[Eagle9a]

From the length of this thread the answer is blindingly obvious..............

48/2(3+9)= whatever you want it to be

[TeaCosy]

Originally Posted by Season 74

Yes but because such a high percentage of people use implied multiplication what makes it wrong? Like English, the normal way to speak is the one most used because more people understand it, but if there was a clearer way to make a point would that not be right?

I would say what makes implied multiplication wrong is when it's used unnecessarily.

From what i remember, all my a level books used implied multiplication. As long as we knew it was that way, then there was no problem. The problem with BODMAS is it doesn't take this into consideration and is a little out of date because back when it was created Maths was quite simple.

The answer is 288, the answer should be 2.

Maths is as simple or complicated as it ever was - the numbers are all still as they ever were

FWIW I agree the answer is 288 but I don't agree that it should be anything else. I suspect half the people who say it's 2 are just a bit put out that it is really 288 when they wanted to show a magic way that it could be 2 if you stood on one foot, squinted and pretended there was algebra and an equation involved or something

[Season 74]

Originally Posted by TeaCosy

Maths is as simple or complicated as it ever was - the numbers are all still as they ever were

I don't think you understood my point. BODMAS was introduced in 1500. At this time, things such as the standard model and other quantum theory would blow even Issac Newton's mind.

I don't think people are put out because no one wants to know about how they think the answer is 2. It is just the way they do Maths. Most of the stuff they will do would blow my mind, so i'm not going to argue you with them if that is the simplest way to solve their equations. I'm not that arrogant.

If anything it is 288ers that seem unwilling to listen. BODMAS is hardly sacred to Maths. Someone still using the division symbol should take a look at their own use of maths before questioning others.

[njp]

Originally Posted by gomezz

Can't seem to find a copy of the original typeset book of Feynman's lectures online. Anyone got a copy on their shelf they can check?

I sense desperation if you think you can blame physicists' mathematical follies on changing typesetting conventions. The Feynman lectures, first printed in 1963, contain both forms of expression. My copy is the fifth printing, from 1975, but there are no indications that any of the content has changed.

I recall seeing equations in both forms written on blackboards by lecturers, and writing them in both forms in my lecture notes. I don't doubt for a moment that these are the forms in which Feynman, and generations of physicists before him, wrote them.

I think you are just going to have to accept that a "rule" which is routinely ignored is not really a rule at all!

[ETA: I've just discovered that there is finally going to be a new edition of the Feynman lectures!]

Originally Posted by curls2006

http://cdn2.knowyourmeme.com/i/000/1...jpg?1302454815

Nice!

[gomezz]

Certainly does not seem to be a rule for non-mathematicians.

[James T]

If someone were to attempt to redefine arithmetic from scratch, I suspect that instigating two subtly different multiplication operators would seem like an illogical and unnecessary complication. That said, mathematical notation is strewn with illogical and unnecessary complications, which presumably weren't built into the fabric of mathematics at an early stage, but have developed over time for the sake of pragmatism. For starters, the very fact that we have more than one notation for multiplication is daft but universally accepted. Then there's the fact that cos²y is taken to mean (cos y)², while the same does not apply when the power is –1, but again this has just become the norm.

The illogicality of such conventions need not be a problem so long as everyone recognises and accepts them. But the problem for this 'precedence of implied multiplication' convention that some are claiming is that it clearly isn't universally recognised and accepted, as evidenced by this poll.

I can see the sense in such a convention: (a/b)c is likely to be written as ac/b, so why not denote a/(bc) as simply a/bc? The trouble is that the use of the horizontal fraction bar in most texts means that the issue doesn't crop up often enough for this convention to become established.

Because an unestablished convention is not really a convention at all, I voted 288.

[njp]

Originally Posted by gomezz

Certainly does not seem to be a rule for non-mathematicians.

It's not a rule for mathematicians either. Kurt Gödel mocks your aspirations.

[gomezz]

Example?

[John259]

Has anyone found a textbook at any level which states the implied multiplication rule?

Has anyone found any examples of unimplied multiplication?

[MR. Macavity]

Can anyone give an example to me of how real life quantities in the part [ 2(9+3) ] could lead to an 'answer' to the sum that is bigger than 48?

What I mean is, say, if you had 48 loaves of bread and you wanted to divide them out amongst a 'number' of recpients, say, 2 famaliies of 9 and 2 families of 3 [ 2(9+3) ] , is there any way in which arithmetically speaking you could end up giving them all 288 loaves each - out of the original 48?

Am I looking at the puzzle the wrong way? Thanks

[njp]

Originally Posted by gomezz

Example?

I refer to Gödel's On formally undecidable propositions of Principia Mathematica and related systems.

Ironically enough, given the subject of this thread, the online translation of his paper commences with this introduction:

"Gödel's famous proof is highly interesting, but may be hard to understand. Some of this difficulty is due to the fact that the notation used by Gödel has been largely replaced by other notation. Some of this difficulty is due to the fact that while Gödel's formulations are concise, they sometimes require the readers to make up their own interpretations for formulae, or to keep definitions in mind that may not seem mnemonic to them."

It seems uncertainty is all around us!

[gomezz]

The only thing that is certain is that this is a discussion that will remain forever incomplete.

[Flufan]

Originally Posted by MR. Macavity

Can anyone give an example to me of how real life quantities in the part [ 2(9+3) ] could lead to an 'answer' to the sum that is bigger than 48?

What I mean is, say, if you had 48 loaves of bread and you wanted to divide them out amongst a 'number' of recpients, say, 2 famaliies of 9 and 2 families of 3 [ 2(9+3) ] , is there any way in which arithmetically speaking you could end up giving them all 288 loaves each - out of the original 48?

Am I looking at the puzzle the wrong way? Thanks

The real-world situation here has the necessary concepts extracted and they're then expressed mathematically. The point that the 288-ers here are making is that they believe the mathematical expression would need to be

48÷(2(9+3))

ie with the extra (red) brackets, for the result to be 2. It's about correctly/unambiguously expressing problems in mathematical terms; if only we could come up with a mathematical "paradox" and hope to turn round and see the real world conform, with a miracle! Unless I'm misunderstanding what you're getting at...?

[Tictac2]

Originally Posted by John259

Has anyone found a textbook at any level which states the implied multiplication rule?

Has anyone found any examples of unimplied multiplication?

From page 125 of this paper on maths typography:

The core of the problem resides in the possible ambiguity of the juxtaposition (see
Fateman and Caspi [8], who bring lots of examples of ambiguous notation in the
context of machine recognition of TEX-encoded mathematics). However, we feel that
by all reasonable criteria, the ambiguity should be kept limited within the denominator,
instead of letting it propagate beyond the fraction, which is exactly what would
happen if we adapted the competitive rule a/bc = (a/b)c. Indeed, the mathematical
interpretation of juxtaposition is context dependent, a good case in point being the
classic a(x+y). Its meaning depends on whether a is a function that may have x+y as
its argument, or not. Under Nath’s rules 1/a(x + y) is invariably equal to 1/(a(x + y)),
while under the competitive rule the meaning of 1/a(x + y) would be (1/a)(x + y)
in case of a = const! But then we conclude that the traditional rule a/bc = a/(bc)
remains the only reasonable alternative for an unthinking machine.

Edit: I should add that this is just one school of thought. I don't think anyone's going to find a definitive answer for this.

[njp]

Originally Posted by Tictac2

From page 125 of this paper on maths typography:

Interesting read. But you should have included this next bit:

Anyway, we must admit that there is currently no general consent on this point. The AIP style manual [1] says: “do not write 1/3x unless you mean 1/(3x),” while the Royal Statistical Society [16] considers the notation a/bc “ambiguous if used without a special convention.” The Annals of Mathematical Statistics even changed its rules from 1/2π to 1/(2π) between 1970 and 1971. Use of programming languages and symbolic algebra systems with different syntactic rules also has a confusing effect.

Are the most ardent 288-ers now willing to concede that this is not at all as clear-cut as they originally thought?

[MR. Macavity]

Originally Posted by Flufan

The real-world situation here has the necessary concepts extracted and they're then expressed mathematically. The point that the 288-ers here are making is that they believe the mathematical expression would need to be

48÷(2(9+3))

ie with the extra (red) brackets, for the result to be 2. It's about correctly/unambiguously expressing problems in mathematical terms; if only we could come up with a mathematical "paradox" and hope to turn round and see the real world conform, with a miracle! Unless I'm misunderstanding what you're getting at...?

Essentially you understand me correctly - I think! For what it is worth, from my angle, I would not require the red brackets to solve this problem with an answer of 2 - I would 'work out' 2(9+3), and then divide 48 by that answer.

I think what I'd like to understand from a '288er' is how to divide 48 by a number greater than 1 and up with a number greater than 48? The puzzle, to me, starts off with the situation of us having 48 somethings, which we are going to divide up. How can we end up with situation where we go from a situation of having 48 of something, dividing it up, and then having 288 of something?

I'm not saying it isn't impossible, just for someone to explain how it can be arrived at in practical terms - we only ever use mathematics to describe situations after all - don't we?

[munta]

Originally Posted by njp

Interesting read. But you should have included this next bit:

Anyway, we must admit that there is currently no general consent on this point. The AIP style manual [1] says: “do not write 1/3x unless you mean 1/(3x),” while the Royal Statistical Society [16] considers the notation a/bc “ambiguous if used without a special convention.” The Annals of Mathematical Statistics even changed its rules from 1/2π to 1/(2π) between 1970 and 1971. Use of programming languages and symbolic algebra systems with different syntactic rules also has a confusing effect.

Are the most ardent 288-ers now willing to concede that this is not at all as clear-cut as they originally thought?

One would hope so. However, I fear not

[Doctor_Wibble]

I continue to be amused by the demands for an explicit written rule when we are arguing over which *convention* is to be used. Evidence thus far indicates that a number of conventions are widely used, even if not *universally* accepted. I suspect that one reason for the lack of an explicit rule may be the sheer volume of papers (mathematical or otherwise) that would potentially be invalidated as a result.

Obviously I remember BODMAS being taught at school, but it certainly wasn't taught as a discipline in its own right - but as a mnemonic for people who couldn't remember how to solve the problems that were given. We weren't given the bizarre contrived examples that you see around the place because the general teaching approach was that if you got something like that the correct approach was to go back and do it again, and put the brackets in this time.

As far as the 'j' word goes, in my own experience a grouped term would always be handled as a single item, primarily because it would be obvious that this was the way to do it - again, if you reach a point where you actually have to stop mid-calculation and look at an expression in this way, then you should go back and do it again, and put the brackets in this time (or format it properly).


[ Edit : if anyone has a copy of Florian Cajori's 'History of Mathematical Notations', that might be interesting - unfortunately the online preview doesn't seem to include the multiplication part... ]

[njp]

Originally Posted by munta

One would hope so. However, I fear not

I think it is time for us all to enjoy once again your Hitler parody, the one truly worthwhile outcome of this thread!

[adodie]

Originally Posted by MR. Macavity

Can anyone give an example to me of how real life quantities in the part [ 2(9+3) ] could lead to an 'answer' to the sum that is bigger than 48?

What I mean is, say, if you had 48 loaves of bread and you wanted to divide them out amongst a 'number' of recpients, say, 2 famaliies of 9 and 2 families of 3 [ 2(9+3) ] , is there any way in which arithmetically speaking you could end up giving them all 288 loaves each - out of the original 48?

Am I looking at the puzzle the wrong way? Thanks

Shhhh!
You aren't supposed to use real world applications for maths. Maths is for computer programming (which seems to be what the majority of posters on here think).

[tellytart1]

Originally Posted by adodie

Shhhh!
You aren't supposed to use real world applications for maths. Maths is for computer programming (which seems to be what the majority of posters on here think).

Lol - and the 288ers are here re-creating the famous miracle of the loaves and fishes from the bible to feed the 40,000

[GrizzyDee]

Originally Posted by MR. Macavity

I think what I'd like to understand from a '288er' is how to divide 48 by a number greater than 1 and up with a number greater than 48? The puzzle, to me, starts off with the situation of us having 48 somethings, which we are going to divide up. How can we end up with situation where we go from a situation of having 48 of something, dividing it up, and then having 288 of something?

From the 288 POV (not saying I steadfastly believe it's 288, just answering your question):

You start with your 48 loaves of bread. You divide those loaves into 2. You then multiply the 24 loaves you have by 12, which is hard to do with bread - make croutons maybe?
The point is, you started out assuming that all the "288ers" somehow divided their bread by 24 as well like you did, which they didn't. They divided it first into 2 and then multiplied it, giving a bigger number at the end.

[Sea_salt]

Originally Posted by TeaCosy

As far as I have seen nobody has given any reason in the thread for why BODMAS shouldn't apply. The only (implied!) reason is that they don't want it to, which is absurd.

That's why some of us have been looking for, and finding evidence to support views one way or another.

If the assertion is that the strict BODMAS, Left-to right rule is the only "correct" way to interpret the expression what evidence do we have?

We know this rule is taught in schools and is commonly accepted.
The rules as stated do not deal with implied multiplication explicitly though there is separate evidence that states 2x, 2(x), 2.x and 2 * x are equivalent.
We haven't found evidence this is *the* way to do it according to any published "standard"
There is evidence that the solidus and obelus have the same meaning.
There is evidence that implied multiplication after the solidus (/) takes precedence in many books, and so far, no evidence of the alternative use.
There is evidence that the form x/(yx) is used in books, particularly recent ones, but also evidence that many writers omit the brackets.
There is evidence that, in the past, grouped expressions took precedence when the solidus is used instead of the horizontal bar and this use was specified by the person who introduced the solidus to mathematics.
There is evidence that the usage of mathematical notation changes over time, and differs between practitioners.
There is evidence that the body that advise on standards in the US are comfortable with the non BODMAS/left-right notation when specifying constants.

As the evidence is contradictory, if you take this evidence at face value the only reasonable conclusion is that the original expression is ambiguous so the original assertion is false. If you consider some evidence to be invalid you may come up a definite 2 or a definite 288 answer.

[John259]

Originally Posted by njp

Are the most ardent 288-ers now willing to concede that this is not at all as clear-cut as they originally thought?

The maths convention is clear-cut enough. What is obvious is that a lot of people don't adhere to the left-to-right aspect of it for various reasons. That's to be expected amongst the general public and even perhaps students. It is surprising though to see some academics not adhering to it.

[Sea_salt]

Originally Posted by gomezz

Certainly does not seem to be a rule for non-mathematicians.

It certainly does not seen to be a rule for mathematicians or scientists.