Quote:
Originally Posted by Bill_Marsden
... I can say I learned something, but their were too many disparaging comments on old school IMO, and directly aimed at people who still use it. Old school got us where we are now, and it doesn't become invalid just because you don't like it.

I apoligise if you believe I was being "disparaging" especially if that is a reference to me doing something bad. I don't see it as "bad" if I criticise an old school doctor for using leeches when there are better modern methods. I didn;t say the doctor was stupid, or evil, or less than myself, I simply criticised an old fashioned mentaility that has become largely obsolete.

Actually I wasn't referring to your post, but if you want the credit...

It is water under the bridge, I regret mentioning it. I like this thread.

 

I always try to avoid division, actually this is why I gave a correct answer to this thread, because I basically stopped using division a while ago, I never use division nor the division sign, I prefer writing x^-1 to signify multiplication by the reciprocal.

Generally, you cannot do this. In some cases, maybe, but not in all. Division is not associative, multiplication is. I've said this before, but I guess you really do not understand. That's okay. You may never need to know. Mathematics is not for everyone.

 

"professional" mathematicians

I guess my degree in mathematics does not count? I did actually work as a mathematician as well. When I worked for the Ontario government's environmental research division, one of my jobs was to check the biologist's math work before they attempted to publish papers.

 

I agree that black/whiteboards have their place, but that place is an extremely small niche generally in a 1 teacher/multistudent educational environment.

This educational environment is a small niche? How are university and college lectures given these days?

In the workplace or real world math use the number of calculations done in computers vs those done by hand on boards/paper is astronomical.

Computers do not do mathematics, they do calculations. There is a huge difference. The mathematics is done first, by humans, then the formulas and algorithms are programmed into a computer.

Some people still drive horse buggies but the road rules have since been adaped for modern transport. Are road rules and math rules so different?

Yes, they are. This analogy does not work. Road rules are legal rules, mathematics rules are not. Road rules change with time and place. Mathematics rules do not. In mathematics the tools may change, but the rules don't.

 

It is apparent here that most of the respondents may never have actually taken any pure mathematics in their educational careers. Most of the replies are all about applications and not mathematics. The talk is about computer and calculators, not about theoretical algebra. Sure 'Math for electronics engineers' or 'Applied mathematics with engineering examples' are great courses, but they do not, evidently, teach you everything about mathematics.

Hands up all those that believe calculation (number crunching) is the same as mathematics?

The next time you see courses like:

Advanced Analysis
Theoretical Algebra
The Calculus of non-linear Manifolds

show up on the course calendar, consider taking one. You may never actually have to do a calculation in any of them.

 

Generally, you cannot do this. In some cases, maybe, but not in all. Division is not associative, multiplication is. I've said this before, but I guess you really do not understand. That's okay. You may never need to know. Mathematics is not for everyone.

Can you give an example where you cannot write:
\(x / y = x \cdot y^{-1}\)?

If I remember my math classes from the first and second year from the uni, algebraic division is defined through the closed '\(\cdot\)' (dot) operation, called "multiplication".

 

I did already. Post 41. Do you need another example?

 

[...]

Right? Now, your saying that division is just the same as multiplication right? So we can go ahead and write:

a/bc = (a/b)c = a/(bc) ... ooops, problem here. Still with me?

(a/b)c is not = a/(bc)

Are you guy's forgetting that multiplication is associative and division is not? You cannot just toss divisive factors around like multiplicative factors

[...]

I don't mean to offend you, but you remind me a bit of an old user, BestFriend.

Why can't you settle with writing a/(bc) as a*(bc)^-1
which doesn't let any misunderstanding about whether you can write a/(bc)=(a/b)c

None of us said that division is associative, we said that if you turn any divident into a multiplied power, you have no doubt about how you can do the multiplication.

 

A very interesting post with some very interesting detours.

BillO I have to thank you for providing the formal standard as applied by the American Mathematical Society to spell out something that some of us can see as implicitly obvious. I wasn't too sure when in post #20 you said "we all knew and understood that implied multiplication was done first. Always. Made life easier when we were all on the same page." Obviously you did your homework to find the standard you linked in post #58 and quoted in post #61:

"We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division"

That is some mouthful and truly took me a awhile to understand what the meaning was; "juxtaposition" just ain't in my vocabulary, and I have a hankering for obscure words. Heck, I even got the correct definition for "hausehole" when playing some word game.

But it is intuitively correct inside my brain that an implied multiplication is always in order and always takes precedence. Thus the two items 2 and (9+3) are intertwined and connected by the implicit operation, and thus the divisor is 24. I'm just happy you found the formal standard that agrees with my brain. Otherwise I would have had to rethink how I think.

You may not prefer blackboards, neither do I, but I still use pencil and paper to do mathematics, to work out the expression for how things relate. There is just no way I would ever write 2*x when I mean 2x, just as there is no way I would write 34 when I mean 3*4. The former is a general standard and the latter is just rubbish. I also know about an old school operation called "division." It is typically indicated by a long horizontal line clearly separating a numerator expression and a denominator expression, and it takes 2 or 3 lines on the paper to describe.

When using a computer to evaluate a given expression I have no predefined standard; do what works the best and costs the least. Most of my work is on embedded systems where both speed and accuracy are necessary, but both inputs and outputs are integer quantities. My last project entailed some curve fitting to some straight lines so I needed to evaluate mx+b; in my case m was less then zero so I actually defined an m'=1/m and instead evaluated x/m'+b in my code to increase my accuracy when doing integer arithmetic. Gave me less then a few bits of error where my curve was defined as a 9th degree polynomial.

So much for eliminating division, or did I just reinvent it?

OK, so the expression as written is trash and obviously subject to misinterpretation. The interpretation is subject to mistakes and should be done to some standard. It is interesting to note that the picture of the Casio calculators in post #1 do no interpretation and enter the expression as defined. If I type the given formula without any interpretation into my TI-30Xa calculator I get 4. Same in window's calculator applet (scientific mode).

If I enter the expression as defined into excel I get an error.

So based upon my actual testing I wish to toss FOUR into the works as a possible answer, as my calculator and computer both agree on this.

But the real answer is two, by the standard definition.

If you believe anything different you have yet to find your error.

 

I don't mean to offend you, but you remind me a bit of an old user, BestFriend.

Why can't you settle with writing a/(bc) as a*(bc)^-1
which doesn't let any misunderstanding about whether you can write a/(bc)=(a/b)c

None of us said that division is associative, we said that if you turn any divident into a multiplied power, you have no doubt about how you can do the multiplication.

Geo (do you mind if I call you that), please do not use your lack of understanding to judge others.

All you have done is chosen an association. This is getting hard for me to explain as my understanding comes from a place where I do not think you have been. Kind of like if you walked into a room full of audiophiles that professed they know electronics because they use it and flatly refuse to accept what you say. Like you said, no insult intended.

Let me try one more time. You are saying that expression:

\(A/B/C\)

is the same as the expression:

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

I am saying they are not. Here is why:

\( (A \cdot E) \cdot F \ = \ A \cdot (E \cdot F) \ \ \ \ \ But.... \ (A/B)/C \ \neq \ A/(B/C)\)

If you want me to go beyond this, please pick up book on theoretical algebra and work your way through it. If at that point you still disagree with me, at least we will both have the tools to effectively argue the point. Agreed?


I've got to go. I'll get to your question in the other forum at a later time.

 

@ErnieM

Thank you. I'm glad you get it. I think that makes 3 of us. The calculator jockeys are still having nothing of it though.

 

Hey! You 're cheating!

Let \(D=B^{-1}\ and\ E=C^{-1}\)
\(
That\ said,\ we\ have:\\
A /B /C = A \cdot D \cdot E\\
and\\
(A \cdot D) \cdot E=(A \cdot B^{-1}) \cdot E^{-1}=(\frac{A}{B}) \cdot \frac1C=\frac{A}{BC}\\
A \cdot (D \cdot E)=A \cdot (B^{-1} \cdot C^{-1})=A \cdot (\frac1B \cdot \frac1C)=A \cdot (\frac{1}{BC})=\frac{A}{BC}\)

Is there something wrong with my reasoning? Associativity isn't a problem as we can see in the example I 've just shown.

I don't have a problem with your implied multiplication precedence as it is. The objection I have is that it doesn't seem to be a strict rule, one mathematics can use to form coherent relations. I don't want someone else to to coach me on how he wrote his math, when he hands me a new mathematical hypothesis on a piece of paper. The word juxtaposition by itself bears a relativity that I don't think is strict enough. Are you willing to define that every time that a multiplication mark is omitted between two terms that particular operation will be carried out before other multiplications or divisions but after any power operetions?

If you do, then ok, that is consistent enough. I won't prohibit you from doing math that way. But please allow me to operate with my set of rules when I do my math.

 

The calculator jockeys are still having nothing of it though.

This has nothing to do with a calculator. It has to do with the basic order of operations. THERE IS NO REAL RULE stating that multiplication by juxtaposition must be completed before "regular" multiplication and division. Perhaps some programs run that way, but that is what some calculators use. I cannot take that link that was posted previously as PROOF that the 2 must be multiplied by what is inside the '()' first. This problem requires only the order of operations, and a basic knowledge of mathematics.

 

I guess my degree in mathematics does not count? I did actually work as a mathematician as well. When I worked for the Ontario government's environmental research division, one of my jobs was to check the biologist's math work before they attempted to publish papers.

I am sorry, I did not mean to offend you. I just do not know where the idea of juxtaposition taking precedence originated and what was behind it. Everyone I have asked, though (except for some people on this forum, obviously ) have agreed that 48/2(9+3) should be treated as 24*12, not 48/24.

 

Is there something wrong with my reasoning? Associativity isn't a problem as we can see in the example I 've just shown.

No, I agree completely. Remember what I said?

Given

\(A/B/C\)

And

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

Then

\( (A \cdot E) \cdot F \ = \ A \cdot (E \cdot F)\)

Which is exactly what you showed. What I am saying is that

\(A/B/C \ \neq \ A \cdot E \cdot F\)

Because, when we try the same associations with

\(A/B/C\)

We get:

\( (A/B)/C \ = \ \frac{(\frac{A}{B})}{C}\)

And

\( A/(B/C) \ = \ \frac{A}{(\frac{B}{C})}\)

Or more simply

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

Therefore

\( A \cdot E \cdot F \ \ \ is\ not\ the\ same\ expression\ as \ \ \ A/B/C\)

I don't have a problem with your implied multiplication precedence as it is. The objection I have is that it doesn't seem to be a strict rule, one mathematics can use to form coherent relations. I don't want someone else to to coach me on how he wrote his math, when he hands me a new mathematical hypothesis on a piece of paper. The word juxtaposition by itself bears a relativity that I don't think is strict enough. Are you willing to define that every time that a multiplication mark is omitted between two terms that particular operation will be carried out before other multiplications or divisions but after any power operetions?

I do, but would never write it that way in the first place.

If you do, then ok, that is consistent enough. I won't prohibit you from doing math that way. But please allow me to operate with my set of rules when I do my math.

No problem.

Also, I'm getting more LATEX practice than I ever wanted.

Edit : But apprantly not enough

 

This has nothing to do with a calculator. It has to do with the basic order of operations. THERE IS NO REAL RULE stating that multiplication by juxtaposition must be completed before "regular" multiplication and division. Perhaps some programs run that way, but that is what some calculators use. I cannot take that link that was posted previously as PROOF that the 2 must be multiplied by what is inside the '()' first. This problem requires only the order of operations, and a basic knowledge of mathematics.

I guess it's a matter of convention. The real problem is that 48/2(9+3) is just not the right way to write the expression as some, like yourself, do not follow the convention, others do. It seems the younger folks here do not follow the convention, so it may indeed be disappearing.

 

I am sorry, I did not mean to offend you.

No real damage done. I have a rather thick skin.

 

BillO I do respect your knowlege and training and have appreciated your posts on this subject. But that does not mean I agree with you on all points.

This educational environment is a small niche? ...

I said that the amount of calculations done on blackboards in comparison with calculations done by machinery is astronomically small. This is a fact. In 50 years from now how many people will be doing fractions on blackboards?

... Computers do not do mathematics, they do calculations. There is a huge difference. ...

I won't argue with that in concept, but I will argue its relevance. The problem at hand is 48/2(9+3) which is definitely a "calculation" in that it has a 4 simple elements and a 3 step sequential process easily resolvable to a result of 1 integer. It is not a high level math "theorem".

... This analogy does not work. Road rules are legal rules, mathematics rules are not. Road rules change with time and place. Mathematics rules do not. In mathematics the tools may change, but the rules don't.

Not true. All rules exist to provide standards and guidelines, systems we can all use to make things go smoothly. In all aspects of reality rules are updated as needed. The reason the math rules exist in their modern form is because they have been updated and revised over the ages, and there WILL be additions and revisions in the future.

...
But it is intuitively correct inside my brain that an implied multiplication is always in order and always takes precedence. Thus the two items 2 and (9+3) are intertwined and connected by the implicit operation, and thus the divisor is 24. I'm just happy you found the formal standard that agrees with my brain. Otherwise I would have had to rethink how I think. ...

I think everyone should constantly rethink how they think. In that way they would be part of the evolution of mankind in developing better methods, better standards and better "thinking". The opposite of that concept would be to do things how they have always been done and that is vastly inferior behaviour.

You opened with a very strong stance that because something is the convention, the "rule" at this point in time that it must be real. What is "real" and what is "rule" are very different concepts and to equate one with another with 100% conviction may be a little hasty.

 

I am not arguing with the concept that the current rules of convention say that 48/2(9+3) SHOULD BE resolved to 2. BillO cleared that up with just one link to the Society of Mathematics. I have been arguing on the topic of what 48/2(9+3) actually IS, why it IS that, and what it may be considered to be in the future.

If someone is capable of being dispassionate and unbiased then it can be possible to evaluate something from different angles and using different techniques to allow something to be seen for what it IS, rather than what the traditional rules say it is. I know it's hard to suspend your preconceptions and old habits but there are a number of powerful logical arguments that have been raised;

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Full beats abbreviation argument;
"Can not" is perfect. "Can't" is an abbreviation (contraction) that is ALLOWED by convention and tradition to make life easier.

But in analysis of what the abbreviation "can't" IS, we all know it is the imperfect form of the concept "can not". We all see the term "can't" and know what it really IS, a reduced-effort imperfect way of expressing the perfect concept "can not". Now tradition, convention and even the modern "rules" allow us to use this imperfect term to make life easier. The rules allow us to work with the imperfection and compensate for its error. But even if the rules did not exist it does not affect the FACT that "can't" IS "can not".

Reality is greater than rule. Real beats rule. Whether the rule this decade allows "can't" or does not allow it, will never be greater than the reality that "can't" IS the concept "can not". Rules help society run smoothly but are largely superfluous in understanding what something IS.

Now in this section 2(9+3) we all know there is a multiplication! In reality this contraction 2(9+3) IS 2*(9+3). The multiplication exists and what we have is an imperfect way of writing a perfect concept 2*(9+3) by lazily leaving out the multiplication symbol. This imperfect abbreviation is allowed (just like "can't") to make life easier because we can form rules through fashion and popular consensus to let us *compensate* for the imprefection and allow faster writing!

The full term is more perfect than the abbreviation. If there is a question of which is right, it should be the full term. The abbreviation exists as an imperfect contracted version of a perfect concept. The perfection of the full concept eliminates ambiguity that the imperfect written form may introduce.

So where 2*(9+3) IS a more perfect way of writing and understanding 2(9+3) and it IS, then we can substitute it;
2(9+3) = 2*(9+3)
so 48/2(9+3) becomes 48/2*(9+3) once the imperfection has been eliminated.

The only exception to this logical reality-based argument "full>abbreviation" is the convention and tradition that "no, we must apply a SPECIAL rule to make it resolve to a different answer".

48/2(9+3) = 48/2*(9+3) = 288
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Multiply and divide are the same in reality argument;

Multiply and divide are both a process where a result is determined by the ratio between the two numbers. Their similarity in reality is very high.

So any version of
A /2 = B
will evaluate identically to
A *0.5 = B

So /2 is the same as *0.5. In reality you could arbitrarily replace /2 with *0.5 at any position within a sequential process and it would not matter as they are the SAME. Similar to the above argument they are just 2 different ways of writing the same perfect concept.

Now math tradition and convention may have reasons for wanting to handle multiplication and division differently. But in reality, with tradition removed, /2 IS *0.5.

The only exception to the logical reality-based argument "/2 IS *0.5" is the convention and tradition that "no, we must apply a SPECIAL rule to force division to be handled differently to multiplication".

48/2(9+3) = 48*0.5(9+3) = 288
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The (futuristic I admit) perfect simplified symbology argument;

If we were to strip away existing rules and start from scratch, it would be logical to assign symbols for processes and in the interest of simplicity there would be one symbol per process, in line with many modern international standards for symbols.

Fractions are used far less these days than they were in the past. Coming from a metric country it seems barbaric to me to measure something as; 11 & 13/64"! Also, in engineering terms in ANY country we don't measure something small as 37/128" as this is disfunctional to the point of being ludicrous. We would use the decimal measurement; 0.289".

There are lots of good reasons for moving forward from fractions and adopting simplified improved systems for writing and expressing math.

If we were to do this, ie move forward, adopt simpler better standards, we would question whether old standards that were good in dealing with imperfect abbreviations and special rules for fractions really have a place.

And if we were to choose perfect simplified symbols for processes I believe these would be a good choice;
* / + - are the best standard for these 4 processes
() is the best standard for order of precedence when it is needed
and left->right evaluation is far better than adopting a right->left evaluation!

So if we stripped away all the old fashioned rules, all the BS, and adopt a clean new standard for expression of sequential math process in the most simplified and BEST way to write that process we would evaluate;
48/2(9+3) left->right apart from specified precedence.

And of course the only exception to an improved, simplified BEST way to handle the sequence would be; "You guessed it! We can't do this the best way we have to follow a traditional rule".

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All the logical reality based arguments seem to support the result of 288. The only thing that supports the result of 2 is the rule. "We have to do it this way because the rule says we have to do it this way".

Maybe it's a good time to discuss the origin of that rule and the benefits it provides, and analyse whether that rule really needs to exist?

It definitely does not NEED to be inside this; 48/2(9+3)

 

"no, we must apply a SPECIAL rule to force division to be handled differently to multiplication".

Interesting post, and I find myself agreeing with a lot of it, one exception being the statement above. Division and multiplication are inherently different. There was no 'special rules' or convention used in post 95. It is a property of division that it behaves differently under association than does multiplication.

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:

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

Will simply not provide the same result because:

\(A/(B/C) \ \neq \ A \cdot (E \cdot F)\)