Math problem

Factual Questions
101

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.

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?

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.

I disagree and this thread is my cite.

102

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.

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

:).

104

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.

105

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

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):


foo.cpp:2: error: ‘2’ cannot be used as a function

or


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

That math problem looks like one of those “skill-testing questions” they put on the back of prize tickets.

108

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! :D:rolleyes:.”

ETA:

No, amazingly enough those are actually usually fairly unambiguous, for this exact reason.

109

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

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

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

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?

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

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

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

116

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

Well, the American Mathematical Society used this convention, even in a LaTeX-ified context, at least as recently as December 2001:

In case it isn’t clear, that would convert



 1
---
2πi

into



(1/2πi)
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 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

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.

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’?