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.
If the author intends j to be a function, he will have to state it up front. Digits are not used as function indicators.
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.
5/(3x +4) = (4x)/(2x+5)
(5 * (2x+5))/(3x+4)=4x
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.
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.
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.
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.
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.
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)
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)
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.
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…]
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)
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.
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.)
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:
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.
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.
For whatever it’s worth, I get 1 when I do the original problem. My thinking is exactly like excavating’s.
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.
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.
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.