48÷2(9+3) = ?

[Malcolm Wright]

Because base 10 is implied when discussing arithmetic. There was no reason to introduce computer programming languages into the discussion when they were not implied in the OP.

How about other domains such as other countries?
Base 10 is implied for everyone on the planet and there is zero confusion over this implication. Not so for the current controversy. So again, why muddy the waters with a completely irrelevant comparison?
You won't find anyone complaining about confusion over bases when presented with an arithmetic expression. You DO find people complaining about confusion with regards to other domains such as programming languages, and teaching traditions.

M.

[Chachma v'Oz]

This was presented as an arithmetic equation, nothing more. Someone tried to find away of validating his opinion that it was expressed ambiguously and brought up programming languages as an example. They didn't apply to anything suggested in the OP, but seemed to stick in some minds as a possibly pertinent argument.

I introduced Japanese into the discussion in the same manner, to suggest an analogy supporting my position on ambiguity, and I shouldn't have. That just further got us off track. Then the idea of cultural imperialism was brought in and that could have really muddied things up.

This is simple arithmetic and the rules of arithmetic are clear on what the expression means. That doesn't suggest that every reader knows the proper way to calculate the answer, but that isn't because there is anything defective about the equation as expressed, which was the only thing the OP was posted for, and what we were discussing.

It looked ambiguous to some. It isn't ambiguous in fact.

[Malcolm Wright]

This was presented as an arithmetic equation, nothing more. Someone tried to find away of validating his opinion that it was expressed ambiguously and brought up programming languages as an example. They didn't apply to anything suggested in the OP, but seemed to stick in some minds as a possibly pertinent argument.

I introduced Japanese into the discussion in the same manner, to suggest an analogy supporting my position on ambiguity, and I shouldn't have. That just further got us off track. Then the idea of cultural imperialism was brought in and that could have really muddied things up.

This is simple arithmetic and the rules of arithmetic are clear on what the expression means. That doesn't suggest that every reader knows the proper way to calculate the answer, but that isn't because there is anything defective about the equation as expressed, which was the only thing the OP was posted for, and what we were discussing.

It looked ambiguous to some. It isn't ambiguous in fact.

Thank you. I will conclude that since you are now conspicuously silent on bases, you agree that that too detracts from the topic.

The rules of arithmetic are not expressed the same way in every domain.
All I am saying is that this causes a degree of ambiguity if you prefer to express an expression such as the one above, without using parentheses to make things completely explicit in all domains. That's all. Either you agree or you don't.

A programmer will be more inclined to arrive at the wrong answer for the expression as it is expressed. Not because he was poorly taught arithmetic, and not because he did not understand the very simple rules that you learned (its kind of condescending to infer otherwise), but because the domain in which he works operates differently. I spoke earlier of adapting to a changing world: I did so because the domain of programming is increasingly large and arguably surpasses that of people carrying out arithmetic with pen and paper in modern times. What this means is that the domain which you accepted as universal and eternal is proving to be anything but. The only question is whether you are able to admit that adapting to changing circumstances can be beneficial.

As expressed multiple times, the evidence is right before us: controversy rages over this simple arithmetic expression. This conclusively points towards the benefits of using parentheses to remove any ambiguity from the expression, regardless of domain.

It is a very simple question of practicality, which you are approaching from the angle of out-dated, and manifestly ineffective dogma.

[Chachma v'Oz]

I didn't mean to sound condescending. I didn't necessarily learn everything I know about arithmetic that is taught in primary school while I was in primary school. I was referring to the subject and the elementary basis of the rules that govern it, not when one could have picked them up.

I have 40-odd credits of upper level mathematics under my belt and the further I got into it as a student, the more it became clear How Things Work. I was trying to explain how things work with respect to the equation in the OP after my post #3 wasn't accepted by some as the last word in the matter.

I faced resistance. It may have been from my arguments. All I do know is that I tutored math at the university level for a year and that taught me that I wasn't destined to be a math teacher.

One way to understand something better is to try to explain it to someone else, but that doesn't mean one can necessarily get through to that other person no matter how right he may be.

[Malcolm Wright]

I didn't mean to sound condescending. I didn't necessarily learn everything I know about arithmetic that is taught in primary school while I was in primary school. I was referring to the subject and the elementary basis of the rules that govern it, not when one could have picked them up.

I have 40-odd credits of upper level mathematics under my belt and the further I got into it as a student, the more it became clear How Things Work. I was trying to explain how things work with respect to the equation in the OP after my post #3 wasn't accepted by some as the last word in the matter.

I faced resistance. It may have been from my arguments. All I do know is that I tutored math at the university level for a year and that taught me that I wasn't destined to be a math teacher.

One way to understand something better is to try to explain it to someone else, but that doesn't mean one can necessarily get through to that other person no matter how right he may be.

I absolutely accept your explanation of How Things Work, within your domain. Within that domain, there is no ambiguity. Universality is what is really at stake, and that's where I point towards parentheses as a practical solution to what we are facing.

[Chachma v'Oz]

Yes, parentheses, properly applied, would make the equation clear to just about anyone. My contention is that the original expression was complete as written. I base that on arithmetic principles that are universal. Mathematical expression is subject to convention and differences in interpretation of unfamiliar convention leads to disagreements, as we saw in the Hitler sketch.

I was tempted to bring up reverse Polish notation as another form of mathematical expression, where operators and terms are grouped in such a way as to make parentheses superfluous, but that would really have thrown a monkey wrench into things for anyone not comfortable with Polish notation, its advantages, and its applications. See http://en.wikipedia.org/wiki/Reverse_Polish_notation.

It boils down to unfamiliarity with notational conventions leading to misunderstand, nothing more. I'm not of the opinion that the misunderstanding is the fault of the writer if the equation is expressed mathematically correct.

[Malcolm Wright]

Yes, parentheses, properly applied, would make the equation clear to just about anyone. My contention is that the original expression was complete as written. I base that on arithmetic principles that are universal. Mathematical expression is subject to convention and differences in interpretation of unfamiliar convention leads to disagreements, as we saw in the Hitler sketch.

I was tempted to bring up reverse Polish notation as another form of mathematical expression, where operators and terms are grouped in such a way as to make parentheses superfluous, but that would really have thrown a monkey wrench into things for anyone not comfortable with Polish notation, its advantages, and its applications. See http://en.wikipedia.org/wiki/Reverse_Polish_notation.

It boils down to unfamiliarity with notational conventions leading to misunderstand, nothing more. I'm not of the opinion that the misunderstanding is the fault of the writer if the equation is expressed mathematically correct.

I am not of that opinion either. It is possible to observe that a statement is ambiguous and simultaneously recognize that the author of the statement is correct in seeing no ambiguity in his notation. Within his frame of reference, what he wrote was crystal clear.

Mathematical notation is a language. Languages evolve. Overly complicated or cumbersome aspects of languages, which the majority struggles with. tend to die off. Kanji use in Japanese has steadily diminished in favor of kana. Part of me is sad about this, but part of me recognizes there is a good reason for it.

At the heart of our disagreement is that fact that something obscure that causes confusion can be and routinely is considered to be ambiguous. Those who, through their profession or atypical course through life, are well-versed in the relatively obscure rules, will tend to wish that everyone else was, and defend an effectively ambiguous statement as being unambiguous.

Rules don't exist simply to be professed as dogma. They should serve the purposes to which they are applied. If they are found lacking in that regard, for whatever reason, then they should be improved.
Parentheses are superior to the left to right rule: they provide immediate visual groupings, and remove all the confusion that has had threads like these making storms in teapots They are more inline with programming era mentalities.

You will agree with this if you write a much longer expression with multiplications and divisions peppered throughout, and no parentheses. Much harder to read than one using parentheses. If roughly half the population is making mistakes with such a simple expression as the one discussed in this thread, a much longer one will only see worse results...

M.

[Chachma v'Oz]

It is possible to recognize that a statement is unambiguous and simultaneously observe that it might appear to some readers to be ambiguous. If the contention were simply, "it looks ambiguous", I would not argue. That's a value judgment. I take issue with stating that it is ambiguous. That isn't stating opinion, that's a false statement. It may look ambiguous to the casual reader, but that that doesn't mean it's ambiguously expressed.

[Malcolm Wright]

It is possible to recognize that a statement is unambiguous and simultaneously observe that it might appear to some readers to be ambiguous. If the contention were simply, "it looks ambiguous", I would not argue. That's a value judgment. I take issue with stating that it is ambiguous. That isn't stating opinion, that's a false statement. It may look ambiguous to the casual reader, but that that doesn't mean it's ambiguously expressed.

This stems entirely from what your view point is. Since yours is firmly routed within the pencil and paper realm of arithmetic rules you have learned, you view it as unambiguous but confusing to those who don't have what you have.
Your subjectivity causes your stance.

Objectively, ambiguity exists solely in the eye of the beholder. If I know that half my audience will find my statement ambiguous, am I wise to think to myself that it is not - that they are confused, but not I? I think the person who insists on embracing unnecessarily confusing expressions is more to blame for the perceived ambiguity, than those who perceive the ambiguity. I see a higher degree of confusion in the dogmatist than I do in the person who fails to decipher his writings. Why? Because writing something infers a desire to communicate it - as such it is the onus of the writer to set the stage for replicating what is in his mind accurately in the mind of his audience.

M.

[Chachma v'Oz]

Objectively, ambiguity exists solely in the eye of the beholder.

I disagree. Statements can be ambiguous.

Ambiguous statement, on a job reference: "I can't rate Mr Johnson high enough." It doesn't mean just one thing. I may mean, "He is peerless. He was the best employee we've ever had." Or, it may mean, "I can't recommend Mr Johnson for this position based on his inadequate performance while under our employ." The statement is ambiguous.

The arithmetic statement as expressed in the OP is not ambiguous. It means only one thing. It may appear to be ambiguous to someone not familiar with the notation used. We can't criticize the writer for not shouldering the burden of understanding because we don't know the audience he was addressing when it was written. We don't know the context the OP took it from.

[Malcolm Wright]

I disagree. Statements can be ambiguous.

Ambiguous statement, on a job reference: "I can't rate Mr Johnson high enough." It doesn't mean just one thing. I may mean, "He is peerless. He was the best employee we've ever had." Or, it may mean, "I can't recommend Mr Johnson for this position based on his inadequate performance while under our employ." The statement is ambiguous.

The arithmetic statement as expressed in the OP is not ambiguous. It means only one thing. It may appear to be ambiguous to someone not familiar with the notation used. We can't criticize the writer for not shouldering the burden of understanding because we don't know the audience he was addressing when it was written. We don't know the context the OP took it from.

I think I'm beginning to agree that the statement was unambiguous but that the chosen notation gives rise to ambiguity.

M.

[Chachma v'Oz]

I think I'm beginning to agree that the statement was unambiguous but that the chosen notation gives rise to ambiguity.

M.

Perfect!

[Malcolm Wright]

Phew, hehe...

[Chachma v'Oz]

Do you remember ever hearing of New Math? It was a change in the way we taught arithmetic for a brief time. It taught the algebraic basis for arithmetic, how arithmetic worked. It was really a good idea, but met with too much criticism and was dropped.

The traditional way of teaching to multiply 37 times 27 was to write one above the other, stick an X to the left of the lower one and a line under it, and then, starting with the numeral in the lower right, start multiplying integers, carrying as necessary, adding lines of products below in a specific fashion, then adding up the products to produce a result. That was an algorithm for multiplication. Like any algorithm, it worked every time, but it was cumbersome and prone to computational error.

New Math taught students to look at 37 times 27 as 40-3 times 30-3, written (40-3)(30-3). That was easy to do in one's head. That's 1200, minus 120, minus 90, plus 9. That's just 999. Easy. A lot easier than trying to do it the old way without pencil and paper.

One trick I learned to do, which is hard to explain by description, is to take pencil and paper and multiply a long number by another long number using the old notation, but producing a single line below it with the right answer (rather than a stack of lines to be added up). Example:

2468
1234
_____

3,045,112



I've raised a few eyebrows doing that in my day.

[Ratbag]

Tries to cypher through a couple of days posts to find the basic rule....

[Chachma v'Oz]

Know Your Audience?

[Ratbag]

Groans.... you're probably right.... but I'm sitting here hating that I'm going to have to teach this to my kids with NO formal training whatsoever.... it's not the sort of thing one wants to get wrong, especially since my son is leaning heavily towards some sort of mechanics.

[Chachma v'Oz]

I quit tutoring after a year. Apparently I wasn't cut out to easily explain what I know.

[Malcolm Wright]

Do you remember ever hearing of New Math? It was a change in the way we taught arithmetic for a brief time. It taught the algebraic basis for arithmetic, how arithmetic worked. It was really a good idea, but met with too much criticism and was dropped.

The traditional way of teaching to multiply 37 times 27 was to write one above the other, stick an X to the left of the lower one and a line under it, and then, starting with the numeral in the lower right, start multiplying integers, carrying as necessary, adding lines of products below in a specific fashion, then adding up the products to produce a result. That was an algorithm for multiplication. Like any algorithm, it worked every time, but it was cumbersome and prone to computational error.

New Math taught students to look at 37 times 27 as 40-3 times 30-3, written (40-3)(30-3). That was easy to do in one's head. That's 1200, minus 120, minus 90, plus 9. That's just 999. Easy. A lot easier than trying to do it the old way without pencil and paper.

One trick I learned to do, which is hard to explain by description, is to take pencil and paper and multiply a long number by another long number using the old notation, but producing a single line below it with the right answer (rather than a stack of lines to be added up). Example:

2468
1234
_____

3,045,112



I've raised a few eyebrows doing that in my day.

No - I never heard of New Math. The method you showed for multiplication reminds me of the sorts of tricks I use when multiplying in my head. I'd never thought of doing the same thing on paper though - fantastic.

You claim to be bad at explaining, but I'd love for you to give it a go with the long number multiplication!

M

[Ratbag]

I'm still wondering if left to right is a myth or not