Order of Operations

BillO

Joined Nov 24, 2008
999
... but the info on this site is simply about how something is programmed.
No, its about how the American Mathematical Society reviews electronically submitted papers. The specific phrase of interest "We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division" is included so that people submitting papers will know that if they submit something like:

\(48/2(9+3)\)

Will be taken to mean:

\(48/(2\times(9+3))\)

I suppose there is no correct answer,
Yes there is. That adopted by the American Mathematical Society.

and there is no way anyone will convince me otherwise.
Yes, I have head that one cannot teach the blind to see.

\(48/2(9+3)\)

Is just plain written badly. However, the American Mathematical Society apparently put the interpretation to rest in 2001. They have determined, that if someone has that expression in a paper submitted to them, they will evaluate it as 2. If the proof rests on it being 288, the will reject the paper and the proof as being wrong. So, go ahead and 'believe' what you will.

Actually I think it's important to discuss things like this. People need to be exposed to the right way of doing things, even if they choose not to follow them.

It's all good :):):):)
 

Wendy

Joined Mar 24, 2008
23,415
Hmmm, what a pack of dinosaurs. An influential pack of dino's at that.

So another case that might get people hot and bothered...

V*W÷X*Y


Any links that verify this?

Googling around a lot of websites (I'd say the vast majority) go with the other style.
 
Last edited:

magnet18

Joined Dec 22, 2010
1,227
You all realize that we've been having a 7 page argument over multiplication, correct? :rolleyes:
If my friends found out about this...
 

THE_RB

Joined Feb 11, 2008
5,438
No, its about how the American Mathematical Society reviews electronically submitted papers. The specific phrase of interest "We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division"
...
Juxtaposition?? :eek:

So you now you are sying there are TWO types of multiplication? ;)

To my mind, in a math *process* there is only one form of multiplication; A*B
That's it.

Now back in the 1700's when people had to use very poor technology like chalk and blackboards (even the students had handheld chalk slates for math) it was quite painful to write a * operator in chalk. An abbreviation of . helped somewhat but was still noisy and irritating to write in chalk on a hard slate in a hurry.

So they settled on allowing an imperfection; the shorthand way of writing A*B as AB. So chalkers can write in an incorrect math form that can be allowed because they have also had to learn compensatory rules that enable this "shorthand" to be understood by everyone else.

An imperfection that people could agree on that could be used by everyone because it had some time saving benefits.

But AB is essentially faulty. You cannot write 2*12 as 212 even if you have memorised the "compensatory rules". The fault-tolerance only goes so far.

But in these days we do not need to cater to very poor technology and chalk written shorthand and compensatory rules. In 2011 for every calc done in chalk with poor tachnology there are probably trillions of calcs done as processes within good math technology (ie computers).

So if you eliminate the focus on hand-written shorthand abbreviations and compensatory rules that enabled those abbreviations, and work in terms of modern sensible math *processes* then there must be an implied multiplication in;
2(9+3)
There must be a multiplication process there. So that it must equate to
2*(9+3)

And the *process* of the formula in question must be;
48 /2 *12
which = 288 (unless you are also arguing a separate argument that 48 / 2 *12 = 2?)
 

THE_RB

Joined Feb 11, 2008
5,438
Maybe someone needs to give them a computer lesson. ;)

I've seen the code inside the computer where REAL math is done and there is only one mulitplication; var1 * var2

Once you start to deviate from that all you have are human user conventions, ie a "special" way humans have been *allowed to write it*. It has nothing to do with the actual math process.

Consider this part of the formula;
2(9+3)

If you forget human conventions for a minute and look at the only possible math *process* it is a clearly defined 2-step sequential process;
step1; temp = 9+3
step2; result = 2 * temp
there is no other way that process can be performed.

And if you were to clarify (rewrite) that process as proper math symbols (with no abbreviation or old-boy's secret handshake conventions) the best you can do is;
2 * (9+3)

which means the actual formula IS;
48 / 2 * (9+3)

Insisting on these silly old chalk and blackboard rules is like insisting electrical phenomenon is a product of the "ether" and sickness is caused by "humours". There comes a time to move on and do it the correct way, not the traditional way.
 

Wendy

Joined Mar 24, 2008
23,415
Computers have had their say how math is written, due to the fact they need formula's written as a line. Contrary to earlier statements however, fractions are alive and well, and generally written in a 2D format as opposed to a 1D format, which is why LaTeX and other layout software and formats exist.

Programming languages still require that 1D format, but math and its concepts are much older, and don't disappear just because they are now inconvenient. This is sloppy thinking, with an element of wishfullness besides. I have never used any math packages, but I am willing to bet they portray math in the classic formats. In other words, the software must adapt to the rules of math as it was written in 1920.

However, for programming applications math must be able to follow classic formulas. This isn't hard with parenthesis, which brings us to the early posts. It is usually not a mistake to have too many parenthesis, but it can be major mistake to have too few. Writing formula too sloppily will cause the problems we were talking about, which I believe is BillO's point.

In spite of the disparaging remarks about black boards, they are alive and well, which should point out an unreasonable bias. They are being replaced with dry erase boards in some cases, a distinction without merit. There is something about using our hands for formula manipulation, or sketching ideas. I suspect it has to do with simplicity, but I remember reading that a computer display uses different parts of the brain than ink on paper. Creativity isn't that cut and dried.

Software has its uses, but it isn't the only way. It should not be considered as such.

Interesting thread. 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.
 

Wendy

Joined Mar 24, 2008
23,415
Say that after your calculus classes. Like I said, the only reason 1D lines of math exists is to make programming easier, it does not make math or math concepts easier. Looked at any datasheets lately? You will find old math standards are alive, well, and are still being published.
 

BillO

Joined Nov 24, 2008
999
If you forget human conventions for a minute ....
Not a chance. Mathematics is a human thing. Computers were created to help do calculations quickly, not to teach us how to do mathematics. In fact, computers have a real hard time with symbolic mathematics. There have been a few specialized artificial intelligence systems created to try to deal with it, but they have not been overly successful.

It is unfortunate that folks these days are learning computer-math first, rather than getting a good grounding in the subject of mathematics first, then moving on to how it's wedged into a computer.

Now, if you're saying the original linearized expression is ambiguous, I'll agree with you. However, the normal disambiguation (not the computer/calculator) will result in an evaluation of 2.

As I said before, writing the expression that way is bogus to begin with. Either for humans or byte chompers.
 

victorhugo289

Joined Aug 24, 2010
49
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.

This is why this "problem" came as no problem to me while it does to some. To me this operation "48/2(9+3) = ?' shows no ambiguaty at all, basically because I limit myself to Multiplication and Addition all the time.

And the answer is 288.

To get a "2", you have to explicitly write parentheses:

48 / [2(9+3)]

-------------------------------
Google has spoken
Magnet18, you forgot Wolfram Alpha website too: www.wolframalpha.com/
:)
 
Last edited:

DerStrom8

Joined Feb 20, 2011
2,390
By the way, I have asked several "professional" mathematicians, and they all say that the answer is 288. Multiplication by juxtaposition TAKES NO PRECEDENCE over ordinary multiplication and/or division in the order of operations.
Der Strom
 

THE_RB

Joined Feb 11, 2008
5,438
...
In spite of the disparaging remarks about black boards, they are alive and well, which should point out an unreasonable bias. They are being replaced with dry erase boards in some cases, a distinction without merit. There is something about using our hands for formula manipulation, or sketching ideas.
...
I agree that black/whiteboards have their place, but that place is an extremely small niche generally in a 1 teacher/multistudent educational environment.

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.

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?

... 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.

BillO said:
... However, the normal disambiguation (not the computer/calculator) will result in an evaluation of 2.
...
I'm not arguing that the *traditional conventions* would resolve it as 2, I am arguing about what it IS.

When the "tradition allowed abbreviation" is removed, 2(9+3) IS 2*(9+3)

victorhugo289 said:
...
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.
...
This is a superb argument. When we are talking about what things ARE, division is just multiplication by a slightly smaller number (a number less than 1). So in reality (when the tradition component is removed) division and multiplication differ only by the size of the number you are multiplying.
(I also mentioned that in my post #29).

This is how I see it;
48/2(9+3) IS 288
but
48/2(9+3) is ALLOWED to be considered as 2 by convention and tradition.
 
Top