I look at it a different way. The problem isn’t that implicit multiplication does or does not take precedence over explicit division. It’s that grouping symbols take precedence over multiplication. And a fraction bar acts as a grouping symbol, in addition to indicating division. And there’s no problem if an expression is properly typeset so that you can clearly see what is or is not below the fraction bar, so that you know whether or not it’s part of the denominator.
But when you type an expression all on one line, using / for division, it looks like a fraction bar, but it’s not clear how far “under the fraction bar” extends.
Um, right. That’s the whole point of the post you are replying to. Changing PEMDAS would be changing SPOKEN/WRITTEN language, NOT MATH!!! :rolleyes:. PEMDAS is a convention of SPOKEN/WRITTEN language, NOT MATH!!! :rolleyes:
Well, okay, if it has an = sign in the middle, and something on either side of the = sign, it’s an equation. But this thread isn’t about the interpretation of an equation as such; it’s about the interpretation of the expression that makes up the left side of that equation.
I’ve seen too many students struggle in algebra, I think, partly because they don’t really get what an equation is: a statement that two separate expressions have the same value.
An expression is not necessarily an equation, but an equation is an expression that evaluates to true or false depending on the equivalence of its subexpressions. So in this case we’re talking about the left-hand subexpression.
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.
Right. No disagreement here. As I said in my first post, it’s about whether you interpret the expression to mean that 3 is up top or down under, which is the equivalent of saying "how far ‘under the bar extends’ ", as you put it.
BZZZT! “Equations” is not a subset of “expressions”. They’re mutually exclusive.
Chessic: First of all, I was explicity replying to etv, not you.
You all are reading way too much into this. “Implicit multiplication by juxtaposition” is why this problem exists: to see if the student can overlook “implicit multiplication by juxtaposition” and follow the order of operations conventions as taught.
There is a very simple way to make the original expression unambiguously equal 1. If they wanted the expression to have the answer 1, it would have been written to make that clear: 6÷(2(1+2))=? The OP is a simple 4th or 5th grade question. I have 4 math textbooks here which say the same thing, and a math Ph.D I just asked to be safe. My kids get this type of problem in homework designed to teach whether or not they learned the order of operations. The problem is only ambiguous if you choose to ignore standard elementary math conventions. Which is why we have the conventions. Suppose this question were on the SATs. You have to pick ONE answer. As the expression is written, there is only one correct answer.
Exactly. 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. But the ÷ makes the expression quite clear.
Are you a computer scientist or computer programmer? This strikes me as a computer-sciencey way of putting it. In (at least some) computer languages, equations do indeed have values. But normally, outside a CS (or perhaps symbolic logic) context, I’d talk about an equation being true or false, not having a value of true or false.
Ah yes, some math students seem to have gotten into the habit of automatically punctuating their solutions with = signs between steps, whether it makes any sense or not. So, for example, a Calculus student finding derivatives will write something like x² + 7x = 2x + 7.
As it happens, math Ph.D.s may disagree with each other. When people may disagree about the intended interpretation of an expression, we call it “ambiguity”.
Then it becomes a guessing game as to what the SAT-writers were thinking when they wrote the expression; forcing someone to guess the intended interpretation of an ambiguous expression doesn’t change the fact that is ambiguous.
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!
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.)
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.
Depends what you’re evaluating. If you said “does 2=3?” Then I’d say you’re evaluating the logical expression “2=3” as either true or false. You’re still evaluating 2 * 3 = x as an expression on the = sign, it’s just given no other information the expression implicitly values true. If, previously, you were given another value for x, however, this expression may evaluate to false, implying you’ve either arrived at a contradiction (such as when solving a linear system) or an error in your logic. Even so, you’re implicitly evaluating a = b every time you see it.
I’m a computer scientist, however I was thinking in more of a symbolic logic context since that’s how we talked about it in my discrete math class.
My calc teacher marked us off for that purely on the grounds of “think about what you’re doing. It makes no sense! I’m not giving you full credit for an answer that makes no sense.”
I think I understand the disconnect here, because I used to have it.
I used to think that the only way to do math was to do it purely symbolically. Every proof, in my imagination, was a list of equations that, transformation by transformation, marched down the page from a start state to a goal state, and thus was the theorem proved. Natural language statements could be enlightening, but they were not math.
(I also imagined a great machine, programmed with all of the relevant transformations, grinding out proof after proof without need for human intervention, eventually exploring and documenting the entire space of everything true and provable. I was a good little formalist, nagging doubts about combinatorial explosions aside.)
I finally realized how dumb this was when I began to study mathematical logic, as opposed to advanced counting (arithmetic, algebra, and calculus). Obviously, the standard notation is more compact than a natural language rendering of the same concepts, but every notation has its shortcomings (just ask any programmer…) and we’re discussing a shortcoming now.
Anyway, you misstate Platonism: The idea of the Platonic Form is that it’s a permanent and inviolable shadow of true meaning behind all of our symbols, whatever our symbols look like. Therefore, one set is equally as far from that truth as any other, so saying ‘2+2=4’ is just as flawed as ‘Two added to two leaves four total.’ My point stands even in the false Platonic conception of the Universe.
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.)