48÷2(9+3) = ?

[Chachma v'Oz]

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

The commutative propery.

The terms are 48, 2 and 12. The operators are / and X. Multiplication and division are commutative, meaning they are not ordered. You can divide 48 by 2 and then multiply by 12, or you can multiply 12 times 48 and divide by by 2. It doesn't matter. Only one answer is produced. You can produce a different answer only by adding parentheses to group the 2 and the 12. That is not the expression given us. The one given us doesn't have parentheses because it is a different expression, presented elegantly without redundant notation. There is only one way it can be properly read.

I'm not denying that adding parentheses as you suggest wouldn't make it clearer to more people, I'm saying that the expression is not ambiguous without them because, using the rules and properties of arithmetic, they aren't necessary. The expression contains all the notation that is necessary and sufficient (which pretty much defines elegance).

Mathematics is pure. Society is not. Mathematical properties do not correspond one-on-one with those in social constructs. There are lots of social and economic theories that work elegantly on paper but never for long when attempts are made to implement them in human society.

The theories are pure. Human behavior is not. I have no problem compartmentalizing my knowledge of mathematical properties separately from my experience with humankind.

[Malcolm Wright]

The commutative propery.

The terms are 48, 2 and 12. The operators are / and X. Multiplication and division are commutative, meaning they are not ordered. You can divide 48 by 2 and then multiply by 12, or you can multiply 12 times 48 and divide by by 2. It doesn't matter. Only one answer is produced. You can produce a different answer only by adding parentheses to group the 2 and the 12. That is not the expression given us. The one given us doesn't have parentheses because it is a different expression, presented elegantly without redundant notation. There is only one way it can be properly read.

Well it seems to me that rather than the commutative property, you are relying on a left to right rule to make your determination, even if unwittingly since you say you too were unaware of any such rule.

Without using parentheses, the only thing you can use to rationalize why you don't group the 2 and the 12 is a left to right direction.

So if we can agree that in fact it is the left to right rule that makes this expression unambiguous, then you have to contend with the fact that said rule is not taught everywhere in America, and seems not to be taught at all in certain other countries - which leaves us with a situation of true ambiguity. This should not come as a big surprise: the controversy is probably raging for a reason.

As a side note, when written as:

48 / 2 x 12

... the left to right order is so intuitive that you would be forgiven for overlooking the need for it to be codified in a rule.
However when written as:

48 / 2(9+3)

It is no longer intuitive at all because we start from the right, evaluating the parentheses first, then have a natural tendency to want to multiply the contents of the parentheses - a tendency exacerbated for people who use programming or scripting languages.

Anyway, that is really a side note because we both agree that we should not be relying on intuition for such things - there needs to be clear cut and unambiguously defined, coherent rules. So either there is a left to right rule or there is not. If there is not, then you have not provided a reason for why 48 / 2 x 12 is unambiguous.

M.

[garyd]

Except that's not the way it was written. Parentheses take precedent.And a number adjacent to a parentheses becomes part and parcel of the same item in the equation. 2(9+3) is one term. Had it been written 2 X (9+3) You wpould be correct.

[Malcolm Wright]

Except that's not the way it was written. Parentheses take precedent.And a number adjacent to a parentheses becomes part and parcel of the same item in the equation. 2(9+3) is one term. Had it been written 2 X (9+3) You wpould be correct.

But that's exactly how I read it too Gary - and it leads to a result of 2.
Chach says the correct answer is 188 and we have started discussing the expression after the parentheses are resolved to isolate the bone of contention, which is the order in which the equally weighted priority of multiplication and division occur.

M.

[Chachma v'Oz]

Well it seems to me that rather than the commutative property, you are relying on a left to right rule to make your determination, even if unwittingly since you say you too were unaware of any such rule.

No. 12 X 48 divided by 2 is not left to right.

Without using parentheses, the only thing you can use to rationalize why you don't group the 2 and the 12 is a left to right direction.

No. Term A operator 1 term B operator 2 term 3 = term 3 operator 2 term 2 operator 1 term 1. Directionality is order and order is immaterial with these operators.

So if we can agree that in fact it is the left to right rule that makes this expression unambiguous,..

Stop right there. That premise is false. And there is no raging controversy, only disagreement based on ignorance of the properties of arithmetiic.

As a side note, when written as:

48 / 2 x 12

... the left to right order is so intuitive that you would be forgiven for overlooking the need for it to be codified in a rule.
However when written as:

48 / 2(9+3)

It is no longer intuitive at all because we start from the right, evaluating the parentheses first, then have a natural tendency to want to multiply the contents of the parentheses - a tendency exacerbated for people who use programming or scripting languages.

That is due only to your choice to separate some characters with spaces and other without spaces, suggesting they are grouped, which they are not. Every term and every operator is independent. The only group is contained within the parentheses provided in the original expression. No other grouping is necessary to produce the single correct answer, and grouping them incorrectly, as you have done with your arbitrary spacing, produces an incorrect answer.

Anyway, that is really a side note because we both agree that we should not be relying on intuition for such things - there needs to be clear cut and unambiguously defined, coherent rules.

Absolutely. That's why I brought up the Monty Hall problem. That is not intuitive for most people.

So either there is a left to right rule or there is not. If there is not, then you have not provided a reason for why 48 / 2 x 12 is unambiguous.

There is not, and for reasons explained yet again above in this post, the original expression is not ambiguous.

If you still don't see it, I suggest you accept that you simply don't see it.

[olecapt]

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.

Again though there is no accepted rule and not an obvious authority. And if you went to get broad approval by multiple authorities the issue of what is the right thing to do will raise and no consensus will be reached.

Without such a consensus the expressions remain ambiguous. The use of the minimal level of parenthesis achieves the best level of elegance available.

I think in the real world you overuse the paranthesis to avoid any possibility of ambiguity. Who was it said they wrote three pages because they lacked the time to write one?

Paranthesis can also turn into a mechanical or even automated tool to determine that the paranthesis are corectly used. In the olden days I used to evaluate complex expressions by pairing the parenthesis somethimes with color. As you get a few hundred it is a nice way to proceed. Good design often suggests other mechanisms but I often worked in the area of intimate code where performance was important..

[olecapt]

Except that's not the way it was written. Parentheses take precedent.And a number adjacent to a parentheses becomes part and parcel of the same item in the equation. 2(9+3) is one term. Had it been written 2 X (9+3) You wpould be correct.

Only in some domains. In other that rule does not exist. Hence the ambiguity.

[Malcolm Wright]

No. 12 X 48 divided by 2 is not left to right.
No. Term A operator 1 term B operator 2 term 3 = term 3 operator 2 term 2 operator 1 term 1. Directionality is order and order is immaterial with these operators.
Stop right there. That premise is false. And there is no raging controversy, only disagreement based on ignorance of the properties of arithmetiic.
That is due only to your choice to separate some characters with spaces and other without spaces, suggesting they are grouped, which they are not. Every term and every operator is independent. The only group is contained within the parentheses provided in the original expression. No other grouping is necessary to produce the single correct answer, and grouping them incorrectly, as you have done with your arbitrary spacing, produces an incorrect answer.
Absolutely. That's why I brought up the Monty Hall problem. That is not intuitive for most people.
There is not, and for reasons explained yet again above in this post, the original expression is not ambiguous.

If you still don't see it, I suggest you accept that you simply don't see it.

Do we agree that:

48 / 2 = 24
24 x 12 = 288

Do we agree that:

2 x 12 = 24
48 / 24 = 2

If we agree upon this, and if we agree that depending on what order you perform 48 / 2 x 12, you follow one of the two paths outlined above, then we agree that the order in which these operations are performed matters a great deal.

What exactly am I not seeing?
Depending on whether you start from the left, or from the right, you get different results. If we are misunderstanding each other over some notion of direction, then I invite you to define in some other terms the difference in order outlined above.

M.

[Ratbag]

Finds it hard to beleive that there is such intense debate on a matter that I would have before this beleived to have been a problem that arises millions of times on a daily basis....

[Chachma v'Oz]

Do we agree that:

48 / 2 = 24
24 x 12 = 288

Do we agree that:

2 x 12 = 24
48 / 24 = 2

If we agree upon this, and if we agree that depending on what order you perform 48 / 2 x 12, you follow one of the two paths outlined above, then we agree that the order in which these operations are performed matters a great deal.

What exactly am I not seeing?
Depending on whether you start from the left, or from the right, you get different results. If we are misunderstanding each other over some notion of direction, then I invite you to define in some other terms the difference in order outlined above.

M.

I'm guessing that the real question you should be asking is, "What does the denominator consist of"?. It is 2, or it is 2X12? It is 2 and can only be 2.

Why? Because for the denominator to contain more than one element, for it to contain a group of elements, they would have to be grouped with brackets.

Why? Because juxtaposition in mathematics does not imply grouping. Juxtaposition means only multiplication. The original equation did not have everything except the 48 in brackets, so those terms are discrete. They are independent of each other. They are separate from the 2. There is no arithmetic justification whatsoever for combining them into an arbitrary group before acting on the 48. If they were meant to be so grouped they would have to be bracketed to form such a group. That was not done because they are not a group. That's why there is one and only one correct answer to this equation: 288.

[Malcolm Wright]

There is no arithmetic justification whatsoever for combining them into an arbitrary group before acting on the 48.

What you are saying here, although you refuse to admit it, is that there is an unspoken left to right rule.
Otherwise one would not need a justification to act on the right hand group before one acts on the left hand one.

Once you are able to see that you have been arguing for a left to right rule all along, we'll get somewhere with the conversation.

M.

[garyd]

Once you do the numbers inside the parenthesis you 2(12) NOT 2X12. 2(12) is treated as a single term 2x12 is not.

another key is this:
2(9+3) = 2x9 + 2x3

[Chachma v'Oz]

What you are saying here, although you refuse to admit it, is that there is an unspoken left to right rule.
Otherwise one would not need a justification to act on the right hand group before one acts on the left hand one.

Once you are able to see that you have been arguing for a left to right rule all along, we'll get somewhere with the conversation.

M.

Still no. We have to approach any string of characters from one direction or another. It doesn't matter whether we work on this string from left to right or right to left.

If it's left to right, then it's the 48/2 term times the 12 term. If it's right to left, it's the 12 term times the 48/2 term. It doesn't matter. There's no rule of directionality. Multiplication is commutative.

A non-rule that has been introduced into this discussion is a "rule" that because some elements are next to each other, namely the 2 and the (9+3), that they can or may or should be grouped into a single term before acting on the 48. There is no such rule. As a matter of fact, the opposite is the rule. If they aren't grouped with brackets, they cannot be grouped together because there's no arithmetic justification for doing so.

The only thing that makes this expression look ambiguous is by making up a rule that says, "If I want to, I can group the terms 2 and (9+3) together arbitrarily with no adverse consequences". That isn't a true statement and will lead to an answer to the equation that is incorrect.

Whether you want to say that 48/2 times (9+3) is 288, or (9+3) times 48/2 is 288, it doesn't matter. You can say 288 = 12 X 24 if you like. We have to read the expression in at least one direction, so pick one and then order the terms on either side of the = sign any way you like. It doesn't matter. There's no left to right rule.

[Malcolm Wright]

But in deciding that the rightmost grouping is a wrongful grouping, you are choosing a left to right direction. Why can't you see that?
You are arguing that because there is no valid reason to give priority to that grouping, the priority defaults to the left-most term. Because otherwise there is NO PRIORITY and the statement becomes ambiguous as I have argued all along.

I could counter argue that since you claim there is no left to right rule at work here, you are arbitrarily and without proper cause choosing to group 48:2 as if there were parentheses around them.

I think the reason why you are failing to see that you are using a left to right "rule" is because it is so hard-wired in your brain to read from left to right: you are not even aware that it is a rule you are operating with.

M.

[Chachma v'Oz]

But in deciding that the rightmost grouping is a wrongful grouping, you are choosing a left to right direction. Why can't you see that?
You are arguing that because there is no valid reason to give priority to that grouping, the priority defaults to the left-most term. Because otherwise there is NO PRIORITY and the statement becomes ambiguous as I have argued all along.

I could counter argue that since you claim there is no left to right rule at work here, you are arbitrarily and without proper cause choosing to group 48:2 as if there were parentheses around them.

I think the reason why you are failing to see that you are using a left to right "rule" is because it is so hard-wired in your brain to read from left to right: you are not even aware that it is a rule you are operating with.

M.

No. You are ignoring the fact that the 2 doesn't "follow" the 48 in a string, the 2 is "under" the 48 in a fraction. That fraction is a unit, an element of this equation. You have a fraction times (9+3) and you know how to solve that - its the numerator of the fraction times (9+3), all divided by 2. No directionality is involved.

[Chachma v'Oz]

This is the way the equation looks written by hand:

48
__ (9+3) = ?

2

How many answers does this have? One answer: 288. Why is there a controversy about the answer in the OP? Because we are limited to writing all the characters on a single line and there is a tendency by some to not follow the rules of arithmetic when characters are presented in a way that doesn't correspond to the way they learned how to work with them.

There is no left to right rule. It could read with the (9+3) before the 48
__

2

it wouldn't matter. the ? could be on the left side of the equation or on the right side of the equation. It doesn't matter. Pick a way to read it and then order the terms any way you like on each side of the = sign. There is no rule that it must be dealt with in any direction, and no rule that says you can arbitrarily combine the 2 and the (9+3) for any reason.

[Malcolm Wright]

No. You are ignoring the fact that the 2 doesn't "follow" the 48 in a string, the 2 is "under" the 48 in a fraction. That fraction is a unit, an element of this equation. You have a fraction times (9+3) and you know how to solve that - its the numerator of the fraction times (9+3), all divided by 2. No directionality is involved.

I'm perfectly aware of that - but my point stands.
Just because the fractional notation takes away ambiguity, it does not follow that the current notation does too. What you are doing is translating the current 'inline' notation into a fractional notation of your choice in your mind, and then deciding there is no ambiguity because you have removed it by choosing one of the two possible meanings of the original notation.

In France, we were taught to always use parentheses in inline statements of this sort, which is the only way to make clear in a division what we are dividing by.

Remember - there is no priority rule differentiating between division and multiplication. So by deciding that 48 should be divided by two BEFORE 2 should multiply 12, you are making an arbitrary decision based on your predilection.

And yes, you are still relying on a left to right direction in assuming what the fractional expression is. If you weren't, you would just as soon assume that 2(9+3) is beneath the fractional bar.

M.

[Charles]

Finds it hard to beleive that there is such intense debate on a matter that I would have before this beleived to have been a problem that arises millions of times on a daily basis....

I was thinking the same thing when I wrote this in post 13:

"Now, the overwhelming majority of people (including mathematicians and engineers) will look at that problem and conclude it is 2 and they'd be wrong. Which makes me wonder, How many designs have failed (after having gone to production) simply because a bunch of designers casually assumed the solution was 2? Wings fall off airplanes due to mistakes such as these."

[Chachma v'Oz]

There is no arithmetic justification whatsoever for combining the 2 and (9+3) into a product because we know that there is a division sign in the expression. The 2 and the (9+3) are right next to each other only because the expression is written on a single line. Because it is written on a single line, if they were a product they would have to be combined within brackets. It's not optional. That is the only way to make 2(9+3) into a product because we know that the 2 is a denominator of a fraction. That is not in dispute, and that requires that adjacent terms be treated exactly as they are expressed. The absence of brackets combining them means only one thing: those terms are not a group. That is not up to interpretation. It's not reader's choice. That's the way math works. Brackets means they are a group, no brackets means they're not. The expression is precise and non-ambiguous.

I'm running out of ways to express this. There is no arithmetic justification to combine the 2 and the (9+3) into a group without bracketing them when an equation is written on a single line and a division sign is involved. Without brackets they are not a group because of the division sign. The existence of that sign means that we have to identify the rest of the terms surrounding it for what they are, not where they are, so that they will be dealt with correctly. That means, if they are to be grouped, they must be bracketed. It's not optional. If they are not grouped, they are not a group.

It's all because of the division sign. Because that sign was introduced, groups must be bracketed to identify them a groups. Otherwise they are not groups. The brackets are not optional. Their presence means the terms within are a group, and their absence means that the terms, because of the division sign, are not in a group. We aren't free to decide whether or not there "should" be brackets to "clarify" the expression. If they are there, the answer is one thing: 2. If they are not there the answer is another: 288.

The OP did not have brackets. There is only one correct answer to the equation as written: 288 because the equation is not ambiguous. It simply appears in a fashion, on a single line, that requires attention to detail when solving. On a single line, everything following a division sign is not in a group unless it is bracketed as a group. If it is bracketed, it's a part of the denominator. If it is not bracketed, it is not a part of the denominator. There's no choice in interpretation. We don't have that luxury. It's either/or. Only the first element, the 2, is the denominator 48 unless it is included with others - in brackets to so indicate that inclusion. There is no mathematical property that deems the fraction be handled differently.

[Chachma v'Oz]

When expressing the denominator of a fraction on a single line, multiple elements must be grouped in brackets to become one term. If there are no brackets, there is no group and only the first term in the series is the denominator. That's the way arithmetic operations are expressed on a single line.