I am curious--Is there a way to set up a poll on this forum? I think it would be interesting to find out how many people do it one way or the other. What do you all think?

 

'When I use a word,it means just what I choose it to mean — neither more nor less.'

BillO
I don't track you. While I completely agree

\(A / B / C \neq A / (B / C )\)

I think something got missed with

\((A \cdot E) \cdot F \neq A \cdot (E \cdot F)\)

As

\((A \cdot E) \cdot F = (A \cdot \fra{1}{B} ) \cdot \fra{1}{C}\)

\(= (\fra{A}{B}) \cdot \fra{1}{C}\)

\(= \fra{A}{BC} \)

and

\(A \cdot (E \cdot F) = A \cdot (\fra{1}{B} \cdot \fra{1}{C})\)

\(= A \cdot (\fra{1}{BC})\)

\(= \fra{A}{BC} \)

Or did I miss something here?

 

I'm not sure Ernie. Where did I say:

\((A \cdot E)\cdot F \ \neq \ A \cdot (E \cdot F) \ ?\)

 

He's getting confused at post #95. I interpreted that statement as part of the proof that they weren't equal.

 

I am curious--Is there a way to set up a poll on this forum? I think it would be interesting to find out how many people do it one way or the other. What do you all think?

seconded
Moderators?

 

I'm not sure Ernie. Where did I say:

\((A \cdot E)\cdot F \ \neq \ A \cdot (E \cdot F) \ ?\)

Ah! Now I get what you said (so plainly too).

Never mind.

 

...
Association by parenthesis is an important concept in mathematics, particularly in applications. Consider a physical system where the unfolding of events requires one expression in the model to be written

\(A/(B/C)\)

Then, converting that to:

\( A \cdot (E \cdot F) \ \ \ \ \ where \ \ E=B^{-1}, \ \ F=C^{-1} \)

...

I think you have made a fundamental error here Bill.

If we are discussing that multiplication can be used in place of division (re my argument in post #99) then;

(2/2) is NOT (0.5*0.5) !!
The division is an operation carried out on the denominator ONLY, not on both numerator and denominator. The division is a property of the denominator as the ONE action performed is; /denominator.
So;
(B/C) = (B /C) = (2 /2) = (2 *0.5)
That is an accurate conversion.

So if you properly convert your example of A/(B/C), (I will use decimal numbers not fractions);
A/(B/C) = A /(B /C) = 2 /(2 /2)
= 2 /(2 *0.5)
= 2 *(1 /(2 *0.5))
=2 *(1 /(1))
=2 *1
=2

and 2 /(2 /2)
= 2 /(1)
= 2

So if you convert all the division events to multiplication events it works! As it must because division and multiplication are generally identical in such simple examples... Like this one; 48/2(9+3)

 

'When I use a word,it means just what I choose it to mean — neither more nor less.'

Well a word means;
1. the modern definition agreed by the most recent majority
2. the local definition agreed by the closest majority
3. the classical definition complying with tradition
4. the definition the person who used the word intended

I'm with Humpty. All the definitions 1-3 are open to interpretation. But Humpty is the ONLY person who knew exactly what he meant when he used the word. Eveyone one else can only guess and argue what the word meant, but Humpty KNOWS.

 

I think you have made a fundamental error here Bill.

If we are discussing that multiplication can be used in place of division (re my argument in post #99) then;

(2/2) is NOT (0.5*0.5) !!
The division is an operation carried out on the denominator ONLY, not on both numerator and denominator. The division is a property of the denominator as the ONE action performed is; /denominator.
So;
(B/C) = (B /C) = (2 /2) = (2 *0.5)
That is an accurate conversion.

So if you properly convert your example of A/(B/C), (I will use decimal numbers not fractions);
A/(B/C) = A /(B /C) = 2 /(2 /2)
= 2 /(2 *0.5)
= 2 *(1 /(2 *0.5))
=2 *(1 /(1))
=2 *1
=2

and 2 /(2 /2)
= 2 /(1)
= 2

So if you convert all the division events to multiplication events it works! As it must because division and multiplication are generally identical in such simple examples... Like this one; 48/2(9+3)

I agree with your calculations here, but its not what I'm saying. You are missing my point and I honestly do not know how else to express it. I need to remove myself from this thread.

 

I liked the proof on post #95, it pretty much covers why you can not treat multiplication and division as equivalent functions.

 

BillO: Oh nooooo, you can't leave. You're the voice of sanity here.

If you check out then I'll have to go too.

Post #95 is a point well taken (even if it took me while to get it) and I thank you for it.

 

New rule:

No talk of Religion, Politics, or Mathematics will be tolerated!

Since the right way CANT be proven in any of these topics, it can only lead to arguments!

 

I agree with your calculations here, but its not what I'm saying. You are missing my point and I honestly do not know how else to express it. I need to remove myself from this thread.
...

Bill, PLEASE continue! I must have misread your point completely.

I believed you were saying because A/(B/C) does not equate to A.(E.F) that multiplication can not be used instead of division. Was that your point?

Because (with my limited math skill) it seems that A/(B/C) to A.(E.F) is nowhere near "converting the divisions to multiplications".

To convert the / to * on this part; (B /C) becomes (B *F).
Likewise A/result converts to A*(1/result).

I liked the proof on post #95, it pretty much covers why you can not treat multiplication and division as equivalent functions.

Again this is probably due to my ignorance of proper math rules but I don't have any problem solving post #95.

A/B/C is a 2 step sequential process;
1) A/B = temp
2) temp/C = result

for 100/2/50;
1) 100/2 = 50
2) 50/10 = 5

To "change the divisons to multiplications" re my argument, can only convert to this 2 step sequential process;
1) A *(1/B) = temp
2) temp *(1/C) = result

1) 100 *(1/2) = 50
2) 50 *(1/10) = 5

My argument is that multiplication is the same as division in these examples because *for any stage in the sequential process the divide can be replaced with the correct equivalent multiplication*.

As another way of expressing my point; "In the real world multiply or divide does not affect the order of the sequence". Multiply and divides are the same thing, it's just the conventional rule that tells what order to put them in.

Or, once the sequence is KNOWN, then the the multiply can be used anywhere the divide is used *as they are the same thing*.

 

@THE_RB

What I was trying to say was the A/B/C is not the same expression as A.E.F (where A.E.F is as I described) and that it is incorrect to assume you can always replace division factor with multiplication of the reciprocal. While in certain cases the results will be equal, it changes the behavior of the expression under association.

 

Thank you for the clarification Bill.

... What I was trying to say was the A/B/C is not the same expression as A.E.F (where A.E.F is as I described)
...

I agree entirely, I would never even have considered those 2 to be the same expression and was surprised at the comparison!

... and that it is incorrect to assume you can always replace division factor with multiplication of the reciprocal. While in certain cases the results will be equal, it changes the behavior of the expression under association.

I think this is getting to the nuts and bolts of it. The way I see it; (A/B)/C and A/(B/C) are two different processes and will produce two different results. But both work and both can have divisions and multiplications exchanged within them, showing the requirements of divide or multiply do not force either version to be right or even "better".

So there is nothing inherent in A/B/C itself that requires it to become (A/B)/C or A/(B/C). Only the rule.

I suppose what I was asking for an answer to is WHY the rule requires specific precedence, or more specifically the origin of the need to force that particular precedence. Why it used to be important.

If we knew the original reason why A/B(C+D) is required to be considered as A/(B*(C+D)) it would be able to see if this reason still exists in modern calculating examples or if it is just a carryover from the days when fractions needed to be transposed on blackboards.

 

Well, I'm just letting everyone know that I am going to back out of this thread now, as well. There is no way anyone can convince me it is not 288 without getting the U.S. Mathematical Society (or whatever it's called ) to contact me directly, and give me the REASON FOR and PROOF THAT there is such a rule that says multiplication by juxtaposition takes precedence. I suppose that is what this entire question boils down to.
I have enjoyed the conversation and have even learned a bit, but it is time for me to let it go. Have fun, everyone!
Best regards,
Der Strom

 

If we knew the original reason why A/B(C+D) is required to be considered as A/(B*(C+D)) it would be able to see if this reason still exists in modern calculating examples or if it is just a carryover from the days when fractions needed to be transposed on blackboards.

Before the advent of computers/calculators with their single line input and the resultant need to linearize equations, there would have been only one interpretation of:

\( 48/2(9+3) \ \ \ and\ that\ is\ \ \ \ 48/(2*(9+3))\)

Anything else would have been written on a blackboard or paper as:

\(\frac{48}{2}(9+3)\)

So now writing 48/2(9+3) if you mean 48/(2*(9+3)) or if you mean (48/2)(9+3), is just plain bad style.

And this is really my final note on this as I am now heavily into my son's racing campaign and will be a infrequent visitor for a while.

 

Thank you Bill for the wonderfully simple example explaining the need for the forced precedence.

I might be the only person who has hung in waiting for some final resolution and explanation of this issue.

Hopefully I can sum this up well enough without my personal bias towards modernism and minimalism overly affecting the result?

The blackboard allows both vertical and horizontal expression, and the ability for vertical expression allows for TWO different ways to write the same 7 elements;

\( 48/2(9+3)\)=2 and \(\frac{48}{2}(9+3)\)=288

Because there are 2 ways to write it this allows precedence to be specified by the use a fraction element; - and vertical placement, or precedence specified the other way by the use of a more linear format and the divide element; /

So it makes sense that in the earlier days of "linearising" math, the typewriter days etc, that the expression; 48/2(9+3) would be continued to be evaluated on the new linear text equipment using the traditional linear standard, to be be considered as; \(48/2(9+3)\)

But LONG after the typewriter days came the pocket calculator days. Especially with early pocket calculators there was an expectation that the user must enter the elements already in the desired linear order of precedence.

I'm old enough (just) to have carried log table books to school, but there are 2 generations now that have only ever entered their linear sequences into calculators. This has developed into a "modern standard" of sorts, where the user must deliberately enter precedence.

I would say the precedence is popularly expected to be "pre-decided" and the linear expression entered into the machine already contains the precedence in its left to right order (apart from deliberate precedence symbols). People in the "calculator generation" may not be "right", but they have definitely moved towards a new linear standard based on popular consensus.

From a computer programming point of view, all expressions are linear typed (with no ability for fractions or vertical operations) and there is a very strong standard for left to right precedence and the use of proper precedence symbols. A process known as the "pre-processor" works through the text and arranges precedence before the code is compiled.

I have three C compilers and all evaluate 48/2(9*3) as 288, probably for the reason that the pre-processor FIRST inserts the * where it was omitted, and then evaluates A/B*(C) and because the pre-processor exists there will never be this function; A/B(C)

So we have a world where the old standard (that is still official) is that 48/2(9+3) = 2, but a whole world of people who will type 48/2(9+3) into their calculator and ALMOST always get 288 as a result, and a world full of computer programmers who know that 48/2(9+3) WILL be evaluated to 48/2*12 = 288 by the machine.

It leaves me with the conclusion that in fifty years from now, only ONE of those three groups is likely to have changed...

 

So much for math being the universal language.

 

It is interesting that there exist some C compilers that actually take that garbage and give an output. I just tried 3 different compilers and all refused to guess what I meant by:

int x = 48/2(9*3);

PIC32 C gave: "error: called object is not a function"

Visual C++ gave: "error C2064: term does not evaluate to a function taking 1 arguments"

BoostC gave: "error: missing semicolon"

Note: these are all excellent calls by the compilers.

So much for math being the universal language.

48/2(9*3) is an expression begging for evaluation. To evaluate such, we should all follow an agreed upon standard. BillO took the trouble to actually find such a standard published by accredited body.

Google has no such accreditation. As much as I appreciate their work don't make me laugh by suggesting Wikipedia has one either.

As there are no counter examples of such a standard I must go with BillOs.

The question is solved. The answer is 2.

Is that your final answer?

Yes, two.

Final answer.