Yurp! :D

There's no lack of one, just misunderstanding in this thread. Math is hard facts, not multiple answers.

What canwll said, basically the problem wouldn't involve multiplying the denominator unless there were some sort of parenthesis to contain the 3 in the denominator and not the the numerator. Otherwise, the 3 is looked at as a whole separate number. As canwll showed, it's 3, not 1/3.

(6/2)*(1+2)=9 or (6/2)*3=9...........18/2=9.............9=9

6/(2*(1+2))=1 or (6/2)*(1/(1+2))=1.........6/(2*3)=1...........6/6=1

There's only one answer and it's 9. Without the parenthesis, at least.

ahahahahahahaa... Man, that's a good one.

Ever heard of Heisenberg's Uncertainy Principle? That you cannot know both the location and velocity of an atomic particle at the same time? Or what about the "Schodinger's Cat" paradox, which states that the cat is both alive and dead at the same time? It can't be both, because maths is hard facts, not multiple answers, right?

Even moving away from physics and into pure maths, x = 5 mod 6. What's x? Well, it's 5. It's also 11, and 16, and 365. In fact, it's 6n + 5, but without any information about n, you can't get a more accurate answer.

That's what's going on here. This is EXACTLY like a function with multiple possible values. Take a quadratic equation, like x^2 + 2x -3 = 0. "x" is both 3 and -1, right? You can't get one specific answer until you are given more information, like x>0. With this lovely equation, 6/2*(1+2) is both 1 and 9 unless more information about the order in which the multiplication and division should be done is given. Once the extra parentheses are added, the answer would be clear.

If anyone responds to this post with "but but but we were taught in school that you do PEMDAS" or "We were taught in school that if you have multiplication and divison in the same equation you just work from left to right", I am just going to ignore you, or assume that you're just completely incapable of reading seeing as I've explained this 5 times before.

THERE IS NO STANDARD CONVENTION FOR THE ORDER OF EVALUATION OF OPERATIONS. THEY ARE MADE UP IN ORDER TO MAKE IT EASIER TO TEACH THE SUBJECT. It just so happens that most of you tend to have been taught one of two common methods. This thread is all the evidence anyone should need to prove this is the case. If you want more, there's currently a facebook poll with about 2 MILLION votes, and about 200k difference between the two.

If anyone wants to argue this, I'm happy to do so, so long as you have an argument other than "look at these links" or "this is what my maths teacher said". I taught maths for 6 months, and the amount of white lies I had to say in order to avoid topics that were confusing the students was slightly soul-destroying. If anyone wants to disagree with me on the principle that you feel that you are right and I am wrong just because, please go and study maths - specifically Group Theory and Real Analysis - for a few years, and then get back to me.

/end rant

I wasn't trying to offend you. When I said multiple answers, I didn't mean a problem can't have two solutions. I meant that you can't go about problems like the one this thread is about in two different ways and get two wildly different answers, because one is technically incorrect. What I mean is, you can't pick and choose how you solve a problem and then call it the correct answer. There are certain guidelines to follow by, otherwise it is incorrect.

Sorry for the misunderstanding, but I still stand by 9 being the only answer. Even if you do multiplication first, this thread has proven the answer is still 9. It's not a matter of how you do the order of operations, it's a matter of multiplying correctly.

And the parenthesis or lack of makes no difference, as the problem stands (and with the parenthesis it currently has), the solution reads as 9. Thats why if you copy & paste it directly into Google it'll give you 9.

It only works out as 9 doing the multiplication first if you assume that there are parentheses around the (6/2).

In order to get any one, unique answer, you have to add one set of parentheses:

(6/2)*(1+2) = 9
6/(2*(1+2)) = 1

There are your two choices. I will add that Google gives the 9 answer because calculators, indeed all computer programs, are designed to evaluate from left to right in the absence of proper syntax, similar to how you can figure out what someone is saying even if they have atrocious spelling. Doesn't make their spelling any more correct, though. I will also point out that if you put the equation into Google, it adds a pair of parentheses in the position it is assuming you mean due to your incorrect syntax:

http://i.imgur.com/pXNeO.png

if you break the equation down into a, b and c, to represent the parts which there is argument about, it's a lot easier to see what's going on. In the following, a=6, b=2, and c=(1+2):

a/b*c = ... ?

Now, one interesting thing worth noting is that Canwllcorfe, among others, has chosen to see this as (a*c)/b, which is interesting as it totally changes the order. It is however equivalent to (a/b)*c, which is one solution to the problem, so it's fine as an answer, but it just goes to show that "left to right" isn't always even correct, as you can manipulate such an ambiguous statement to have the inverse order of operations.

Your choice, however, boils down to (a/b)*c or a/(b*c). If you substitute back in the values, you will (hopefully) note that this equals 9 and 1 respectively.

Do you still feel that the "parentheses or lack thereof makes no difference"?

I just add up the number of empties in my bin the following morning to deduce whether I have had a good night or not.

Yes, because the parenthesis you showed that Google had on them are unneccessary. They only make it easier for people to see how the process is. Lack of parentheses implies that it's a 6/2 fraction.

That's what I was thinking. I mean, I know the inclusion of another pair of parentheses would certainly alleviate any real confusion, but I don't think they're necessary. I certainly didn't look at the problem and read it as 6/(2*(1+2)), because those parentheses aren't there. So I figure, by default, it should be (6/2)*(1+3). It also seems, to me anyway, more logical.

EDIT: But do note that I'm just explaining my mindset. You could be completely right Moonlit (in that, ironically, there is no definite answer).

I do agree that maths are not cold, hard facts that never change and are designed to only ever have specific answers. What a lot of people may be forgetting is that convention is set to arrive at a particular point consistently, to achieve a particular goal. If you desire to arrive at another point consistently, the convention must be altered, but it doesn't invalidate the outcome. Maths serves a purpose based on your application of it.
I say all that to say, while I'm not educated very much in this department, it seems logical to me that what you're saying is correct.
(I could be off, but I was just trying to verbalize my general take on what you ultimately meant.)

I had no idea this would turn into a huge discussion. This is madness.

One thing about Mathematics is that it is always consistent, 2+2=4. The fault is not in the numbers or not necessarily the way the problem is written, nor is it the case that it is possible the equation is able to yields two different results. The problem is how one interprets the equation. "Schrödinger's Cat" is only a paradox when the box is closed. Once it is open then the answer is quite clear. Same with the equation, it is only a paradox when someone unaware of proceed with the function of the equation. The conventions in math are not arbitrary and put in place so it's possible the equation can yield two different answers. Once some one understand the rules involved in Algebra there isn't a paradox. If you open the box even Schrödinger's cat will say 6/2*(1+2)=9

When one takes the problem 6/2*(1+2)
6/2 could be seen as a fraction (as six halves) or the equation 6÷2 in either case the parentheses are not the problem, both will equal "9." If one take into consideration it is a fraction, it only yield "1" when one considers 2*(1+2) the denominator as a coefficient of (1+2), and one ignores the numerator as also a coefficient of (1+2). And it one considers 6/2 as 6÷2 then the problem 6÷2(1+2) or 6÷2×3, it will only yield "1" when one does not follow convention and doesn't do operations from the left to the right. One the about Math is that one can tackle a problem six ways to Sunday and still end up with the correct answer provided if one follows proper procedure and convention. If you take consistency and convention out of Math, and make things in it relative you will have 2+2=5.

Neapolitan, that is impressive. You have again quoted one of my posts, completely failed to read it, and restated your argument.
Mathematics is not always consistent. If it is defined properly and is being used within the bounds that have been shown to be consistent, then yes, it is consistent, but it is not magically always right.


2+2=4 for a number of reasons, but the easiest axiom to take that proves it is that 2 is defined as 1+1, and 4 as 1+1+1+1, so 2+2 = (1+1)+(1+1) = 4. That's an axiom of addition. However, nowhere in the axioms of addition Or multiplication does it say "Multiplication and Division are of equal precedence and so if there is ambiguity do them from left to right to find the proper answer". The simple fact that so many people disagree with this is ample proof by contradiction that there is no one standard convention. How hard is this to see?



What you stated about Schrodinger's Cat means either you misunderstood the point I was making, or you are claiming that the Cat will always be dead or always be alive when we "open the box". i.e, when parentheses are added (giving the necessary information to find a unique answer) the answer will always be 9. Which is, as I have proven multiple times in this thread, completely wrong.

Ok, first of all, there is no "fraction" in this equation. The simple act of taking 6/2 to mean the fraction "6 over 2" is the first mistake you're making there. Unless, of course, you want to argue that the convention of using / to represent ÷ is non-standard? Because I'd hate to think that there might be ambiguity in mathematical convention, wouldn't you? Secondly "Both will equal 9"? Again, you're blindly refusing to even consider that there might be a different convention on how this can be solved, and just assuming that your way is right. Of course if you use your convention, no matter what way you solve it you're going to get 9, that's part of the point I'm making. If you step outside your beloved convention just for a second, you might see what I'm talking about.

[QUOTE]If one take into consideration it is a fraction, it only yield "1" when one considers 2*(1+2) the denominator as a coefficient of (1+2), and one ignores the numerator as also a coefficient of (1+2). [QUOTE]

I actually have no idea what you're saying here. I'll happily address it if you want to explain what you mean, but I would imagine that it doesn't really affect the overall verity of the post.

I've bolded the important part in that statement. I must say, congratulations on your impressive level of blind faith in convention. Even when someone states and proves that it might not be standard and infallible, you don't for a second even think about trying to show that it's right. You just keep on using it. What is your justification for assuming that your convention is the right one?

Can you prove your convention is correct from Base Principles? If you can't, then it's not a mathematical constant, it is simply a convention, and a non-standard one at that. A common one, yes, but nowhere near universal. "One thing" about Maths is that when it is taught at 1st and 2nd level, a lot of shortcuts are taken in order to make it easier to teach. "Rules" are given so that the concept of ambiguity can be avoided, because maths is difficult enough when people are sure that there is one right answer.

Your final statement about 2+2=5... well, I've already addressed that point. Until you can show me an axiom which states the order of operators (even only defined on the Field of Real Numbers), that argument is invalid.


@Tuna: It does? Really? That's the first time I've heard that particular convention. Can you show me where it states that that's the case, in the axioms of multiplication?

I would like to remind anyone still arguing this that I'm not saying that your answer is wrong. By all means, it's the most popular way of solving the equation. It's just not Syntactically correct, which means that even though the most common convention states that the first pair of parentheses can be assumed around (6/2), it is an assumption, and so long as they are not there, you can claim that the = 1 answer is just as correct.

@Freebase: That's part of what I'm saying, certainly. It's more that nothing in maths can be taken to be certain unless it is defined that way, given as an axiom (a statement so obvious that it cannot be proven), or provable from the definitions and axioms. Nowhere in any of the axioms that I have been taught regarding the Real Numbers (the number line, essentially) is there an axiom or a definition which states the universal order of priority of operations. There is a standard, which is almost universal, but as is evident from this thread, not entirely so. That's what my point boils down to, that a convention that is not a definition, not universal and not provable from the axioms is being taken as irrefutably true.

@CanwllCorfe: I understand your belief, but I'm afraid that it's as a result of the way maths is taught: Certainly among at least 50% of the population of at least the western world, you would be safe in making such assumptions, but mathematics is not a subject where you can say "because it holds for the majority, it holds for all". Indeed, there are still some theorems out there that remain unproven even though we know that it holds for every case that we can possibly calculate, just because we can't prove it for a general case. Does that make sense?

When you google "order of operations" it doesn't seem like there's much disagreement at all. Every link I click on seems to say that multiplication and division are of equal rank, as are addition and subtraction.

And generally, that's fine, as multiplication and addition are associative operations (it doesn't matter what order you do them in). It's where there's ambiguity over which elements the inverse is being applied to that there's a problem.

Consider a situation where you have a/b/c. Is that (a/b)/c, or a/(b/c)? take an example, 2/3/4. (2/3)/4 = 2/12 = 1/6, but 2/(3/4) = 8/3. The two answers are very, very different, but how do you define which one is the fraction? Which equal priority operation needs to be done first? The same problem holds when you have a/b*c. Is that (a/b)*c, or a/(b*c)? The fact that most people assume it is (a/b)*c does not make the syntax any more correct: If someone writes "I hae that" you could assume they left out a t or a v; depending on which it is they've left out, it makes a big difference to the meaning of the sentence, but you can usually figure out which they mean by the context of the sentence. It doesn't however, make the sentence, if isolated from the context, any less ambiguous. Does that make sense?

The answer is 9, because Matlab said so.

Wolfram Alpha would be a more acceptable God, but I guess beggars can't be choosers :P However, I refer to the "why google gives 9" argument regarding choosing the most common convention to solve the ambiguity. Calculators are programmed to add parentheses on a left to right basis if the user has not entered a sufficiently unique statement.

It's 9 because I have done a partial maths degree. :laughing:

...you guys are just deliberately trying to annoy me now, aren't you :P

Matlab is pretty powerful though, I think it can be trusted.

Why assume they left out only one alphabet?

It doesn't change any of my arguments though. Even the simplest calculator can be programmed to allow for bad syntax.

@chiron: I would assume that they were posting in the language of the rest of the sentence :P

Are you saying that Matlab is programmed to allow for bad syntax?

At any rate, the problem question is underspecified. Therefore, to solve this problem we would have to further specify it. Generally, the simplest solution is always best, as given by Occam's razor. Hence,

For 6/2*(1+2),

It is simpler to say (6/2)*(1+2)

Instead of: 6/(2*(1+2)).

... Which is essentially what I've been saying since the first page. While the most common answer is 9, it is ambiguous without more parentheses. Occam's Razor is an interesting argument in favour of 9, though, but I'm glad you feel the original problem is underspecified!

oojay has a good wiki link and comes up with the better answer, in my opinion, although his method relies on a mnemonic which shouldn`t be thoughtlessly applied to every case.

I think MoonlitSunshine has the clearest analysis so far.
I can`t remember what I was taught (in England) about precedence and order of operations, but we were taught to use parenthesis to avoid ambiguity.
So, if I had wanted the answer to be 9, I would have written : (6/2)(1+2) = 3 x 3 = 9
and if I had wanted the answer to be 1, I would have written : 6/2(1+3) = 6/(2 x 3) = 1

So,it`s an interesting question, EvilChuck, but my answer to the guy would be," Go back and write what you mean properly."

I think this is the best example of the two possible solutions yet.

I haven't read all of the posts but I think something that people are forgetting with PEMDAS is that

Multiplication and Division get the same priority

So does Addition and Subtraction.

You always do the order of operations from left to right.

Was this a reply to me? Because if so it doesn't really seem to relate to what I was saying.

It was. I was showing why "equal priority/rank" is a minefield.

Why is it a mine field? When they're equal rank you just move from left to right. Seems easy enough.

Because it's all very well to say that, but there's absolutely no guarantee that the person you're communicating with feels the same way... as is evident from this thread.

That's incorrect. The answer to that is still 9. The way you wrote it hides the multiplication sign making it appear as if you'd multiply the denominator by (1+3). But in actuality, you still multiply the 6 by (1+3)

It looks easier if you actually keep the multiplication sign in: 6/2*(1+2) and 6/2(1+2) are the same thing. And the answer to both is 9.

So you want people to just throw out well established math principles because they "feel" like it? Bottom line, you do it the way oojay did it and you wouldn't be taken seriously in the math community, you'd get the question wrong on a test, etc. Because the math community abides by some universal guidelines where parentheses are assumed, or not assumed depending on the situation. And it's a matter of learning when they're necessary

Solving it oojay's way is the equivalent of putting a period in the middle of a sentence; no one's stopping you from doing it. But the rules that govern basic math that were established to keep a standard would say it's incorrect. You don't get to choose how a problem's solved if it breaks the rules of math laid down just because you feel like it. Unless you want the wrong answer, of course.

Do you want to reference a paper or some sort of quote from "The mathematical community" to back up these wild, wild statements?

But if there's an established way it's handled, as there appears to be, what does it matter how someone feels about it?

Because the "established way it's handled" isn't universally established. They've been doing polls on this recently, the answer given isn't universally equal. As I was saying earlier, you might as well say "Well, people know what I mean if I spell words incorrectly, so there's no point in correcting my spelling".

Well, no one answer my question from earlier in the thread which was...

If we can agree that there isn't an inherent way to complete these problems and that the Order of Operations (OoO) is something to communicate more effectively as a globe, then why such flimsy guidelines sitting within the law?

It isn't logical to have order, where in there is disorder. Math is often seen as hard truths to achieve answers. But most of what we learn in math is philosophy. Its filled with theory and abstract constructs which we represent with numbers but you're not always told what you're multiplying or finding the radical root of.

All Oojay did, and myself since I got 1 also, was assign logical law to an area thats illogical. Why are certain ranks considered equal if there isn't an inherent way to find an answer, and we're making one up? If we're using PEMDAS, then it will determine rank where the system has failed.

I don't think that's what EvilChuck is looking for, regardless of if it's being challenged. I think he just wanted the answer that's taught in school and practiced in practical life.

And it seems pretty established to me, that's why Google calculator knew what I meant even when I didn't add the parentheses. Just because people are doing the problem incorrectly doesn't mean we should accept that as an alternative answer.

Candidate for most boring thread in the history of MusicBanter?

I'm not actually looking for the answer, as I said I'm certain the answer is 9. I just thought it was interesting that despite what should be simple mathematics it had spawned (and has again) a big debate, over what oojay said on the first page or so is '6th grade maths'.

I was taught BODMAS at school, others were taught PEMDAS. This is proof enough imo that the multiplication and division are interchangable, and all thats left is to say you should work from right to left and whatever comes forst is what you do first.

*yawn