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davoid
Android Expert
Then why does it exist?
The
But, as I see it, there is no ambiguity. There's only PERCEIVED ambiguity by people who don't understand how math problems are supposed to be solved!
What's the effective difference between ambiguity and perceived ambiguity? Either way, people are needlessly confused.
I took one look at this expression and knew that the correct answer was 9, but I also saw how it was intentionally written to be tricky. That's why I posted the XKCD comic on the previous page. People complain that math is obtuse. I think those people are wrong (I have a computer science degree with a math minor), but things like this certainly don't help our cause.
One extra set of parentheses around the 6/2 eliminates most of the ambiguity displayed here (even though the ambiguity results from a flawed understanding of the order of operations) at the cost of being a little bit redundant. I think that's a reasonable trade.
CafeKampuchia
Android Expert
FYI, Python says 9 as does PEMDAS and everyone else who answered correctly 
Seriously, it seems to me that this is less a math question and more a sociology question.
Seriously, it seems to me that this is less a math question and more a sociology question.
huh
Android Expert
hmmm note to self...add math to the list of topics
I should probably stay away from ..
speaking of sociology...
I love a good.. intelligent back and forth banter...and YES i just said banter!!!
but since forum is avatar's .. not in person..psuedo-anonymous ... and I'm a girl.. I don't have use of all my "weapons" so to speak...
...I am at a disadvantage
unless anyone wants to discuss game theory..
I should probably stay away from ..
speaking of sociology...
I love a good.. intelligent back and forth banter...and YES i just said banter!!!
but since forum is avatar's .. not in person..psuedo-anonymous ... and I'm a girl.. I don't have use of all my "weapons" so to speak...
...I am at a disadvantage
unless anyone wants to discuss game theory..
Bob Maxey
Android Expert
Hi all. I researched this topic well for about 2 weeks now and have come to the following conclusions. I will summarize what I have said in other forums with respect to the notations, then I will address other points.
First,
if you want to say 0.5x, then you HAVE to write (1/2)x with parentheses or, x "all over 2" with a horiztonal fraction bar, or write x/2. I have never seen (1/2)x before I researched this equation, but since searching online, I HAVE seen fractional coefficients written this way, only because computers are limited to the horizontal typing space.
Therefore:
x/2 = (1/2)x = 0.5x
1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2.
I have always used ab/cd to mean (ab)/(cd) and I topped almost all of my calculus classes since high school through university.(moot point, I know)
Just to re-iterate, to use 6/2 as a fraction, parentheses are REQUIRED. Every book will tell you this.
Now consider the Identity Law:
a = 1a = 1(a)
We know there is ALWAYS an 'invisible' 1 as a ceofficient of a variable if no other number is there. Therefore:
a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using 'pemdas', some folks would do this:
a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.
Now, on to my second point:
consider: factoring, simplifying equations, and the distributive property.
Lets start with the number 6.
6 = (4+2). There is a common factor here: 2. So let's factor it out of both terms.
(4+2) = 2(2+1). The outside 2 remains a part of of the 2 inner terms at all times. It cannot be used in an operation by itself without the rest of (4+2). The reverse of factoring is distribution, so, 2(2+1) = 6. This has to be true always. The argument I have seen to this is that (6/2) can be distributed. This is true ONLY is 6/2 is in parentheses, otherwise, the 6 and 2 are separated by a division slash, and the 2 is a factor of 2+1.
So, let's prove the initial equation:
6/6 = 1
6/(4+2) = 1
6/2(2+1) = 1
the same can be done for other factors:
6/6 = 1
6/(3+3) = 1
6/3(1+1) = 1
Distribution is actually a part of "Simplifying Equations" and is not bound to the order of operations as "multiplication", since it is in fact "removing parentheses by distributing". This can be googled and several references found.
Simplifying 2(2+1) + 3(2+1) = 5(2+1). We "combined like terms" here, by adding, and did not perform the "parntheses" part of order of operations, nor did we multiply, which is also higher priority than adding, because we only simplified.
Lastly, I hear the argument that "This is strictly numbers and you don't use algebra rules since there are no variables". That is the most asinine arguement I have heard yet. All axioms, laws, and properties use variables, meaning that they hold true for "any number", hence the proofs with variables.
I welcome thoughts on this, in an intellectually formed response. I am tired of the 'flaming' that goes on by imbciles on some other forums with rebuttals like "it is 9. go back to grade 3 you moron", or "google says it is 9", when google changes the equation to (6/2)*(2+1), and wolfram contradicts itself with 2n/2n = 1, and 6/2n = 3/n, but then says 6/2(2+1) is 9. wolframs "terms" state that any answer should be verified with common sense and accuracy should also be verified.
First,
if you want to say 0.5x, then you HAVE to write (1/2)x with parentheses or, x "all over 2" with a horiztonal fraction bar, or write x/2. I have never seen (1/2)x before I researched this equation, but since searching online, I HAVE seen fractional coefficients written this way, only because computers are limited to the horizontal typing space.
Therefore:
x/2 = (1/2)x = 0.5x
1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2.
I have always used ab/cd to mean (ab)/(cd) and I topped almost all of my calculus classes since high school through university.(moot point, I know)
Just to re-iterate, to use 6/2 as a fraction, parentheses are REQUIRED. Every book will tell you this.
Now consider the Identity Law:
a = 1a = 1(a)
We know there is ALWAYS an 'invisible' 1 as a ceofficient of a variable if no other number is there. Therefore:
a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using 'pemdas', some folks would do this:
a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.
Now, on to my second point:
consider: factoring, simplifying equations, and the distributive property.
Lets start with the number 6.
6 = (4+2). There is a common factor here: 2. So let's factor it out of both terms.
(4+2) = 2(2+1). The outside 2 remains a part of of the 2 inner terms at all times. It cannot be used in an operation by itself without the rest of (4+2). The reverse of factoring is distribution, so, 2(2+1) = 6. This has to be true always. The argument I have seen to this is that (6/2) can be distributed. This is true ONLY is 6/2 is in parentheses, otherwise, the 6 and 2 are separated by a division slash, and the 2 is a factor of 2+1.
So, let's prove the initial equation:
6/6 = 1
6/(4+2) = 1
6/2(2+1) = 1
the same can be done for other factors:
6/6 = 1
6/(3+3) = 1
6/3(1+1) = 1
Distribution is actually a part of "Simplifying Equations" and is not bound to the order of operations as "multiplication", since it is in fact "removing parentheses by distributing". This can be googled and several references found.
Simplifying 2(2+1) + 3(2+1) = 5(2+1). We "combined like terms" here, by adding, and did not perform the "parntheses" part of order of operations, nor did we multiply, which is also higher priority than adding, because we only simplified.
Lastly, I hear the argument that "This is strictly numbers and you don't use algebra rules since there are no variables". That is the most asinine arguement I have heard yet. All axioms, laws, and properties use variables, meaning that they hold true for "any number", hence the proofs with variables.
I welcome thoughts on this, in an intellectually formed response. I am tired of the 'flaming' that goes on by imbciles on some other forums with rebuttals like "it is 9. go back to grade 3 you moron", or "google says it is 9", when google changes the equation to (6/2)*(2+1), and wolfram contradicts itself with 2n/2n = 1, and 6/2n = 3/n, but then says 6/2(2+1) is 9. wolframs "terms" state that any answer should be verified with common sense and accuracy should also be verified.
jhawkkw
Chinchillin'
Just to make everyone aware, to finally put this question to rest, there are two pitfalls in this question that will lead you to the wrong answer.
1) Because it's written as PEMDAS, that implies that multiplication has higher precedence than division. But in actuality, they have the same precedence.
2) 1/2x = 1/(2x). This is actually not true because as state above, multiplication and division have the same precedence, and thus must be done in order from left to right. Therefore, 1/2x = (1/2)x = 0.5x.
Your understanding of math does seem to be quite good. And your arguments hold true under the assumption of 1/2x = 1/(2x). Unfortunately this is not true and is a common misconception due to the limitation of typing. I'm sure if it was written out on paper that many people here would not be falling into these pitfalls because it would be much more obvious because you can actually use fractions lines as well as showing actual numerators and denominators. Maybe this is what the OP meant when he wrote the equation and forgot to add the parenthesis, or maybe they left out in order to create this pitfall. The reason Wolfram or any graphing calculator gives the answer of 9 is because they follow the order of operations down to the individual character. By adding the parenthesis to the (2x), you are not using the associative property of multiplication, you are actually altering the problem. If you were to change the equation to a pure multiplication question using the inverse property real number field under multiplication, the problem 1/2x would change to 1*(0.5)*x . This makes it much easier to see that the x is multiplied in the numerator due to both the commutative and associative property of multiplication.
You're example using the identity property of a/1a equals a^2 under the standard order operations. However a/(1a) does equal 1 as long as a =/= 0. But once again, this relies on the condition mentioned above holding true. As for the cases with pi and e, this is commonly accepted due to laziness, but a real stickler of a professor like myself would take off points for that
It is interesting that wolfram contradicts itself like that. Should raise that with their programmers. If I type 2x/2x on my ti-89, I get x^2
1) Because it's written as PEMDAS, that implies that multiplication has higher precedence than division. But in actuality, they have the same precedence.
2) 1/2x = 1/(2x). This is actually not true because as state above, multiplication and division have the same precedence, and thus must be done in order from left to right. Therefore, 1/2x = (1/2)x = 0.5x.
Your understanding of math does seem to be quite good. And your arguments hold true under the assumption of 1/2x = 1/(2x). Unfortunately this is not true and is a common misconception due to the limitation of typing. I'm sure if it was written out on paper that many people here would not be falling into these pitfalls because it would be much more obvious because you can actually use fractions lines as well as showing actual numerators and denominators. Maybe this is what the OP meant when he wrote the equation and forgot to add the parenthesis, or maybe they left out in order to create this pitfall. The reason Wolfram or any graphing calculator gives the answer of 9 is because they follow the order of operations down to the individual character. By adding the parenthesis to the (2x), you are not using the associative property of multiplication, you are actually altering the problem. If you were to change the equation to a pure multiplication question using the inverse property real number field under multiplication, the problem 1/2x would change to 1*(0.5)*x . This makes it much easier to see that the x is multiplied in the numerator due to both the commutative and associative property of multiplication.
You're example using the identity property of a/1a equals a^2 under the standard order operations. However a/(1a) does equal 1 as long as a =/= 0. But once again, this relies on the condition mentioned above holding true. As for the cases with pi and e, this is commonly accepted due to laziness, but a real stickler of a professor like myself would take off points for that
It is interesting that wolfram contradicts itself like that. Should raise that with their programmers. If I type 2x/2x on my ti-89, I get x^2
Discussing the "software issue" is another thread of its own. My point there was to prove that google and wolfram are not valid 'arguments' for proof of 9. The notation issue, meh, it was something we used in the 90s during my higher level education. Although, I did mention I have a textbook reference that specifically states:
1 "all over 2n" (using a horizontal fraction line) is simply 1/2n
I didn't make it up.
All that aside, care to comment on the rest of my post (simplifying, eliminating parentheses with distribution, the 'proof' where 6
1 "all over 2n" (using a horizontal fraction line) is simply 1/2n
I didn't make it up.
All that aside, care to comment on the rest of my post (simplifying, eliminating parentheses with distribution, the 'proof' where 6
jhawkkw
Chinchillin'
jhawkkw
Chinchillin'
I didn't overthink it all. The thinking part was to try and figure out why anyone would say 9, which led me on a crusade of research. I still have found anything difinitive, especially when you think about like:You have 6 apples. You divide them among 2 groups of kids, each group with 2 girls and one boy. how many do each get? It is the grouping of the parentheses, with distribution, and lack of ( ) around the 6 & 2. 2 groups of 2+1 kids are 6.
Also, isn't 2 of 2+1 = (2+1)+(2+1) ?
There are just so many "logical" ways of looking at, which is what math is supposed to represent
Cheers!
jhawkkw
Chinchillin'
Yeah, my grammar was a little terrible. Basically what I meant is if the whole 2(2+1) is in the denominator, then the answer is indeed 1.I think there was a typo in there
Indeed that is the case. The easiest way to see that by the use of the definition of division of real numbers, and the associative property of multiplication. That definition is a
davoid
Android Expert
jhawkkw
Chinchillin'
Oh, I know the principles and laws of reciprocity, etc
Heh heh! That is exactly what I am saying. Order of operations: I am completely fine with. It is the things that you are operating on that I am also fine with, but many others are not. They are doing just that: Translating everything into an operator. 2x ? It is 2 x's. 2 cars? yup, 2 cars. We don't say "Hey, did you see the 2 times red cars drive by" ! It is a quantity, just like one mole of something, or a dozen.
24g
jhawkkw
Chinchillin'
Thanks pal. And you too! Nothing worse than "that's stupid, it's 9 you moron"I think you are the first person I've come across here that actually cited analysis, other than myself. It tends to scare most people off.
Cheers :beer:
That's it? Say something intelligent at the very least !!
Did you happen to open that PDF and check out page 53 on it? NO big deal if you didn't. The authors say that 1 "all over" 2n is simply 1/2n. it was used in a Series: 1/2n where n is a member of Natural numbers. It was one example I found. Most text use horizontal fraction lines, as they should.
Regards my friend...
CafeKampuchia
Android Expert
Why the continued fuss? The correct answer was given several posts ago.
