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On April 08 2011 17:09 realills wrote:For the real numbers, multiplication is both associative and commutative. Multiplication and division happen "at the same time" but not necessarily from left to right; you can actually do these in any order and will arrive at the same answer if you've done it correctly. ab/c = a/c*b = 1/c*ab =ba/c = b/c*a = 1/c*ba. EG, 12= 8*6/4 = 8/4*6 = 1/4 * 8*6 = 6*8/4 = 6/8*4 = 1/4*6*8 The people who think that whether you multiply or divide first, or whether you start at the left, right, or somewhere else first, makes a difference in the final answer are wrong. The fact that they arrive at different answers when following different "orders" within multiplying/dividing is proof that they're wrong: it would imply the real numbers are not a field, but we know they are. Their confusion is that they can't consistently decide whether 9+3 is in the numerator and the expression is telling us to MULTIPLY by 9+3, or whether 9+3 is inverted, is in the denominator, and the expression calls for us to DIVIDE by 9+3. Order of operations doesn't resolve the question. All that matters is the syntax, so that we understand what the question is asking. The correct syntax when you want to divide by multiply terms, but use only one divisor operator, is to group them together inside parenthesis: 48/(2*(9+3)). That is not the syntax in this question, and the correct answer here is 288. Similarly, 1/2x actually reads as "one half x, one half times x" BUT it's even more likely with that example, than with this numerical example, that someone who wrote that syntax made a mistake and actually *meant* 1/(2x). If I were a tutor or teacher or reviewing some casual work I would certainly expect "1/2x" to have meant to be 1/(2x), but the technically correct reading is (1/2)x. So some mistakes people are making: Show nested quote +The distributive property of multiplication CLEARLY states that the 2(9+3) is an entire term The distributive property does NOT tell us that (9+3)/2 is the same as (18+6). If you want to distribute in this problem, you have to use the two terms in the numerator, 48(9+3)=432+144, and then divide by 2 as the last step. You will get 288. No, there is no difference, according to the syntax, between these two terms. However, one is much easier to misinterpret. 2*(9+3) is exactly the same as 2(9+3), but in this case, one is a better choice because it more clearly conveys what the question is asking; they are equivalent, though. But (9+3)/2 is not 24. Division is not associative, so you cannot "move" the operator away from the 2--it needs to stick to the front of the 2, always. 1/2 is not the same as 2/1. Ultimately I agree that this is a failure of the writer (and that no one would seriously write this question this way). The syntax does give one correct answer, but the writer had multiple choices of how to write the question and chose the most misleading (though still technically correct) way.
this.
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The only reason there's confusion is because people who went past age12 maths began to completely lose the use of the ÷ sign. So that sign no longer has meaning.
![[image loading]](http://imgur.com/GKZd9.png)
The first column shows what you could wrongly interpret the question to be. Having not seen the division sign for a long time you assume it means over all.
In the second/right column you have what the question is actually asking. The top picture is how you should probably interpret the question in your mind and the second picture is how anyone with maths experience would write it down.
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On April 08 2011 17:32 Klive5ive wrote:The only reason there's confusion is because people who went past age12 maths began to completely lose the use of the ÷ sign. So that sign no longer has meaning. The first column shows what you could wrongly interpret the question to be. Having not seen the division sign for a long time you assume it means over all. In the second/right column you have what the question is actually asking. The top picture is how you should probably interpret the question in your mind and the second picture is how anyone with maths experience would write it down.
I think people were more confused by the lack of multiplication sign between the 2 and (9+3) than they were about the division sign (although it might have added to the confusion)
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On April 08 2011 17:09 realills wrote:For the real numbers, multiplication is both associative and commutative. Multiplication and division happen "at the same time" but not necessarily from left to right; you can actually do these in any order and will arrive at the same answer if you've done it correctly. ab/c = a/c*b = 1/c*ab =ba/c = b/c*a = 1/c*ba. EG, 12= 8*6/4 = 8/4*6 = 1/4 * 8*6 = 6*8/4 = 6/8*4 = 1/4*6*8 The people who think that whether you multiply or divide first, or whether you start at the left, right, or somewhere else first, makes a difference in the final answer are wrong. The fact that they arrive at different answers when following different "orders" within multiplying/dividing is proof that they're wrong: it would imply the real numbers are not a field, but we know they are. Their confusion is that they can't consistently decide whether 9+3 is in the numerator and the expression is telling us to MULTIPLY by 9+3, or whether 9+3 is inverted, is in the denominator, and the expression calls for us to DIVIDE by 9+3. Order of operations doesn't resolve the question. All that matters is the syntax, so that we understand what the question is asking. The correct syntax when you want to divide by multiply terms, but use only one divisor operator, is to group them together inside parenthesis: 48/(2*(9+3)). That is not the syntax in this question, and the correct answer here is 288. Similarly, 1/2x actually reads as "one half x, one half times x" BUT it's even more likely with that example, than with this numerical example, that someone who wrote that syntax made a mistake and actually *meant* 1/(2x). If I were a tutor or teacher or reviewing some casual work I would certainly expect "1/2x" to have meant to be 1/(2x), but the technically correct reading is (1/2)x. So some mistakes people are making: Show nested quote +The distributive property of multiplication CLEARLY states that the 2(9+3) is an entire term The distributive property does NOT tell us that (9+3)/2 is the same as (18+6). If you want to distribute in this problem, you have to use the two terms in the numerator, 48(9+3)=432+144, and then divide by 2 as the last step. You will get 288. No, there is no difference, according to the syntax, between these two terms. However, one is much easier to misinterpret. 2*(9+3) is exactly the same as 2(9+3), but in this case, one is a better choice because it more clearly conveys what the question is asking; they are equivalent, though. But (9+3)/2 is not 24. Division is not associative, so you cannot "move" the operator away from the 2--it needs to stick to the front of the 2, always. 1/2 is not the same as 2/1. Ultimately I agree that this is a failure of the writer (and that no one would seriously write this question this way). The syntax does give one correct answer, but the writer had multiple choices of how to write the question and chose the most misleading (though still technically correct) way.
Great post but you're wrong about one thing. The writer was 100% successful. The question is designed to trick people and a lot of intelligent people got it wrong. Definitely a great troll question 
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I thought it was 288, then I thought.. "hey.. there must be something strange about this or he wouldn't post it", got unsure about which order the calculations should be handled in and failed..
I blame it on just waking up.
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I nearly fell for it, shame on me. Then I thought "This is Team Liquid, there are no 'simple' math problems" and solved it correctly. 
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Germany2896 Posts
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar.
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On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? Since this is an argument about the grammar of mathematical notation the only way to resolve it absolutely is finding a normative version of that grammar. I asked this before, and the only external link we got so far shows that the result is 2:
http://www.purplemath.com/modules/orderops2.htm
(5th example)
I know the 'correct' answer is 'depends on the syntax'. But now I'm kinda curious to know what the "official" notation says, if there's any.
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I'd like to quote planet math, which will hopefully serve to educate the MASSive amount of ignorance in this thread.
(full link): [url=http://planetmath.org/?op=getobj&from=objects&id=3951[/url]
The order of operations is a convention that tells us how to evaluate mathematical expressions (these could be purely numerical). The problem arises because expressions consist of operators applied to variables or values (or other expressions) that each demand individual evaluation, yet the order in which these individual evaluations are done leads to different outcomes.
A conventional order of operations solves this. One could technically do without memorizing this convention, but the only alternative is to use parentheses to group every single term of an expression and evaluate the innermost operations first.
The nearly universal mathematical convention dictates the following order of operations (in order of which operators should be evaluated first):
Factorial.
Exponentiation.
Multiplication.
Division.
Addition.
Any parenthesized expressions are automatically higher ``priority'' than anything on the above list.
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I want to stress that order of operations and symbolic representation of operators has nothing to do with actual math. These are tools used to help other people understand your mathematical idea. When I'm brainstorming a math problem I use my own set of symbols and notation to streamline some of the more repetitive chores.
As a rule of thumb, if you can't write a fraction with a horizontal bar, do it using a parentheses and an exponent; "/" is always ambiguous, and the 3rd grader division sign is even worse. Remember guys, when using a keyboard to write down math, be extra careful and remember to use these () to contain an operation or even add a few extra spaces to separate: (42/2) * (9+3). I think you can even do without using * or /. (2^(-1))(42)(9+3) remove all ambiguity and removes the need for order in doing operations. Your new mnemonic can now just be P for parentheses.
People need to understand that the algorithmic part of math (i.e arithmetic, taking integrals, calculating eigenvalues) is worthless when you don't know what the idea as a whole means. I think i'm starting to understand why so many people hate math. It's because what they think is math, is nothing more than what a compute/abacus/sand table/calculator does.
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On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar.
Order of operations.
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x =2
x = 48 / 24
x = 48 / (2*12)
x = 48 / 2(9+3)
but 48 / 2(9+3) = 288 ?!
And thats why we always (commonly) treat single groups like 2(x+y) with priority. Also DO note that equal operations can be done in random order and the result will NOT change, regardless if it's done left-to-right, right-to-left or by god-know-what-order.
For example: a*b*c = b*a*c.
Yes, you can argue all day long, that technically the answer is 288 (I am forced to agree here), but nothing changes the fact, that equation is poorly written and ambiguous. Therefore is also semantic issue as well.
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Bosnia-Herzegovina114 Posts
I voted 2 because ... *shrug* ... who thinks when surfing?
Correct answer is 288.
(source)
http://www.wolframalpha.com/input/?i=48÷2(9+3)
Ultimately I agree that this is a failure of the writer (and that no one would seriously write this question this way). The syntax does give one correct answer, but the writer had multiple choices of how to write the question and chose the most misleading (though still technically correct) way.
Agreed.
The only reason there's confusion is because people who went past age12 maths began to completely lose the use of the ÷ sign. So that sign no longer has meaning.
I even thought that was a + for a moment!
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On April 08 2011 17:44 space_yes wrote:Show nested quote +On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar. Order of operations. That doesn't solve the question :S It's not a matter of order of operations it's a matter of interpretating "1/2x" as "1/(2x)" or "(1/2)x". You can get either 2 or 288 using the right order of operations. Those quoting the order are just completely missing what the ambiguity is here.
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Germany2896 Posts
Wikipedia is not normative. And the planetmath link doesn't even touch the subject of omitted multiplication before brackets/variables.
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On April 08 2011 17:49 VIB wrote:Show nested quote +On April 08 2011 17:44 space_yes wrote:On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar. Order of operations. That doesn't solve the question :S It's not a matter of order of operations it's a matter of interpretating "1/2x" as "1/(2x)" or "(1/2)x". You can get either 2 or 288 using the right order of operations. Those quoting the order are just completely missing what the ambiguity is here.
Isnt it a matter of interpretating the order of quotations?
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On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar.
That's the whole...like 100% core point of this thread. Same as in any language, there can, by definition, never exist any "official" notation of anything. Why?
Because, let's say we all agree that 2(9) is the same as 2*(9). But over the course of time, people distinguish between those two, ignoring the "rule". Then the rule itself loses all its meaning...same with grammar/etc. Language - and here ALSO the language of math - is always reliant on the society, the people who use the language. Therefore the OP has rightly shown that the language used in the OP is misleading because it can, in fact, be interpreted both ways. Depending on the "school of thought", if you wanna call it that way, that you are following.
In this respect this is a great thread, because it shows the uselessness of official notational rules if the "users" themselves partially ignore them and get so used to a "wrong" notation, that this "wrong" notation in fact becomes "correct".
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On April 08 2011 17:47 shadowy wrote:
x =2
x = 48 / 24
x = 48 / (2*12)
x = 48 / 2(9+3)
but 48 / 2(9+3) = 288 ?!
And thats why we always (commonly) treat single groups like 2(x+y) with priority. Also DO note that equal operations can be done in random order and the result will NOT change, regardless if it's done left-to-right, right-to-left or by god-know-what-order.
For example: a*b*c = b*a*c.
Yes, you can argue all day long, that technically the answer is 288 (I am forced to agree here), but nothing changes the fact, that equation is poorly written and ambiguous. Therefore is also semantic issue as well.
Note to thread: the question is supposed to be "poorly written and ambiguous." The point of the question is to resolve the ambiguity. Do you really think the questions (48/2)(9+3) or (1/2)(x) warrant their own thread?
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On April 08 2011 17:47 shadowy wrote:
x =2
x = 48 / 24
x = 48 / (2*12)
x = 48 / 2(9+3)
but 48 / 2(9+3) = 288 ?!
And thats why we always (commonly) treat single groups like 2(x+y) with priority. Also DO note that equal operations can be done in random order and the result will NOT change, regardless if it's done left-to-right, right-to-left or by god-know-what-order.
For example: a*b*c = b*a*c.
Yes, you can argue all day long, that technically the answer is 288 (I am forced to agree here), but nothing changes the fact, that equation is poorly written and ambiguous. Therefore is also semantic issue as well.
x = 48 / (2*12)
x = 48 / (2(9+3))
Why do you eliminate the parentheses?
btw 48(9+3)÷2 is imo a better way to write the term without adding parentheses
And for the 1/2x. I would always write x/2 if I mean (1/2)x therefore I see it as 1/(2x) even though it probably doesn't make too much sense.
For me:
48÷2(9+3) = 288
48÷2x while x = (9+3) = 2 -> if it should be 288 here I would write 48x÷2
48÷2(x) while x = (9+3) = 288
48÷2(12) = 288
I wonder if anyone else thinks the same.
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Germany2896 Posts
On April 08 2011 17:53 sleepingdog wrote:Show nested quote +On April 08 2011 17:39 MasterOfChaos wrote:
Is there any official specification of mathematical notation? This is an argument about the grammar of mathematical notation. "What operator precedence does an omitted multiplication sign in front of a opening bracket have?" So the only way to resolve it absolutely is finding a normative version of that grammar. That's the whole...like 100% core point of this thread. Same as in any language, there can, by definition, never exist any "official" notation of anything. Why? Because, let's say we all agree that 2(9) is the same as 2*(9). But over the course of time, people distinguish between those two, ignoring the "rule". Then the rule itself loses all its meaning...same with grammar/etc. Language - and here ALSO the language of math - is always reliant on the society, the people who use the language. Therefore the OP has rightly shown that the language used in the OP is misleading because it can, in fact, be interpreted both ways. Depending on the "school of thought", if you wanna call it that way, that you are following. In this respect this is a great thread, because it shows the uselessness of official notational rules if the "users" themselves partially ignore them and get so used to a "wrong" notation, that this "wrong" notation in fact becomes "correct".
For natural languages that's obviously true. For programming languages it's almost never true.
And it would make sense for some mathematical association to define a well defined grammar for mathematical notation. In absence of a normative specification some convention becomes correct one most influential practitioners interpret it the same way.
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I got 288 and 1/(2*x) and im a 10th grade dropout.
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EPT
