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#101
04-30-2011, 02:00 AM
 sich_hinaufwinden Guest Join Date: Jan 2003
Quote:
 Originally Posted by needscoffee There is a very simple way to make the original expression unambiguously equal 1.
I think most of us are aware of this. The question remains, what was intended by the person who wrote the original equation/expression. For instance, what if the equation's author was developing questions for the SAT that you would be taking and they didn't happen to agree with you on this. Some reasonably not-stupid people have weighed in on this in disagreement with you.

Quote:
 Originally Posted by needscoffee Suppose this question were on the SATs. You have to pick ONE answer. As the expression is written, there is only one correct answer.
Don't they sometimes throw out questions? The crappy ones? The ones that some reasonably not-stupid people including some experts (not me) on both sides of the argument could discuss at length?

Quote:
 Originally Posted by needscoffee If the original equation had had used a slash mark instead of the ÷, then you would have a slightly better argument of not knowing what the equation as presented indicated.
I don't remember ever explicitly learning about the disntinction between the two symbols but now that it's been brought up I'll admit that I have been thinking along the lines of the slash mark interpretation.

Quote:
 Originally Posted by needscoffee But the ÷ makes the expression quite clear.
I disagree and this thread is my cite.
#102
04-30-2011, 02:29 AM
 needscoffee Guest Join Date: Oct 2009
Quote:
 Originally Posted by sich_hinaufwinden I think most of us are aware of this. The question remains, what was intended by the person who wrote the original equation/expression.
It is irrelevant what the author intended. He wrote what he wrote, and according the the order of operations it means one and only one thing. It only means something different if you don't follow the O-O-O. If he intended something else, we'll never know it if he doesn't follow standard elementary school pre-algebra conventions. If he intended it to mean something different, it is upon him to use the appropriate brackets or number order to indicate it as such. It's not up to us to decide he made a mistake in forgetting brackets and to second guess alternative intentions. It is ambiguous only if you don't follow the conventions exactly. I think this is being way overthought. It's not a philosophy problem. It's a trick question intended to trip up people who don't strictly follow the order of operations. I can pull out homework given to my kids with pages of this type of problem. One of my kids is in remedial math and this was a huge unit for her.

I fully understand how the original expression can be seen both ways, and in fact we naturally want to solve it to equal 1, because of the parentheses telling us to multiply. It was designed to be that way.

Quote:
 For instance, what if the equation's author was developing questions for the SAT that you would be taking and they didn't happen to agree with you on this.
Anyone writing standardized math questions for the SATs will use this type of question to determine who can remember the order of operations and who can't. It's basic pre-algebra.

#103
04-30-2011, 02:31 AM
 needscoffee Guest Join Date: Oct 2009
Quote:
 Originally Posted by sich_hinaufwinden I disagree and this thread is my cite.
.
#104
04-30-2011, 02:37 AM
 AndyMatts Guest Join Date: Jun 2005
Quote:
 Originally Posted by Saint Cad So if I write "The Conferderate States of America succeeded." then my sentence is miswritten and does not say what I intended it to because I don't want to say they won the Civil War? Just because something is miswritten doesn't make it correct and if someone wrote 1/2x meaning 1/(2x) but it is properly parsed as (1/2)x, it is not ambiguous or correct anymore than the South won the war because I miswrote "seceeded"
But that's assuming you MEANT seceeded. What if you meant that they succeeded in starting a civil war? Or succeeded in causing relatives to kill one another? By leaving room for doubt by the way you structured what you said ("the Confederate States of America succeeded" - with no specific object of the action), it may not be correct to interpret it another way, but that's not a mistake on those making the interpretation, it's your mistake for how you structured what you said. You may know, as the speaker, but others can't unless they make assumptions.

Now, if we assume that the person writing the formula meant for 2(1+2) to mean 2 X (1+2), we might be wrong. It is ambiguous because they fully use the operator symbols in part of the equation, but not in that part. Why not? Can we assume they were just stupid and inconsistent, or do we assume there is meaning in the inconsistency? The inconsistency in the use of operators means that we can't be sure of the intent. It's ambiguous because it's poorly written.

Last edited by AndyMatts; 04-30-2011 at 02:39 AM..
#105
04-30-2011, 03:39 AM
 Francis Vaughan Guest Join Date: Sep 2009
Quote:
 Originally Posted by needscoffee From Wikipedia http://en.wikipedia.org/wiki/Order_of_operations (Bolding mine):
This demonstrates exactly why Wikipedea is a third rate resource and should never be considered as a final arbiter in such discussions. Less than 24 hours after the quoted text was seen on Wikipedia, some of it has gone. In particular, the explicit reference to the expression being discussed has been removed. I would wager that the reference was only placed there very recently by a random edit to the text. Wikifiddling is a great way to try to win an argument. A very dishonest way, but it fools a great many. (I'm not suggesting that needscoffee was the culprit, rather that he has been duped as many others may have been.)
#106
04-30-2011, 04:09 AM
 Francis Vaughan Guest Join Date: Sep 2009
My take on this discussion is this.

The argument arrives not in the operator precedence at all. It arrives in the definition of the juxtaposition operator.

A good way of thinking about it to note that despite arguments that computer languages have no such problem with the expression, the reality is that they all do. Simply because the expression as written is not legal in pretty much all common languages. It is syntactically ill formed, and the compiler will reject it.

Try compiling the expression, you get something like this (gcc, and python):
Code:
foo.cpp:2: error: ‘2’ cannot be used as a function
or
Code:
TypeError: 'int' object is not callable
Surprise surprise.

Indeed you will find it very hard to convince any language to ever compile the expression. The mechanism by which you will, in languages that do provide a back door, is typically by overloading the () operator, which means that the expression will actually end up with the answer 1, because the function call (that implements the multiply) will take precedence. That still doesn't solve the juxtaposition issue, since most languages won't let you change their syntax. Since 2 is a literral, and not a variable, most languages still won't let you make it compile since 2 will evaluate to a base type. This hints at the wider issue.

So it then comes down to what has been thrashed out here. What is the actual definition of the juxtaposition operator? Moaning about PEMDAS doesn't help. PEMDAS does not include a definition of juxtaposition, and thus cannot be invoked to decide the question. In order to create compileable code, or an expression to evaluate, you need a clear and unambiguous definition of juxtaposition. And the bottom line is that there isn't one. As has been demonstrated earlier, even computer systems that can parse the expression (Wolfram Alpha) use context to decide on the nature of the operator. It partly depends upon the type of the operands. And this is probably the root of the issue. Argument rages on a statically defined semantics that works irrespective of the operand types, and yet it is also clear that no such definition exists.
#107
04-30-2011, 07:56 AM
 orcenio Guest Join Date: Sep 2008
That math problem looks like one of those "skill-testing questions" they put on the back of prize tickets.
#108
04-30-2011, 07:59 AM
 Hypnagogic Jerk Guest Join Date: Apr 2005
Quote:
 Originally Posted by Indistinguishable If this question were on the SAT, I would look for a fair amount of context to help me resolve what was being intended. If I couldn't find any context, then, yes, I probably would settle on the (6 / 2) * (1 + 2) interpretation... not because it would be unambiguously clear, but because that would be my best guess as to the thought process involved in designing the question. But I would be full of hesitancy... hesitancy indicative of ambiguity!
I agree with you, and I'd probably roll my eyes at how obvious it is that the person who wrote the question isn't a mathematician, and is trying, as needscoffee says, to trick me, while himself lacking a thorough understanding of the conventions of mathematical writing. It shows how divorced what kids do in school is from what the subjects they supposedly see actually are. (And also I feel for needscoffee's kids, if this kind of problem is what they see in "math" class.)

It also reminds me of something else. I don't know if you've got this in English, but in French, a question grade school kids will often get is be given a word (in writing) and asked how many syllables it contains. But the "correct" answer is usually one more than the number of syllables native speakers actually pronounce when they say the word. Why? Because those who write these questions consider the final 'e' in the word, which isn't actually pronounced, to be part of an extra syllable. Say the word is "patate" (potato) for example. It's usually pronounced [patat] with a syllable break after the first a, and so it has two syllables. (Is there a way to show syllable breaks in IPA?) But in this formalised context of "counting syllables", it becomes [pa'ta'tə] with syllable breaks -- or even pauses -- after both a's, so three syllables. I find it incredibly stupid that those who came up with such questions are so clearly not linguists and have no idea what a syllable actually is, but if I ever was on Are You Smarter than a Fifth Grader and was asked how many syllables the word "patate" contains, I'd smile and say "Three, of course! ."

ETA:
Quote:
 Originally Posted by orcenio That math problem looks like one of those "skill-testing questions" they put on the back of prize tickets.
No, amazingly enough those are actually usually fairly unambiguous, for this exact reason.

Last edited by Hypnagogic Jerk; 04-30-2011 at 08:01 AM..
#109
04-30-2011, 09:48 AM
 Pasta Charter Member Join Date: Sep 1999 Posts: 1,762
Quote:
 Originally Posted by needscoffee
Without even clicking on those links, I'd be willing to bet that the juxtaposition operation is not mentioned in them. It is that operator that brings in the ambiguity. If the original expression used "*" or "x", then there would be nothing to discuss.
#110
04-30-2011, 10:15 AM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by BigT I don't know why they teach PEMDAS at all. Just remember that the higher order operations come first, and that parentheses change that. Oh, and I was always told to write, for example, y = 2x/3 rather than y = 2/3x. I was explicitly taught by every teacher I've ever had that implied multiplication provides its own scope.
As was I. I was genuinely surprised this was a matter for debate--I thought "everyone knew" that you do implied multiplication after parens and exponentiations and before anything else (including division and explicit multiplication."

This is what I have consistently been taught.

Above, St Cad said that 1/2x means the same thing as (1/2)*x ?! This is contrary to every bit of mathematical practice I've ever witnessed.

To be honest, though I can appreciate the possibility that there genuinely are two different conventions being authoritatively taught concerning this matter, I still strongly suspect that the one I follow is the one that is really authoritatively taught. Gradeschool teachers who don't teach my convention are, I strongly suspect, mistaken in the sense that once their students go to college and take math classes they will be taught differently.

Has anyone ever been taught St Cad's interpretation at the College level?
#111
04-30-2011, 10:27 AM
 Saint Cad Guest Join Date: Jul 2005
I'm just not buying into this whole "juxtaposition makes it equal to 1" argument. Juxtaposition is simply a shorthand way to write multiplication so 2(2+1) is the same as 2x(2+1). If anyone has a cite that juxtaposition joins the two operands like siamese twins beyond the basic multiplication operator please give it out because I certainly can't find it.

And I'll state again, someone writing something incorrectly does not make it ambiguous, it makes it wrong. If I write "The ball is blue." but I meant to write "red", the statement is not ambiguous - it clearly states the color of the ball as blue. It just so happens that the statement is wrong.
#112
04-30-2011, 10:30 AM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by Mijin This is all fine when using literals, but in a more common algebraic expression, I think it gets confusing because functions are also represented with parenthesis. So the expression: a ÷ b(c + d) We would have to evaluate b(c + d) first, if b is a function. We could argue that there is still no ambiguity once we know what the each of the terms are. But the potential for mistakes is definitely there.
I would be extremely surprised if anyone here could find someone who uses math on a daily basis (except gradeschool math teachers I guess?!) who would agree that

a ÷ b(c + d) = (a ÷ b)(c + d)

In fact, if someone were to get a mathy professional to sign off on that equation, testifying that it is unambiguously correct, I'd give them a cookie. Any takers?

Last edited by Frylock; 04-30-2011 at 10:35 AM..
#113
04-30-2011, 10:34 AM
 Frylock Guest Join Date: Jun 2001
A cite from a gradeschool textbook in contemporary use which explicitly says something to the effect of "Be careful! Don't treat juxtaposition as any different from explicit multiplication when it comes to the order of operations!" I'd be surprised as well.
#114
04-30-2011, 10:37 AM
 Saint Cad Guest Join Date: Jul 2005
Quote:
 Originally Posted by Frylock As was I. I was genuinely surprised this was a matter for debate--I thought "everyone knew" that you do implied multiplication after parens and exponentiations and before anything else (including division and explicit multiplication." This is what I have consistently been taught. Above, St Cad said that 1/2x means the same thing as (1/2)*x ?! This is contrary to every bit of mathematical practice I've ever witnessed. To be honest, though I can appreciate the possibility that there genuinely are two different conventions being authoritatively taught concerning this matter, I still strongly suspect that the one I follow is the one that is really authoritatively taught. Gradeschool teachers who don't teach my convention are, I strongly suspect, mistaken in the sense that once their students go to college and take math classes they will be taught differently. Has anyone ever been taught St Cad's interpretation at the College level?
In all fairness, I doubt that any problem would be written 1/2x in college. The teacher would either write it using a math writing program like LaTEX or would make it clear that it is a fraction such as
1 x
2
or
1
2x
#115
04-30-2011, 02:03 PM
 Snarky_Kong Guest Join Date: Oct 2004
Quote:
 Originally Posted by Saint Cad In all fairness, I doubt that any problem would be written 1/2x in college. The teacher would either write it using a math writing program like LaTEX or would make it clear that it is a fraction such as 1 x 2 or 1 2x
And why would they not write it that way? Because then it would be ambiguous.
#116
04-30-2011, 02:38 PM
 Saint Cad Guest Join Date: Jul 2005
Quote:
 Originally Posted by Snarky_Kong And why would they not write it that way? Because then it would be ambiguous.
Ummm no.
Because why write fraction on a single line if you have a white board or LaTEX? I doubt any math teacher in college says, "I was going to write it on one line, but I should probably write it in the standard way to avoid being ambiguous."
#117
04-30-2011, 03:05 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by Saint Cad If anyone has a cite that juxtaposition joins the two operands like siamese twins beyond the basic multiplication operator please give it out because I certainly can't find it.
Well, the American Mathematical Society used this convention, even in a LaTeX-ified context, at least as recently as December 2001:

Quote:
 Originally Posted by AMS Guide for Reviewers Formulas: You can help us to reduce production and printing costs by avoiding excessive or unnecessary quotation of complicated formulas. We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division. For example, your TeX-coded display $${1\over{2\pi i}}\int_\Gamma {f(t)\over (t-z)}dt$$ is likely to be converted to $(1/2\pi i)\int_\Gamma f(t)(t-z)^{-1}dt$ in our production process.
In case it isn't clear, that would convert
Code:
 1
---
2πi
into
Code:
(1/2πi)

Last edited by Indistinguishable; 04-30-2011 at 03:08 PM..
#118
04-30-2011, 03:46 PM
 excavating (for a mind) Guest Join Date: Nov 2010
Interesting thread. I am an engineer and I use algebra such as this to calculate capacities, weights, dimensions, etc..., in my sleep. I looked at the expression in the OP and evaluated it as being equal to 1. Anything else would be just wrong.

One thing I haven't seen in three pages of responses is that mathematics is not taught to be a purely academic science. It is taught to be a tool to be used to solve real-world problems.

Consider:

Case 1
There are six boys going on a camping trip. Each tent will hold two boys. One tent stake is needed for the rear of each tent and two are needed for the front. How many tent stakes are needed?

Case 2
There are six cookies in the cookie jar. There is a group of 3rd graders and a group of 4th graders eating lunch. Each group of kids consists of one boy and two girls. If the cookies are divided equally, how many cookies will each kid get?

Now the expression in the OP could be used to determine the answer each case, but they would be evaluated differently. Nobody would evaluate Case 1 and come to the conclusion that only one tent stake is needed. Neither would anyone evaluate Case 2 and imagine each kid could somehow get 9 cookies.

Back when I was in school, I was taught My Dear Aunt Sally. Parentheses were always evaluated first (otherwise you wouldn't need parentheses). Multiplication and division were evaluated next (from left to right), then the additions and subtractions. Multiplication and division are really the same operation (as are addition and subtraction), so not only would it make no sense to do all the multiplications before the divisions, but it shouldn't matter. Same with additions and subtractions. Exponents weren't addressed because, in the 4th grade, we weren't dealing with exponents. By the time we has to deal with exponents, we had to understand the order of operations well enough to know that you had to square the radius before you multiplied by pi; the other way just wouldn't work.

Now, if you made it through that, you would have to conclude that the correct evaluation would have to be 9, since if you add the 1 & 2, then perform the operations as they appear from left to right, you get 9. But, there is no multiplication sign between the 2 and the (1+2). This makes a difference; it makes 2(1+2) a function. While 2(1+2) is evaluated as 2 x (1 + 2), it really means you have two of the quantities (1 + 2). With this understanding, the expression is clearly equal to 1.

Now, up-thread, did I really see PEMDAS being referred to as a pneumonic? What is this, a plague?

excavating (for a mind)
#119
04-30-2011, 04:00 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by excavating (for a mind) Multiplication and division are really the same operation (as are addition and subtraction), so not only would it make no sense to do all the multiplications before the divisions, but it shouldn't matter. Same with additions and subtractions.
But it does matter, as you no doubt realize, because subtraction isn't just addition, it's addition combined with negation, and what's really going on when you decide whether to add or subtract "first" is that you're deciding the scope of the negation. E.g., 3 - 2 + 1 could be parsed as 3 - (2 + 1) or (3 - 2) + 1, and those are different, because they amount to 3 + N(2 + 1) and 3 + N(2) + 1, respectively, where N is the negation operation. Those of course differ by the difference between N(1) and 1; the ambiguity (without an order of operations convention) is in the choice as to whether 1 falls under the scope of the negation or not.

And the same thing, of course, with multiplication and division.

So the key question is, what is the scope of the reciprocation involved when one writes a division symbol. And different people follow different conventions, not always fully formally codified, for resolving that question, as we have seen.

Last edited by Indistinguishable; 04-30-2011 at 04:03 PM..
#120
04-30-2011, 04:27 PM
 TriPolar Guest Join Date: Oct 2007
*(÷(6,2),+(1,2)) is the correct answer for how to express 6÷2(1+2)=?, where ?=9, and using some interpretations of notation that may be in use in this thread. And also where 'correct' means 'my preference for', which is the definition that has been used in this thread. Or does 'unambiguous' mean 'has different interpretations but only mine is correct'?
#121
04-30-2011, 06:27 PM
 New Deal Democrat BANNED Join Date: Jul 2010 Location: north east USA Posts: 1,992
Quote:
 Originally Posted by copperwindow I've seen this on facebook, and I'm afraid I don't understand why there is confusion. 6÷2(1+2)=? Is it 1 or 9?
6 / 2(1 + 2) =
6 / 2 x 3 =
3 x 3 =
9

My understanding is you compute what is in parenthesis first. If you have a series of divisions and multiplications, you go from left to right.
#122
04-30-2011, 06:33 PM
 Irishman Guest Join Date: Dec 1999
Quote:
 Originally Posted by Mijin For me the OP was counter-intuitive because another use of parentheses is to denote functions. So if we look at the following expression: 6 ÷ j( 1+2 ) = ? Say j is a function, j(x) = x ^ 2. Surely in this situation we need to evaluate j(3) first. Or am I confusing myself?
If the author intends j to be a function, he will have to state it up front. Digits are not used as function indicators.

Quote:
 Originally Posted by Jragon On an unrelated note, it always bugged me when people did something like 5/(3x +4) = (4x)/(2x+5) = (5 * (2x+5))/(3x+4)=4x. It really should be more like (2+3)/(3x +4) = (4x)/(2x+5) => (5 * (2x+5))/(3x+4)=4x, since 4x/(2x+5) =/= 4x, which transitivity would imply if you were to just use all equals signs instead of an implication sign.
Yes, I was taught that was an error. You can't put equals to translate one equation to the equivalent equation. I would usually put in on the line below.

Code:
        5/(3x +4) = (4x)/(2x+5)
(5 * (2x+5))/(3x+4)=4x
Quote:
 Originally Posted by needscoffee But there is no ambiguity to this expression when you follow the order-of-operation convention. More brackets would make that expression more explicit, but the o-o-o spells it out plainly. That is the point of the OP's expression, to demonstrate whether a person has learned it or not. It's not a guessing game, it's following the conventions of the game. (Using the ÷ symbol instead of the / makes it simpler to see. The / makes it impossible to determine which is being expressed.)
As I see it, this is a context problem. In context of a class unit discussing order of operations, it should be clear to most people that the intent is to assess one's ability to follow the OOO. In that context, I would probably catch that the grouping is a multiplication and should not be treated special. Probably.

But in real world math use, I have written and performed expressions numerous times where I did precisely that = used clustering as a form of offsetting, and not included extra parenthesis.

And now we have a journal cite that explicitly states they would default to clustering being default grouping like parenthesis.

Quote:
 Originally Posted by Saint Cad And I'll state again, someone writing something incorrectly does not make it ambiguous, it makes it wrong. If I write "The ball is blue." but I meant to write "red", the statement is not ambiguous - it clearly states the color of the ball as blue. It just so happens that the statement is wrong.
Not an equivalent example. "The ball is Indian." Does that mean it is Native American, or comes from Southern Asia? It doesn't matter which I meant, or if I was incorrect and it was actually Spanish, the statement is still ambiguous.

Quote:
 Originally Posted by Saint Cad Ummm no. Because why write fraction on a single line if you have a white board or LaTEX? I doubt any math teacher in college says, "I was going to write it on one line, but I should probably write it in the standard way to avoid being ambiguous."
Seems you are missing the point. The teacher would explicitly avoid that form by using LATeX or a white board or whatever precisely because that form is ambiguous. A teacher doesn't have to say that sentence, he/she is going to automatically use the less confusing presentation.

Last edited by Irishman; 04-30-2011 at 06:34 PM..
#123
04-30-2011, 06:53 PM
 Mijin Guest Join Date: Feb 2006
Quote:
 Originally Posted by Frylock I would be extremely surprised if anyone here could find someone who uses math on a daily basis (except gradeschool math teachers I guess?!) who would agree that a ÷ b(c + d) = (a ÷ b)(c + d)
Erm, I'm confused. Isn't (a ÷ b)(c + d) the surprising but correct PEMDAS interpretation?
I'm not pushing that interpretation, I'm giving a reason why I found it counter-intuitive.

Quote:
 Originally Posted by Irishman If the author intends j to be a function, he will have to state it up front. Digits are not used as function indicators.
My point was that letters can be used to represent either values or functions.
There is potential for confusion because a visually identical equation needs to be parsed in a different order based on which it is.
#124
04-30-2011, 07:39 PM
 excavating (for a mind) Guest Join Date: Nov 2010
Quote:
 Originally Posted by Indistinguishable But it does matter, as you no doubt realize, because subtraction isn't just addition, it's addition combined with negation, and what's really going on when you decide whether to add or subtract "first" is that you're deciding the scope of the negation. E.g., 3 - 2 + 1 could be parsed as 3 - (2 + 1) or (3 - 2) + 1, and those are different, because they amount to 3 + N(2 + 1) and 3 + N(2) + 1, respectively, where N is the negation operation. Those of course differ by the difference between N(1) and 1; the ambiguity (without an order of operations convention) is in the choice as to whether 1 falls under the scope of the negation or not. And the same thing, of course, with multiplication and division. So the key question is, what is the scope of the reciprocation involved when one writes a division symbol. And different people follow different conventions, not always fully formally codified, for resolving that question, as we have seen.
While I fully understand your concepts, I don't agree with your using them in this argument.

Bringing up the concepts of negation and reciprocation only adds to the confusion and is why many people "have trouble with math"; they start off just trying to solve a problem then find themselves trying to figure out why 3-2+1 is the same as (3-2)+1 but isn't the same as 3-(2+1). Subtraction is merely the addition of a negative number, plain and simple. In your example, 3-2+1, you can add the 1 to -2 before or after you add the -2 to the 3 and it will make absolutely no difference. You can even change the order; add the 1 to the 3, then add the -2. When you try to re-write this as 3-(2+1), you are changing the problem; you change the -2 to a +2, add 1, then change it back to -3 before trying to add it to the 3. Now, you can either try to either teach someone a bunch of rules for negation and parsing, or you can get them to understand that subtraction is merely the addition of a negative number. That is a+b+c = (a+b)+c = a+(b+c) and this is true if a=3, b=-2 and c=1 just the same as it is true if a=3.14159, b=2.71828 and c=1.41421.

Same thing with reciprocation. Dividing a number by 5 is merely multiplying it by one fifth. The order in which you multiply or divide makes no difference because it is all multiplication. The reason people get confused in the same as with the addition example. 20/5 x 3 is the same as 20 x 1/5 x 3. If you try to argue that it changes if you parse this as (20 x 1) / (5 x 3), you would be making the same mistake as you did with the addition example. Express it as a decimal and nobody gets confused. 20 x 0.20 x 3 = 12 no matter how you parse it because it makes no difference on the order.

I really wish just one of the people who tried to teach me math and algebra as a kid would have explained this to me this way; I guess one reason I had so much trouble with algebra (Cs and Ds) I but sailed through calculus (straight As) in high school was I had to figure it out on my own, and that took me a while.

excavating (for a mind)
#125
04-30-2011, 08:02 PM
 excavating (for a mind) Guest Join Date: Nov 2010
Quote:
 Originally Posted by Frylock I would be extremely surprised if anyone here could find someone who uses math on a daily basis (except gradeschool math teachers I guess?!) who would agree that a ÷ b(c + d) = (a ÷ b)(c + d) In fact, if someone were to get a mathy professional to sign off on that equation, testifying that it is unambiguously correct, I'd give them a cookie. Any takers?
Well, if you made that 9 cookies instead of just one, I guess I could bend my principles.

But, I really think as you do. If you wanted it to mean that, you'd write

a ÷ b x (c + d)

because although b(c+d) is evaluated as "b times (c+d)", it really means "b of (c+d)", meaning it is a function and you have to evaluate the function before you perform the other operations.

excavating (for a mind)
#126
04-30-2011, 09:50 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by Mijin Erm, I'm confused. Isn't (a ÷ b)(c + d) the surprising but correct PEMDAS interpretation? I'm not pushing that interpretation, I'm giving a reason why I found it counter-intuitive.
If the convention is to treat juxtaposition exactly like explicit multiplication for Order of Operations purposes, then yes that would be the right interpretation. But what I'm saying is that I doubt that is the convention among those who could plausibly be thought of as the "authorities" for mathematical convention. I would bet that they would not treat juxtaposition and explicit multiplication in the same way.
#127
05-01-2011, 01:33 AM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by excavating (for a mind) If you wanted it to mean that, you'd write a ÷ b x (c + d) because although b(c+d) is evaluated as "b times (c+d)", it really means "b of (c+d)", meaning it is a function and you have to evaluate the function before you perform the other operations.
When people write "2X", is the intention for 2 to be thought of as a function being applied to the input X? How about when people write "2(Y + Z)" to mean the same thing, only with Y + Z in the place of X? I think it's fairly clear that they generally don't think of these notations as working in this manner; that's just an ad hoc, back-rationalization for what's actually going on, which is that, by the caprice of history, the notation "something(something else)" has ended up being used to mean different things in different contexts: sometimes it means function application, while sometimes, unrelatedly, it means multiplication. When people write "b(c + d)" to mean "b times (c + d)", it really does mean "b times (c + d)", and not some other thing involving a re-interpretational detour through function application notation.

[It is possible in various ways to view function application and multiplication as both instances of some unifying abstract concept, if you like, but that's not actually what inspires this notational coincidence...]

Last edited by Indistinguishable; 05-01-2011 at 01:37 AM..
#128
05-01-2011, 10:07 AM
 excavating (for a mind) Guest Join Date: Nov 2010
Quote:
 Originally Posted by Indistinguishable When people write "2X", is the intention for 2 to be thought of as a function being applied to the input X? How about when people write "2(Y + Z)" to mean the same thing, only with Y + Z in the place of X?
Perhaps not as a conscience intention, but that is what it means. The function is multiplication.

When 2(Y + Z) is written, it essentially always meant to mean 2 x (Y + Z). That is what I would assume it means in the OP; that is, the (Y + Z) value is to be multiplied by 2. That is why, for me, the expression evaluates to 1, because, as you said above, when people write b(c + d) it really means b x (c + d) and when the expression is evaluated with that meaning, you get 1.

excavating (for a mind)
#129
05-01-2011, 12:05 PM
 Irishman Guest Join Date: Dec 1999
Quote:
 Originally Posted by Mijin Erm, I'm confused. Isn't (a ÷ b)(c + d) the surprising but correct PEMDAS interpretation? I'm not pushing that interpretation, I'm giving a reason why I found it counter-intuitive.
That's the thing, and the point Frylock was making. Many people who do mathematical operations on a daily basis use clustering of "b(c + d)" to mean [b(c + d)]". This is not a rule that is written anywhere. It might be considered a sloppy adherance to PEDMAS, but it is a common convention of use that is picked up over time. Perhaps some college professor uses it and his students pick it up from him. Perhaps some high school teacher uses it and her students pick it up. Perhaps it is more common in physics and engineering classes than in mathematics classes. I don't know. But the reality is that many people who see that have an interpretation that that is a form of clustering, just like parenthesis/brackets. And clustering takes precedence over M/D and A/S.

Maybe it's a mistaken expectation picked up through the most common use. Just thinking about it, we were much more likely to see "a + b(c + d)" than "(a ÷ b)(c + d)". In fact, after elementary school, we almost never saw ÷ again. Now evaluate the first of those, you see that by correct PEDMAS, the multiplication takes precedence, right?

Well, if I only ever see examples where the clustering is multiplication and the combination is with addition or subtraction, I get an inherent expectation to do the clustering. It appears that clustering is the precedence, when really it's the multiplication.

If that expectation is ingrained, and coupled with ÷ not being used, but rather / to represent division, and most problems do not use single line presentation, but ratios, then I almost never confront a case like the problem we have here, where the clustering multiplication rule is shown to be incorrect.

When single line ratios are presented, it is already ingrained in the math teacher's behavior not to give goofy things like "6/2c" to mean "6/2 * c". They are taught, like I was taught, to clump the c in the numerator. So if c is written after the 2, especially without a space, it is assumed to be part of the denominator.

And with that expectation, the clustering rule is reinforced.

Quote:
 My point was that letters can be used to represent either values or functions. There is potential for confusion because a visually identical equation needs to be parsed in a different order based on which it is.
Yes,I can see that, but it's a bit unfair to pull out the function rule, because you have to define functions. Otherwise, how is the reader to know it is a function and not a variable? The default assumption is that letters are variables. (Unless you're in a physics class that has already defined j to be the vector direction or something. Special cases define their own situations.)
#130
05-02-2011, 11:58 AM
 sich_hinaufwinden Guest Join Date: Jan 2003
Quote:
 Originally Posted by excavating (for a mind) One thing I haven't seen in three pages of responses is that mathematics is not taught to be a purely academic science. It is taught to be a tool to be used to solve real-world problems.
While I wasn't trying to make that exact point in post #72, I did note that a real world problem would have removed the ambiguity of what was intended:

Quote:
 Originally Posted by sich_hinaufwinden ...if this was a real world problem there would likely be units involved that would indicate how the numbers should be crunched.
FWIW, the example I concocted, but decided to delete before I posted, was to evenly divide 6 apples among the people from 2 married couples with 1 child each. But I figured anyone discussing this issue at this level wouldn't need an example to understand what I was getting at.

However, since you're the only other poster to mention it (as far as I remember), perhaps the point itself was too obvious for anyone else to bother with.

Quote:
 Originally Posted by excavating (for a mind) Now, up-thread, did I really see PEMDAS being referred to as a pneumonic? What is this, a plague?
I can't tell from what you've written if you understood that I was quoting from the acronymfinder.com link because I thought the error in their definition was amusing to me.
#131
05-02-2011, 12:26 PM
 Barkis is Willin' Guest Join Date: Jan 2010
For whatever it's worth, I get 1 when I do the original problem. My thinking is exactly like excavating's.
#132
05-03-2011, 11:23 PM
 copperwindow Guest Join Date: Sep 2005
Is PEMDAS taught in other countries (like Canada)?
#133
05-04-2011, 03:17 AM
 j_sum1 Guest Join Date: Jul 2003
Quote:
 Originally Posted by copperwindow Is PEMDAS taught in other countries (like Canada)?
I have only heard PEDMAS from US sources.
In New Zealand the commonly used mnemonic is BEDMAS (brackets).
Less commonly used is BIDMAS - the I is for indices
Also less common is BEMA (Division and subtraction being considered as multiplication and addition of an inverse.)
Australians go for BODMAS. The O stands for "Of". I haven't yet met a student, teacher or textbook that does a halfway adequate explanation of what that is supposed to mean.

Personally I hate all of these. I prefer the general principle that operations are done in order from most powerful to least powerful. I know that this sucks as a full explanation, but lets face it -- do you actually need a mnemonic to remember the order of three things?

Much has been made in this discussion of the use of juxtaposition to indicate multiplication and citing this as the source of the confusion. Point taken, but I don't see this as the only problem. The use of / or ÷ also contributes. IMO the only real justification for using these symbols is if you need to present your working on a single line -- which is increasingly rare with today's technology. Use of fraction notation removes all ambiguities (I think) and is more accessible from a visual standpoint.

It is interesting to note that modern calcultors are designed to handle full multi-level fraction notation. This makes what you enter in the calculator exactly correspond with what you would write on a piece of paper. Unfortunately this has had the effect of weakening understanding of order of operations. Students will tend to type in the whole expression as they see it (and often make mistakes) and do not recognise the separate operations occurring and the sequence in which they are applied. Old-style calculators that can only handle one operation at a time do not have this problem since they force the user to decode what they are presented with.
That said, I do get frustrated with the generic cheapo calculators that do not prioritise multiplication over addition. 1+2×3 comes out as 9 and not 7.
#134
05-04-2011, 08:11 AM
 Thudlow Boink Charter Member Join Date: May 2000 Location: Springfield, IL Posts: 18,472
Quote:
 Originally Posted by j_sum1 Much has been made in this discussion of the use of juxtaposition to indicate multiplication and citing this as the source of the confusion. Point taken, but I don't see this as the only problem. The use of / or ÷ also contributes. IMO the only real justification for using these symbols is if you need to present your working on a single line -- which is increasingly rare with today's technology.
Agreed. But wasn't technology responsible for / in the first place? The first place I remember seeing the slash used as a division symbol was in computer programming (e.g. BASIC, back in the 80s), where there was a need for a division operator symbol that was found on a standard keyboard. But in those contexts, multiplication was indicated by *, never by juxtaposition.
Quote:
 That said, I do get frustrated with the generic cheapo calculators that do not prioritise multiplication over addition. 1+2×3 comes out as 9 and not 7.
It's not a matter of genericness or cheapness so much just different kinds of calculators: "standard" or "four-function" calculators that don't follow the order of operations vs. scientific and graphing calculators that do (although, indeed, the generic, cheapo calculators tend overwhelmingly to be of the former type). The Calculator accessory that has long been included with Windows has a "Standard" mode and a "Scientific" mode that you can choose between.

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