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#101




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I disagree and this thread is my cite. 
#102




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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:
http://www.khanacademy.org/video/int...st=Prealgebra http://www.khanacademy.org/v/moreco...?p=Prealgebra 
#103




#104




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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; 04302011 at 02:39 AM. 


#105




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#106




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 Code:
TypeError: 'int' object is not callable 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




That math problem looks like one of those "skilltesting questions" they put on the back of prize tickets.

#108




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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:
Last edited by Hypnagogic Jerk; 04302011 at 08:01 AM. 
#109




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#110




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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




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




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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; 04302011 at 10:35 AM. 
#113




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




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1 x 2 or 1 2x 


#115




And why would they not write it that way? Because then it would be ambiguous.

#116




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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




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1  2πi Code:
(1/2πi) Last edited by Indistinguishable; 04302011 at 03:08 PM. 
#118




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 realworld 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, upthread, did I really see PEMDAS being referred to as a pneumonic? What is this, a plague? excavating (for a mind) 
#119




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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; 04302011 at 04:03 PM. 


#120




*(÷(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




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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




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5/(3x +4) = (4x)/(2x+5) (5 * (2x+5))/(3x+4)=4x Quote:
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:
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; 04302011 at 06:34 PM. 
#123




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I'm not pushing that interpretation, I'm giving a reason why I found it counterintuitive. Quote:
There is potential for confusion because a visually identical equation needs to be parsed in a different order based on which it is. 
#124




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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 32+1 is the same as (32)+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, 32+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 rewrite 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




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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




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




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[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; 05012011 at 01:37 AM. 
#128




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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




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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:



#130




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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. 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




For whatever it's worth, I get 1 when I do the original problem. My thinking is exactly like excavating's.

#132




Is PEMDAS taught in other countries (like Canada)?

#133




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 multilevel 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. Oldstyle 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




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