48÷2(9+3) = ?

[Malcolm Wright]

We're saying the same thing.

Or, I agree with you at least.

Ah cool, hehe - the internet is weird. I bet if we had been in the same room, we would have immediately understood we thought the same way.

[Ratbag]

Sooo to sum up.... in a case where there is no other reason to place one particular part of a sum before the other there is actually a bona fide rule that says left to right.

And we all agree it'd be simpler to add extra parenthisis to such a sum in any case where the person reading it may become confused (ie your average Joe) to save a HEAP of time.

Unless of course your trying to trick someone into getting it wrong and not indeed just trying to work out how many tanks you need to win a war

?

(heck is this the first board in all cyberspace to agree on the answer?)

[jwreck]

Sooo to sum up.... in a case where there is no other reason to place one particular part of a sum before the other there is actually a bona fide rule that says left to right.

And we all agree it'd be simpler to add extra parenthisis to such a sum in any case where the person reading it may become confused (ie your average Joe) to save a HEAP of time.

Unless of course your trying to trick someone into getting it wrong and not indeed just trying to work out how many tanks you need to win a war

?

(heck is this the first board in all cyberspace to agree on the answer?)

[Chachma v'Oz]

You've failed to address half of my post.

It is far from absurd to tweak a system. It is not working perfectly well, as evidenced by the confusion that exists even amongst educated people.
Face it, it would be much clearer to tell people to always use parentheses in these cases. Problem solved.

M.

The system works perfectly well. Not understanding the rules of the system causes wrong answers. There is only one correct answer to the equation as written. One can change the equation, and the answer, by adding parenthesis. That doesn't clarify it; it changes it into something else.

[caddis]

OK - But that likely requires a half hour or more of effort.

Tell you what. I will do it if you agree to eat a guppie or the equivalent when I demand it in compensation.

How is that for a fair deal?

Whatever. I simply have NEVER heard of an official source telling you that you must perform multiplication before division. I personally think you and others have assumed it to be true because of the acronym PEMDAS.

To throw Malcolm a bone, that acronym is what needs to be "tweaked" because it is what is responsible for the "controversy"

[Chachma v'Oz]

Whatever. I simply have NEVER heard of an official source telling you that you must perform multiplication before division. I personally think you and others have assumed it to be true because of the acronym PEMDAS.

To throw Malcolm a bone, that acronym is what needs to be "tweaked" because it is what is responsible for the "controversy"

PEMDAS may be what is misunderstood. It's P - E - MD - AS. P comes first. Then E. Then M and D in either order. Then A and S in either order. Multiplication and division (like addition and subtraction) are at the same level and are not ordered within that level. It doesn't matter which is done first within the same level.

The system works fine. People who don't understand how to use the system come up with wrong answers.

Thus, the original equation is not ambiguous. It has only one correct answer - 288. To change the answer to 2 you have to introduce parentheses that pull one of the terms from the third level and group it in the first level. The absence of that modification means that 2 is not the answer to the equation.

[jwreck]

Whatever. I simply have NEVER heard of an official source telling you that you must perform multiplication before division. I personally think you and others have assumed it to be true because of the acronym PEMDAS.

To throw Malcolm a bone, that acronym is what needs to be "tweaked" because it is what is responsible for the "controversy"

Well I will swear to all things Holy that I was taught in school to multiply first. Granted, it wouldn't be the only thing that a teacher ever got wrong in the course of my education, but that's what I was taught.

[Chachma v'Oz]

Well I will swear to all things Holy that I was taught in school to multiply first. Granted, it wouldn't be the only thing that a teacher ever got wrong in the course of my education, but that's what I was taught.

I believe it. For years it was taught (and maybe still is) that one should never use a preposition to end a sentence with. There's no basis for that rule. English grammarians don't have such a rule, but many schoolteachers thought it was carved in stone.

The focus of education should be less on what to think and more on how to think (and research, and analyze). We'd have fewer numbskulls cluttering our world if we'd concentrate on what's important to teach about life.

[Lulu]

Well I will swear to all things Holy that I was taught in school to multiply first. Granted, it wouldn't be the only thing that a teacher ever got wrong in the course of my education, but that's what I was taught.

Same here. I am 99% positive I was taught multiplication before division. And I was in California so it isn't a Texas thing. I also think I was taught that the multiplication next to parentheses (as in this wonderful example) is done first before anything else. I even had a financial management professor who solved a problem like this and did the 2(xxx) part before the other parts of the formula. He added parentheses so maybe he caught the mistake as he was solving it, but it gave me the impression that it was assumed you do that first. Thank god I don't need math in my work is all I gotta say.

[caddis]

Well I will swear to all things Holy that I was taught in school to multiply first. Granted, it wouldn't be the only thing that a teacher ever got wrong in the course of my education, but that's what I was taught.

I don't doubt you were taught it in elementary school because those teachers are teaching multiple subject....Jacks of all trades and masters of none.

[Xavier Tremely]

When programming, I have a tendency to over-use parentheses for the sake of certainty over pitfalls such as these.

[olecapt]

Whatever. I simply have NEVER heard of an official source telling you that you must perform multiplication before division. I personally think you and others have assumed it to be true because of the acronym PEMDAS.

To throw Malcolm a bone, that acronym is what needs to be "tweaked" because it is what is responsible for the "controversy"

Try this which you may find meets your requirements in its references. It is also an interesting read if you care about this junk.

http://mathforum.org/library/drmath/view/57021.html

[Ratbag]

So scratch the definitive conclusion and redefine that no-one ACTUALLY knows.... man that stinks, can't beleive that there's no succinct answer. Maybe one will arise from all this Hoohah eventually, I for one will be teaching my kids there is no right answer when we get to it, and be sure to teach them tha value of brackets, only another 2 and a half years till we get to this stuff.

Hang on... does that mean I was actually completely correct in the first post?

[Chachma v'Oz]

So scratch the definitive conclusion and redefine that no-one ACTUALLY knows.... man that stinks, can't beleive that there's no succinct answer. Maybe one will arise from all this Hoohah eventually, I for one will be teaching my kids there is no right answer when we get to it, and be sure to teach them tha value of brackets, only another 2 and a half years till we get to this stuff.

Hang on... does that mean I was actually completely correct in the first post?

Yeah. The problem is with the "math"...

[olecapt]

So scratch the definitive conclusion and redefine that no-one ACTUALLY knows.... man that stinks, can't beleive that there's no succinct answer. Maybe one will arise from all this Hoohah eventually, I for one will be teaching my kids there is no right answer when we get to it, and be sure to teach them tha value of brackets, only another 2 and a half years till we get to this stuff.

Hang on... does that mean I was actually completely correct in the first post?

I am not sure it needs to be solved. Just a notice to those who know with certainty that science ain't really that way.

You had the question down very well. But no you did not understand what the answer was.

Than again I am a mere Noob while you are a Sr. Member and others have truly exalted positions. So perhaps I go too far and presume too much.

[Malcolm Wright]

The system works perfectly well. Not understanding the rules of the system causes wrong answers. There is only one correct answer to the equation as written. One can change the equation, and the answer, by adding parenthesis. That doesn't clarify it; it changes it into something else.

As previously mentioned the system works perfectly well for those who know all the rules and have no confusion about them.
As previously mentioned, that group seems to be the minority.

As previously concluded, this shows the system does not work perfectly well, for most people.

Using parentheses can be done in different ways to solicit both answers - as such, it is a way to clarify, rather than to modify, considering the aforementioned situation.

Please recall we are not only considering North America (which already exhibits much confusion on the subject). Mathematics is supposed to be a universal language, and as such, the rules must be universal. I was partly schooled in the french system, and there is no left to right rule there. I was not taught it, and the french wiki page on the order of arithmetic operations does not list it:

Les règles de priorité sont :

les calculs contenus entre parenthèses (ou crochets) sont prioritaires sur les calculs situés en dehors de ces parenthèses. La barre d'une fraction ou d'une racine carrée joue le rôle d'une parenthèse;
les exposants sont prioritaires sur les multiplications, divisions, additions et soustractions;
les multiplications et divisions sont prioritaires sur les additions et soustractions.

The result hinges entirely on whether, after computing the contents of the parentheses, you either first perform the multiplication or the division. Since they are of equal priority, and since some countries have no rule for left to right or right to left sequencing of operations, and since even the countries that allegedly do can't get even half of their population to learn that rule - the system is well and truly useless.

Hence: use parentheses to return us to the universality that is required of the discipline.

M.

[Chachma v'Oz]

As previously mentioned the system works perfectly well for those who know all the rules and have no confusion about them.
As previously mentioned, that group seems to be the minority.

As previously concluded, this shows the system does not work perfectly well, for most people.

Not knowing how to solve a problem does not imply that the problem is ambiguous. It implies that the student lacks the tools to solve the problem. That is not the fault of the system that provides the correct answer.

Mathematics is supposed to be a universal language, and as such, the rules must be universal.

It is, and they are.

I was partly schooled in the french system, and there is no left to right rule there. I was not taught it, and the french wiki page on the order of arithmetic operations does not list it.

Nor was I. There is proper order for operations, but I'd never heard of a "left to right" rule before reading this thread.

The result hinges entirely on whether, after computing the contents of the parentheses, you either first perform the multiplication or the division. Since they are of equal priority, and since some countries have no rule for left to right or right to left sequencing of operations, and since even the countries that allegedly do can't get even half of their population to learn that rule - the system is well and truly useless.

Both your premise and your conclusion are wrong. 48 ÷ 2 X 12 is the same as 12 X 48 ÷ 2, and the system that utilizes that fact to always provide the same, correct answer is not useless.

Hence: use parentheses to return us to the universality that is required of the discipline.

The system also requires elegance of expression and the parentheses you would like to see used are mathematically redundant. The only change in the expression that has a sound reason is replacing the ÷ symbol with a /. I haven't seen ÷ used as an arithmetic symbol except in the most casual of ways since I was in grammar school, and that was a long time ago.

There is only one correct answer to that equation and making a procedural mistake in how it is calculated may produce an incorrect answer. One way of making a procedural mistake is ignorance of the proper order to deal with the terms in the equation. It is not an error in either the equation nor in the system that is universally used to calculate the correct answer if one arrives at an incorrect answer.

There is nothing ambiguous about the equation and there is nothing wrong with the system used to solve it. What is wrong is misapplication of the system.

[Malcolm Wright]

Nor was I. There is proper order for operations, but I'd never heard of a "left to right" rule before reading this thread. Both your premise and your conclusion are wrong. 48 ÷ 2 X 12 is the same as 12 X 48 ÷ 2, and the system that utilizes that fact to always provide the same, correct answer is not useless.

I'm not sure why you wanted to shift around the terms - that does not illustrate our quandary.
The two expressions that illustrate our quandary are:

(48 / 2) x 12
or
48 / (2 x 12)

Without using parentheses ( 48 / 2 x 12 ), and without a left to right rule, how do you propose that there is only one correct answer, and how can the expression not be ambiguous?

I do not accept your elegance prerogative: there is nothing inelegant in the use of parentheses in this instance, and as I see it, they are necessary to avoid confusion. To demonstrate that they are not, you would have to demonstrate that people are not confused! They are.

M.

[Chachma v'Oz]

Use of parentheses in the manner you describe to (also) arrive at the correct answer is redundant, thus reducing the expression's elegance. They are redundant because, if you follow the rules of arithmetic, they are unnecessary to arrive at the correct answer. Ignorance of the rules does not result in ambiguity of expression but can lead to an incorrect conclusion by the student.

It's no different than someone speaking to you in Japanese and you being not sure what exactly was meant. It could mean this, or it could mean that, the way you see it. If that occurs, you don't direct the speaker to rephrase it because "the way you stated it, it's ambiguous". It wasn't ambiguous at all to someone who understands the language.

You learn the rule he was using to mean one and only one thing. He isn't obliged to alter the rule or change his expression for your benefit. . He wasn't ambiguous. His statement wasn't ambiguous. You were just unsure how it should be dealt with to extract the one correct meaning because of inexperience with the language. That's on you, the novice, not on the speaker properly expressing his thought.

[Malcolm Wright]

Use of parentheses in the manner you describe to (also) arrive at the correct answer is redundant, thus reducing the expression's elegance. They are redundant because, if you follow the rules of arithmetic, they are unnecessary to arrive at the correct answer. Ignorance of the rules does not result in ambiguity of expression but can lead to an incorrect conclusion by the student.

Which rule of arithmetic makes 48 / 2 x 12 unambiguous?

It's no different than someone speaking to you in Japanese and you being not sure what exactly was meant. It could mean this, or it could mean that, the way you see it. If that occurs, you don't direct the speaker to rephrase it because "the way you stated it, it's ambiguous". It wasn't ambiguous at all to someone who understands the language.

That's a terrific example actually. Japanese is ambiguous in a number of ways, and don't take my word for it. Take the word of native speakers who readily confess to it. For instance it is common place to omit using a pronoun, causing Japanese people to have to inquire to make sure who is being talked about - me, you, him? And yet omitting use of the pronoun is not only correct, it is in many ways more polite.

Your stance presupposes that the system is perfect. It often isn't, and considering the amount of confusion surrounding something so simple as the order of arithmetic operations, it stands to reason that the system could be improved. Do you dispute that?

Do you really think that the benefits of using parentheses explicitly (removing confusion for perhaps half of the population), would be outweighed by what you perceive as a loss of elegance?

I don't really care about this in and of itself, but it is symbolic of more important things, and I fear your position springs from a sort of conservatism that holds us back from making common sense improvements to adapt to a changing world. As mentioned before, I believe the reason why I and many others want to compute 2 x 12 and then divide 48 by the result of 24 is because of the order of syntactic evaluation in such popular scripting languages as Python. The mind is evolving and current circumstances are actually pushing ambiguity towards incorrectness. A perfectly ambiguous situation does not favor either answer, but here we have circumstances in our environment that are causing people to favor the WRONG answer, which is worse than a merely ambiguous situation.

M.