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Damn, i was working upstairs and read the thread on my iphone and i come back and someone already posted the solution to all the havoc.
I guess i'll reiterate. People are getting the wrong answer because of how they are visuallizing the equation.
2(9+3) = (2*9 + 2*3) - This is correct if the equation allowed it to be possible as you can see
48
--- (9 + 3) does not allow this to be so.
2
48
_________ = This is what most of you are doing which is wrong since there's no brackets in the
2(9+3)
original equation. Simple mistake but will give you the wrong answer of 2.
This is why bedmas or pedmas is used in early grades so no rearranging of equations is needed.
With this equation any PROPER algebraic arrangements can be done
48
--- (9 + 3) FOR EXAMPLE
2
(48*9 + 48*3)
--------------------- = 288
2
I hope we solved this thread!
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In the UK schooling system, it is taught to be
Brackets
Indices
Division
Multiplication
Addition
Subtraction
So from that view it's 288.
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On April 09 2011 06:12 AcrylicMass wrote:
Aren't the two equations from the last poll equivalent? I didn't really interpret it one way or another. Its just two ways of writing the same thing.
If you draw that on paper it's two different things. Number one "1 / (2 * x)" is:
1
----
2 x
and "(1 / 2) * x" is:
1
--- x
2
which is the same as:
x
---
2
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Well, I do a PhD in math...
For notational convenience it is common to consider 1/2x as 1/(2x), unless you want to write bad papers.
No professional mathematician would rely on other professional mathematicians "knowing" that multiplication and division have the same precedence and are left-associative.
I write "knowing" because I know more math than most, and I certainly didn't know.
I'd call this a question of obscure technical details rather than math.
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I'm reading formulars like this like I type them into my calculator.
And in my calculator for example if I type:
1/2x
It will just calculate from left to right so:
(1/2)x
If you read it separately it sounds like:
one divided by two x
which would indicate:
1/(2x)
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On April 09 2011 05:23 MasterOfChaos wrote:At least one reputable source, namely the American Mathematical Society used high priority for omitted multiplication signs in their publications. Show nested quote +We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division. For example, your TeX-coded display $${1\over{2\pi i}}\int_\Gamma {f(t)\over (t-z)}dt$$ ![[image loading]](http://latex.codecogs.com/gif.latex?$${1%5Cover{2%5Cpi%20i}}%5Cint_%5CGamma%20{f(t)%5Cover%20(t-z)}dt$$) is likely to be converted to $(1/2\pi i)\int_\Gamma f(t)(t-z)^{-1}dt$ ![[image loading]](http://latex.codecogs.com/gif.latex?$(1/2%5Cpi%20i)%5Cint_%5CGamma%20f(t)(t-z)^{-1}dt$) in our production process. http://replay.waybackmachine.org/20011201061315/http://www.ams.org/authors/guide-reviewers.html
It's interesting that that when the moment variables get substituted by actual numbers in the question, the actual notion has to change. Remember that the fundamental is how do we interpret a question when a variable gets plugged in.
Say I ignored the second question and saw the first one. First off, I noticed that the notion is incorrect, so like the majority of people do, follow BEDMAS (or any other fancy order of operation method) to solve the question, from left to right, because you don't know if 2(9+3) was based upon a substitution.
The second question, (1/2x), is based on the notation that the 2x part is completely enclosed in parentheses, and in the denominator, because it's usually it's the constants or variables that will modify the 2 value, and are treated as varaibles to scale the number 2. Otherwise, if that was not the case, the notation would be wrong and would be either (x/2), or (1/2)x. I get these types of notion problems myself especially when punching in formuls in excel documents, as it follows a very strict order of operation method, and doesn't allow these juxtaposition notations stated in the quote, but the idea is that you have to assume that any variables that come after a constant will cause that constant to change as if the entire product was in parentheses. Otherwise, notation is bad.
Finally, it's always assumed that whenever you use a division signal, it's always the following number that gets divided into. Whenever you use a fractional sign, it's usually whatever is the products after it before the operation sign. Notation wants you to assume that it's all in the denominator, while using a division sign suggests the next number. Hence, if the division sign used in question 1 was a fractional sign, I could have assumed that the entire 2(9+3) could have been the denominator (and would have been 2), because it would have just been bad notation otherwise.
Example: Correct Notation: (1 + s)/2s, Bad Notation: (1 + s)/2*s
Probably why we never division sign in engineering in favor of the fractional sign... we never use it anyways.
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United States10328 Posts
On April 09 2011 06:26 TheBB wrote:
Well, I do a PhD in math...
For notational convenience it is common to consider 1/2x as 1/(2x), unless you want to write bad papers.
No professional mathematician would rely on other professional mathematicians "knowing" that multiplication and division have the same precedence and are left-associative.
I write "knowing" because I know more math than most, and I certainly didn't know.
I'd call this a question of obscure technical details rather than math.
I agree--if you were writing a paper, you'd just write $\frac{1}{2x}$ and there'd be absolutely no ambiguity whatsoever.
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On April 09 2011 05:15 Severedevil wrote:Show nested quote +On April 09 2011 04:29 Perscienter wrote:
I suppose grammar is just for beginners, too. The internet elite apparently doesn't need it.
Yes, in point of fact, strict grammatical adherence IS for beginners. There are circumstances where not using strict grammar is correct for a variety of possible reasons. (Including situations such as fiction writing, in which time is not a factor.)
Yes, Periscienter, your supposition is 100% correct. In a very real sense, it is correct to say that grammar is just for beginners.
(Now Severedevil already covered this, but I wanted to elaborate a little more because it’s highly germane to this thread.)
A beginner thinks that there is one, overarching grammar for a language—some uber-grammar that encompasses everything, is always appropriate, and contains no internal contradictions. A beginner is wrong about this. An expert user, on the other hand, recognizes that “grammar” is simply the name that we give for the conventions that we use to express ourselves. An expert user understands that these conventions vary from place to place and from time to time, sometimes subtly and sometimes dramatically. In other words, the expert knows that there is no one “correct” way to write things, that there are instead only clear and unclear ways to write things. If, however, you elevate grammar to the status of a hard-and-fast rule rather than a description of consensus use that differs in different discourse communities, you are missing the point of grammar entirely.
The Wikipedia article on grammar offers some good starting points for an improved understanding of the grammar phenomenon:
The term "grammar" can also be used to describe the rules that govern the linguistic behaviour of a group of speakers. The term "English grammar," therefore, may have several meanings. It may refer to the whole of English grammar—that is, to the grammars of all the speakers of the language—in which case, the term encompasses a great deal of variation. A fully explicit grammar that exhaustively describes the grammatical constructions of a language is called a descriptive grammar. Linguistic description contrasts with linguistic prescription*, which tries to enforce rules of how a language is to be used. Recently, efforts have begun to update grammar instruction in primary and secondary education. The primary focus has been to prevent the use outdated prescriptive rules in favor of more accurate descriptive ones and to change perceptions about relative "correctness" of standard forms in comparison to non standard dialects.
As you can see, there is widespread and widely-backed movement from a strict prescriptivist approach to grammar, one that includes a fanatical loyalty to rules, to a more descriptivist approach, one that recognizes the historical and geographical contingency of rules. This new understanding acknowledges that “rules” have a place in the language insofar as they facilitate intercommunication but at the same time insists that a belief that “rules” are unalterable and universal laws only results in confusion, elitism, and bickering by partisans in the camps that various rules generate. What many of the people in this thread (Severe, VIB, MasterofChaos, et. al.) have been arguing against (magnificently, in fact) is ths prescriptive insistence—the insistence that if one “only understood the rules of math,” one would recognize that the answer is so-and-so or such-and-such.
What this insistence amounts to is a misunderstanding of the role of convention in grammars of any kind, including mathematical ones.
What this insistence ignores is that ambiguity and confusion likely exists in the language (or notation) itself rather than in the minds of its readers.
What this insistence causes is a trainwreck of a thread like this, where people participate in elaborate gymnastics of self-congratulation and express anxiety over the fact that the rest of the world isn’t as smart as they are.
It’s a bad scene all around.
----------------------------------------------------------
* some of the more relevant problems with prescriptivism, as noted by Wikipedia:
A further problem is the difficulty of defining legitimate criteria. Although prescribing authorities almost invariably have clear ideas about why they make a particular choice, and the choices are therefore seldom entirely arbitrary, they often appear arbitrary to others who do not understand or are not in sympathy with the criteria. Judgments which seek to resolve ambiguity or increase the ability of the language to make subtle distinctions are easier to defend. Judgments based on the subjective associations of a word are more problematic.
Finally, there is the problem of inappropriate dogmatism. While competent authorities tend to make careful statements, popular pronouncements on language are apt to condemn. Thus wise prescriptive advice may identify a form as non-standard and suggest it be used with caution in some contexts; repeated in the school room this may become a ruling that the non-standard form is automatically wrong, a view which linguists reject.
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Germany2896 Posts
On April 09 2011 06:42 ]343[ wrote:
I agree--if you were writing a paper, you'd just write $\frac{1}{2x}$ and there'd be absolutely no ambiguity whatsoever.
You don't always do that. Since once it's gets nested it takes a lot of vertical space and becomes harder to read. So it's rather common to flatten the fractions. And that's exactly the rationale they gave in the AMS guideline.
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Wow the last tons of pages are so far over my head. At first I was sure it was two now I am just scared and confused.
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On April 09 2011 06:12 AcrylicMass wrote:
Aren't the two equations from the last poll equivalent? I didn't really interpret it one way or another. Its just two ways of writing the same thing.
(1/2)*x = x/2
1/(2x) = 1/(2x)
So if x = 4
(1/2)*x = 2
1/(2x) = 1/8
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omg so many 2s. and 1/(2x) is actually winning
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There is no ambiguity at all. Consider the following:
48 - 2 + (9 + 3) = ?
Everyone knows this is 58, and not 34. Now compare it to the problem in the OP:
48 ÷ 2 · (9 + 3) = ?
You have to use the same order to solve both of these problems. The main reason people say the answer is 2 and not 288 is because people are not used to the division symbol being written in this manner. But the fact that it was deceiving, that people aren't used to it, doesn't mean that this way of writing is incorrect or ambiguous.
People are also being mislead by PEMDAS (Parentheses Exponents Multiplication Division Addition Subtraction) which is actually: Parentheses, Exponents, Multiply and Divide Left to Right, Add and Subtract Left to Right.
1/2x should always be read like 1 ÷ 2 · x which is (1/2)x.
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On April 09 2011 06:53 Crais wrote:
Wow the last tons of pages are so far over my head. At first I was sure it was two now I am just scared and confused.
Don't be intimidated it's more a matter of philosophy and use of conventions. I say philosophy because what we are getting in to is the "correctness" of a convention. It makes no sense to think of a convention in that manner, but saying 288 is correct implies that pemdas/bemdas or whatever you want to call it is 'correct' . The reason I was talking about accepted standards is because you can think of them like a 'default go-to convention' when nothing else is specified.
An even easier way to put it is, we are talking about the different possible ways of handling ambiguity of an expression such as this, and the final answer is dependent upon which of these ways is chosen.
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On April 09 2011 07:06 Zealot)KT( wrote:
1/2x should always be read like 1 ÷ 2 · x which is (1/2)x.
why exactly? i read it as X/2, and there are no clues to the opposite.
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On April 09 2011 07:10 gerundium wrote:Show nested quote +On April 09 2011 07:06 Zealot)KT( wrote:
1/2x should always be read like 1 ÷ 2 · x which is (1/2)x. why exactly? i read it as X/2, and there are no clues to the opposite.
(1/2)x = x/2
you both said the same thing bro 
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On April 09 2011 07:10 gerundium wrote:Show nested quote +On April 09 2011 07:06 Zealot)KT( wrote:
1/2x should always be read like 1 ÷ 2 · x which is (1/2)x. why exactly? i read it as X/2, and there are no clues to the opposite.
He is asserting that division and multiplication have the same precedence and that they need to be applied left to right. Again it gets back to using this convention which apparently isn't a standard in all countries. It's just boring semantics and it can't be asserted as correct until an international standard is in place 
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Here in brazil we don't use flat fractions much, but I can see it can be a little tricky if you do in case of 1/2x. I would read it as (1/2) x but I'm used to 1/2 being vertical.
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Germany2896 Posts
On April 09 2011 07:06 Zealot)KT( wrote:
Everyone knows this is 58, and not 34. Now compare it to the problem in the OP:
48 ÷ 2 · (9 + 3) = ?
This is not identical to the problem in the OP. The OP omitted the "·". The difference in interpretation hinges exactly on the effect of this omission.
Many people give this implicit multiplication a higher priority. Even the publication guidelines of the American Mathematical Society contained that convention, so it's not just people who're too stupid to know maths.
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My script kiddie programming experience in my ancient past taught me to read that as "One divided by two X", or 1/(2x). I don't really think any of these questions have blatantly wrong answers though.
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EPT
