BODMAS, or however you spell that. This acronym seems to say that we should do division before multiplication, and addition before subtraction.

That's just plain flat-out wrong. No wonder our children am confuzzed.

Nava
11-23-2014, 12:11 AM
Because for many places multiplication is performed before division. Are you outside the US?

In Spain it's "whatever is in the parenthesis before anything else", but with the formula as originally given we'd still run into that issue of "where is that parenthesis supposed to be". We also learned that it was better to pile up the parenthesis (using different symbols, preferably) than to leave a formula unclear. There was no precedence between multiplication and division, nor between addition and substraction. Since multiplication-division and addition-substraction are inverse commutative pairs (division is multiplication by the inverse, substraction is addition of a negative value), you should get the same result no matter which of the two you begin with - if you don't, you've made a mistake.

This same week I had to ask a Swedish customer to clarify one of his units, because as written it wasn't clear whether its dimentions included T-2 or T2. The solution came in the form of a parenthesis showing that the "*s2" was part of the denominator.

Chessic Sense
11-23-2014, 12:11 AM
You keep going on about BEDMAS. Why do you assume the author is using it, and why do you assume the reader is? Further, why do you assume that either of them knew the other was using it?

BEDMAS is not universal. In fact, I, for one, didn't even know it existed until a few years ago and I still think it's as strange as saying "Why do you think this?" or "I'm off to phone me mum."

So let me get this straight: In the expression "F/ma=1," you think the a is in the numerator? That is, you think it has a positive exponent?

Senegoid
11-23-2014, 12:19 AM
Meant to add this to my latest post above but missed edit window:

ETA: BTW, nowhere in all my years of math education (1st grade through lower-division college math with DiffEq) did I evah hear or see any variant of that mnemonic. And now that I have seen it, I'm glad I never did before now.

"Dolly Madison Says Compare" (or "Daughter, Mother, Sister, Brother; Divide, Multiply, Subtract, Bring Down") are okay with me (although each one seems to leave out a step that the other includes).

I'm also glad I never got taught SOH-CAH-TOA which I would have had as much trouble remembering as I would have had simply, you know, memorizing the actual definitions.

Nancarrow
11-23-2014, 12:20 AM
Is the purpose of BEDMAS not to make it so that equations can not be misunderstood?

Yes. Just like the purpose of the USSC is to resolve unambiguously and for all time the meaning, intent and application of the US constitution.

And just like the analogy, it performs decently a lot of the time, but there's always people setting out to bend it until it breaks. :)

Point is, you'll probably never get utterly perfect notation rules because the better they are, the more determined people will be to find a way to screw them up. For a more mathematical analogy, I suppose it's a bit like the 18-19th century attempts to rigorously define the idea of a 'function'. Every time someone thinks they've nailed it, along comes some smartass like Weierstrass (http://en.wikipedia.org/wiki/Weierstrass_function). In fact now that I think of it, mathematics must be one of the few professions in which trolling is not only tolerated but actually insisted upon.

Getting back to your problem, the best way imo to resolve the ambiguity is to use a horizontal fraction line (apparently a 'vinculum', as opposed to a 'solidus'... huh). Those are really good at making really clear what's on the top and what's on the bottom. Of course you can't use plain ASCII, have to go to LaTeX or something, but the takeaway lesson is that ASCII is not for maths.

DaveBfd
11-23-2014, 12:41 AM
You keep going on about BEDMAS. Why do you assume the author is using it, and why do you assume the reader is? Further, why do you assume that either of them knew the other was using it?

BEDMAS is not universal. In fact, I, for one, didn't even know it existed until a few years ago and I still think it's as strange as saying "Why do you think this?" or "I'm off to phone me mum."

So let me get this straight: In the expression "F/ma=1," you think the a is in the numerator? That is, you think it has a positive exponent?

I don't mean BEDMAS as in the actual acronym, I mean the proper OOO. The OOO is universal, is it not? I was only saying BEDMAS as a simple way to refer to order of operation. I would like to assume that all parties involved have an understanding of the order of operations. If they didn't, the whole conversation becomes meaningless.

As I noted above, physical sciences are the one field which specifically and intentionally go against the standard OOO in this way. If this were a physical science question rather than a simple math one, you may have a point in your f/ma comment. I still want to know why this is so.. Anyone?

Nava
11-23-2014, 12:43 AM
The OOO is universal, is it not?

No it's not, see my previous post. My math teachers would have a collective apoplexy if they encountered a student who thought you need to perform division before multiplication or addition before substraction.

DaveBfd
11-23-2014, 12:52 AM
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.

Derleth
11-23-2014, 01:10 AM
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.

DinoR
11-23-2014, 03:19 AM
I don't mean BEDMAS as in the actual acronym, I mean the proper OOO. The OOO is universal, is it not?

It's not.

From the wikipedia link earlier (thanks GamerHat):
Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally", with the first letter of each word creating the acronym PEMDAS. Canada uses BEDMAS, standing for Brackets, Exponents, Division, Multiplication, Addition, Subtraction. Most common in the UK and Australia[8] are BODMAS and BIDMAS.

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.

Senegoid
11-23-2014, 03:22 AM
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.

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!

naita
11-23-2014, 03:29 AM
For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,
http://en.wikipedia.org/wiki/Order_of_operations#Exceptions_to_the_standard

Personally I'd limit that to implied multiplication. I want 1/2a to be 1 over 2a, but I want 1/2*a to be 0.5*a. 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(2*2) entirely. If you mean 20/(5*2*2) you're saving one symbol that way, if you mean 20/5*2*2 you're using one extra just to make things ambiguous.

Jragon
11-23-2014, 05:02 AM
http://en.wikipedia.org/wiki/Order_of_operations#Exceptions_to_the_standard

Personally I'd limit that to implied multiplication. I want 1/2a to be 1 over 2a, but I want 1/2*a to be 0.5*a. 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(2*2) entirely. If you mean 20/(5*2*2) you're saving one symbol that way, if you mean 20/5*2*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

1
---
2x

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.

naita
11-23-2014, 05:18 AM
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.

The wikipedia article mentions this change explicitly. ;)

DaveBfd
11-23-2014, 08:24 AM
It's not.

From the wikipedia link earlier (thanks GamerHat):
[/B]

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.

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..

Pasta
11-23-2014, 11:12 AM
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.

Thudlow Boink
11-23-2014, 11:52 AM
I larnt me my basic arithmetic in grammar school, circa 1958-1963, in the United States. The rule was:
Evaluate any parenthesized group before combining that result with anything outside those parentheses.
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!
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.

I think we should all begin by thoroughly purging our minds, and the world's textbooks, of that abomination of a mnemonic PEDMAS, or BEDMAS, or BODMAS, or however you spell that. This acronym seems to say that we should do division before multiplication, and addition before subtraction.

That's just plain flat-out wrong. No wonder our children am confuzzed.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.

DaveBfd
11-23-2014, 12:21 PM
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.

Evidenced a few posts above.

Pasta
11-23-2014, 12:31 PM
DaveBfd, do you agree that there are notation conventions that the expression violates, independent of order-of-operations considerations?

Great Antibob
11-23-2014, 12:47 PM
BEDMAS is not universal. In fact, I, for one, didn't even know it existed until a few years ago and I still think it's as strange as saying "Why do you think this?" or "I'm off to phone me mum."

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:

Well, the Ancient Society of the No Homers has decreed it is actually 4 and a squiggle.

But seriously, it's both.

Sorry about the ambiguous answer, but that's the best we can do. The confusion comes from some textbooks and teachers teaching that if you see an implied multiplication, like 2(1+2), you should do that first. If that's the case, the answer is 1.

Other textbooks and teachers teach that you should do the division before the implied multiplication, as you would under a normal order of operations order. Then, the answer is 9.

Of course, if there's no agreement even among different teachers, that means there's no hard and fast rule.

The problem is the the order of operations is itself a convention. It's a way of codifying how most people approach arithmetic. In real life (even for mathematicians), we'd probably ask for clarification or curse the problem poser for giving us an ambiguous expression.

Most of us would use an extra set of parentheses to clarify the expression or use a rational expression with a numerator and denominator, instead.

We've been seeing this question submitted a lot on the "Ask Dr Math" site, and it's the 2nd round of it. A similar question was posed a couple weeks ago and also spread like wildfire. These things come in waves.

The way we've been explaining it is to use the phrase "American history teacher". When heard, is that a teacher of American history? Or a history teacher from America? It's an ambiguous expression with no established rule to resolve the ambiguity.

Actually, thinking on it some more, the "real" answer is whatever your teacher says it is. Doesn't change the ambiguity in the real world, but it does mean the difference between good and bad marks in class.

(bolding newly mine)

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.

(bolding newly mine)

Chessic Sense
11-23-2014, 01:32 PM
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.

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."

DaveBfd
11-23-2014, 02:13 PM
DaveBfd, do you agree that there are notation conventions that the expression violates, independent of order-of-operations considerations?

It depends. Obviously, bracketing 2*2 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 2*2, 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 5*2*2 as being the denominator, you must change the formula as it has been written.

Topologist
11-23-2014, 03:25 PM
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.

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 + x2, which is unambiguously (1 - x) + x2, not 1 - (x + x2). For multiplication and division, this convention simply doesn't exist among mathematicians.

Pasta
11-23-2014, 03:30 PM
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.
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.

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.
(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.

Topologist
11-23-2014, 04:19 PM
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).

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.

Jragon
11-23-2014, 04:24 PM
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.

Topologist
11-23-2014, 04:34 PM
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.

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