View Poll Results: 6÷2(1+2) = ?

9


94 
54.34% 
1


77 
44.51% 
7


2 
1.16% 
27Likes


January 11th, 2013, 08:55 AM

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Quote:
Originally Posted by huh
yikes!! I was just coming back on to say I think thatI may have sounded all stiff and snooty and maybe even a little b I didnt mean it that way.. in fact I don't even like math!!

No worries



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January 11th, 2013, 09:12 AM

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Quote:
Originally Posted by Petrah
I slept in math, so used Google Now to do it for me.

what does Google Now say?
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January 11th, 2013, 11:13 AM

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Quote:
Originally Posted by MoodyBlues
Then why does it exist?

The ÷ symbol is basically used for young children. I found a document on the berkley.edu site, discussing 'order of operations' in schools, of which this is an extract:
And notice the reference to a kind of parlor game designed to trap people by phrasing in "unreasonably convoluted" terms.
Quote:
Originally Posted by MoodyBlues
The 'order of operations' handles linear formulas just fine.

The document also says that the Rules for the Order of Operations
Quote:
should be applied judiciously and never at the
expense of clarity.

For me, clarity is the issue here.
Quote:
Originally Posted by MoodyBlues
Neither is the way it was written in this thread.

Well... I personally read it the wrong way so it's a matter of subjective opinion.
Quote:
Originally Posted by MoodyBlues
It's not that unusual to see something written like that.

Well it's the first time for me...
Quote:
Originally Posted by MoodyBlues
Hey, I said "as I see it"!




January 11th, 2013, 02:52 PM

#304 (permalink)

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Quote:
Originally Posted by MoodyBlues
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.



January 11th, 2013, 04:09 PM

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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.
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January 11th, 2013, 09:24 PM

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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..psuedoanonymous ... 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..


Last edited by huh; January 11th, 2013 at 09:24 PM.
Reason: wait!!! Also because I suck at math!!!!


January 12th, 2013, 06:46 PM

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Quote:
Originally Posted by MoodyBlues
So the statement is that there are "three words in the English language that end in 'gry.'" Not "there are three words in 'the English language.'" And the question being asked refers to the three words that end in "gry""Angry" and "hungry" are two. What's the third?
Unless I'm having a major brain fart (and that's ENTIRELY possible!), it just doesn't make sense.

However, the problem still remains: what is the answer:
An÷gry(2+hungry) = ? and remember, we are talking about an equation that only can be solved in a partial vacuum.
I say it is 12, but I failed InterMath.



January 13th, 2013, 02:34 AM

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huh?



January 15th, 2013, 11:30 AM

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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 reiterate, 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.



January 15th, 2013, 12:13 PM

#310 (permalink)

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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.
Quote:
Originally Posted by mathman26
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.
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.

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 ti89, I get x^2
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Last edited by jhawkkw; January 15th, 2013 at 12:17 PM.


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January 15th, 2013, 12:31 PM

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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÷6=1 originates, etc) ?
PS: I also believe there are more "9" answers since most people simply know "pemdas" and don't remember distributive property nor how to properly eliminate parentheses and the like. It is like asking grade 2 students what is the answer to 2+1*2. "Most" if not all will say 6, but this isn't a "majority wins" contest. Parentheses are a "grouping" symbol, and the question is read aloud as "What is 6 divided by 2 groups of 2+1"



January 15th, 2013, 10:15 PM

#312 (permalink)

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by mathman26
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÷6=1 originates, etc) ?
PS: I also believe there are more "9" answers since most people simply know "pemdas" and don't remember distributive property nor how to properly eliminate parentheses and the like. It is like asking grade 2 students what is the answer to 2+1*2. "Most" if not all will say 6, but this isn't a "majority wins" contest. Parentheses are a "grouping" symbol, and the question is read aloud as "What is 6 divided by 2 groups of 2+1"

Under modern day standard, the / and ÷ are used interchangeably as division. This is probably due to the increase in computer programming. Because of this the / sign has lost it's meaning of fraction when written in horizontal form and more assumed to be division. However since the original problem used the ÷ and not the / sign, it should assumed that it is not actually a fraction. Therefore there should be no issue simply applying the standard order of operations.
For a little off topic banter, I recently did a puzzle where the order of operations precedence changed, where AS comes before MD and the results are quite interesting. In one question, a linear equation actually produces TWO and only two valid solutions.


Last edited by jhawkkw; January 16th, 2013 at 12:37 AM.


January 16th, 2013, 02:01 PM

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I cannot accept 6 ÷ 2x as 3x, though. Blindly using order of operations, I guess you could write 6 ÷ 2 * 3, but that is not what it asking. "2x" is a single term, as defined by many algebra books, therefore, there are 2 terms here. 6 & 2x. hence, 6 ÷ 2x = 3/x. People are translating 2x into 2 * x. 2 * x however, are 2 terms separated by an operator, and 2x is one term. Evaluated the same? Yes. But they mean 2 different things from every book I have used in my studies. And, if 2x is one term, then 6 ÷ 2x = 6 "allover 2x" or 6/(2x). Let x = 1+2, and the answer is one. One would have to disprove 2x is a single term.
How about a problem with area of a rectange:
http://i45.tinypic.com/v43qiu.jpg
The area is 2(2+1) sq ft. When asked how many times it can fit another area, the units should cancel out.
6 sq ft ÷ 2(2+1) sq ft = 1
If you tried it the other way:
6 sq ft ÷ 2 ft * (2+1) ft = 9 ft. It can fit into 6 sq ft, 9 ft times ?
I am trying to drive home the point that 2(2+1) is one value, along with the example of factoring/distributing.
In that picture, I am also trying to reinforce that a fractional coefficient absolutely requires parentheses. This notation is undoubtedly used in all texts that use the / as a fraction bar.
Thoughts?



January 16th, 2013, 02:06 PM

#314 (permalink)

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by mathman26
I cannot accept 6 ÷ 2x as 3x, though. Blindly using order of operations, I guess you could write 6 ÷ 2 * 3, but that is not what it asking. "2x" is a single term, as defined by many algebra books, therefore, there are 2 terms here. 6 & 2x. hence, 6 ÷ 2x = 3/x. People are translating 2x into 2 * x. 2 * x however, are 2 terms separated by an operator, and 2x is one term. Evaluated the same? Yes. But they mean 2 different things from every book I have used in my studies. And, if 2x is one term, then 6 ÷ 2x = 6 "allover 2x" or 6/(2x). Let x = 1+2, and the answer is one. One would have to disprove 2x is a single term.
How about a problem with area of a rectange:
http://i45.tinypic.com/v43qiu.jpg
The area is 2(2+1) sq ft. When asked how many times it can fit another area, the units should cancel out.
6 sq ft ÷ 2(2+1) sq ft = 1
If you tried it the other way:
6 sq ft ÷ 2 ft * (2+1) ft = 9 ft. It can fit into 6 sq ft, 9 ft times ?
I am trying to drive home the point that 2(2+1) is one value, along with the example of factoring/distributing.
In that picture, I am also trying to reinforce that a fractional coefficient absolutely requires parentheses. This notation is undoubtedly used in all texts that use the / as a fraction bar.
Thoughts?

There is a lot of fact in there, so I'll address everything both correct and incorrect.
2x is indeed one term, otherwise know as a monomial. However, 6/(2x) and (6/x)2 are both also considered to be monomials. What is the splitter of terms is addition/subtraction. Just because there is an operator between them, doesn't mean that it breaks it into 2 terms. A perfect example is exponentiation. x^2 is considered to be one term, but x^2 = x*x. The same applies to 2x; 2x = 2*x = x + x. Simplifying x+x is referred to as combining like terms. However simplifying x*x is not because multiplication is not a splitter of terms. It's weird and seems arbitrary, but it is what it is. Blame ancient Babylonians and ancient Greeks.
I'm am not disputing at all that 6/(2x) where x = 2+1 isn't 1 at all; it is absolutely equal to 1. The discrepancy occurs because the way that the equation is written, the (2+1) is actually on the side of 6/2 fraction and thus making multiplication of the numerator and not the denominator. Also, there is no such thing as "blindly" using order of operations. They're absolute unless second hand stipulations have been stated in the problem. It's not something that comes with conditions on where it works and where it doesn't. Division and exponentiation do have conditions where they fail (division by zero, negative base of a rational exponent with an even denominator), but the order of operations do not. If logic dictates that the order of operations are wrong, then you add parenthesis. That is why they are given highest priority, and their purpose over all: to alter the order of operations.
As for your example, they are actually two different questions that are trying to be posed as the same. In fact, there is an inconsistency that causes a contradiction. In the left picture, the width is 2ft, but in the right picture it is 3. The left picture calculates the area beforehand and poses the question and asks how many times the area can be divided by itself. The right picture asks to calculate the area of a new rectangle. The logic of the first question is correct, but the notation is wrong. Even though the problem does require the multiplication to occur before the division, the notation signals that the division must occur before the multiplication. The correct equation should be written as 6 ÷ [2(2+1)].


Last edited by jhawkkw; January 16th, 2013 at 02:39 PM.


January 16th, 2013, 04:56 PM

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Quote:
Originally Posted by jhawkkw
I'm am not disputing at all that 6/(2x) where x = 2+1 isn't 1 at all; it is absolutely equal to 1.

I think there was a typo in there Please clarify
Quote:
As for your example, they are actually two different questions that are trying to be posed as the same. In fact, there is an inconsistency that causes a contradiction. In the left picture, the width is 2ft, but in the right picture it is 3. The left picture calculates the area beforehand and poses the question and asks how many times the area can be divided by itself. The right picture asks to calculate the area of a new rectangle. The logic of the first question is correct, but the notation is wrong. Even though the problem does require the multiplication to occur before the division, the notation signals that the division must occur before the multiplication. The correct equation should be written as 6 ÷ [2(2+1)].

The picture I showed you is 2 different problems. I was only trying to show how different the equations really mean. I didn't mean for you to try and understand them as the same. Sorry about that.
So then, what I gather, from your post, is 6/2(2+1) = (6/2)(2+1) ??
I have a REALLY hard time wrapping my head around that. Every reference I have seen, and I mean every, uses parentheses for a fractional coefficient.
And... if n = 1n, then n/1n = n^2. Just because we do not write "1" as a coefficient, does not mean it is not there. Therefore 1n/1n SHOULD = n/n = 1, but, according to 'order of operations' 1n/1n = n^2 ?? This is why I cannot get my head to accept 6/2n = 3n. To me it is 3/n. Just as 6/2(2+1) is 6/6. Why would anyone write this: 6/2(2+1) to mean 9, when it would be written either
1  (6÷2)(2+1) = 9
2  6(2+1)÷2 = 9
Those are clear as the day is long , just as 6÷2(2+1) is one is clear to me.
I look forward to you response on:
6 ÷ 3x = ?
a/a or 1a/1a; and
the rectangle problem with respect to the "multiplication" required to be first. The question I have is: Isn't it already computed ? 2(2+1) square feet?



January 16th, 2013, 04:57 PM

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OK I know I said I wouldn't....but
sometimes we "over think" things and make them more complicated than they actually are
and then we don't see the forest for the trees..so to speak
It is what it is..
9
:0



January 16th, 2013, 05:04 PM

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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!



January 16th, 2013, 05:29 PM

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January 16th, 2013, 06:14 PM

#319 (permalink)

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Quote:
Originally Posted by mathman26
I think there was a typo in there Please clarify

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.
Quote:
Originally Posted by mathman26
The picture I showed you is 2 different problems. I was only trying to show how different the equations really mean. I didn't mean for you to try and understand them as the same. Sorry about that.
So then, what I gather, from your post, is 6/2(2+1) = (6/2)(2+1) ??
I have a REALLY hard time wrapping my head around that. Every reference I have seen, and I mean every, uses parentheses for a fractional coefficient.

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÷b = (a)*(1/b) where (1/b) is a fraction also known as the reciprocal of b. This is the common technique often taught for dividing fractions. (1/2) ÷ (2/3) = (1/2) * (3/2) = (3/4) is an example of this. So applying this property, 6÷2(2+1) = 6*(1/2)(2+1) = 6*.5(2+1). From here, it's much easier to see how one comes up with 9. Using the associative property, which states that you can change the parenthesis order for multiplication only and it doesn't change the outcome, one can come up with (6*.5)(2+1). Going backward across the definition of division, you would get back to (6/2)(2+1). As for using parenthesis for a fractional coefficient, just take a look at the original problem in the title of this thread; they use the ÷ sign, not the / sign. So the OP didn't even signal fraction and meant actual division either by accident or design.
Quote:
Originally Posted by mathman26
And... if n = 1n, then n/1n = n^2. Just because we do not write "1" as a coefficient, does not mean it is not there. Therefore 1n/1n SHOULD = n/n = 1, but, according to 'order of operations' 1n/1n = n^2 ?? This is why I cannot get my head to accept 6/2n = 3n. To me it is 3/n. Just as 6/2(2+1) is 6/6. Why would anyone write this: 6/2(2+1) to mean 9, when it would be written either
1  (6÷2)(2+1) = 9
2  6(2+1)÷2 = 9
Those are clear as the day is long , just as 6÷2(2+1) is one is clear to me.
I look forward to you response on:
6 ÷ 3x = ?
a/a or 1a/1a; and
the rectangle problem with respect to the "multiplication" required to be first. The question I have is: Isn't it already computed ? 2(2+1) square feet?

As for 1a/1a or 6 ÷ 3x, these are cases where the letter of the law completely contradicts logic. If you were to follow PEMDAS exactly, 1a/1a = a^2; just like 6 ÷ 3x = 2x. But just like calculators, the order of operations doesn't think logically and we as humans can see that the intent of the 1a/1a is suppose to be (1a)/(1a) = 1. That is the only way PEMDAS or a calculator or a computer is going to get the correct answer, just like 6 ÷ 3x should be written/typed 6 ÷ (3x). It's stupid, and I have been the victim of these stupid semantics like this typing up programs to solve problems in the past.
I think what is making it difficult to accept is even though 2x is considered to be 1 term, it really is 2*x. Let's say you were given the value of x for example to be 3. You wouldn't say the answer is 23, you would say it's 2*3 = 6. So even though it's written next to it as a coefficient, it's multiplication and not concatenation. Therefore 6 ÷ 3x is actually equal to 6 ÷ 3 * x. It looks dumb and is not how it's normally written, but that's what it is really saying. Extending that to the original question, the 6 ÷ 2(2+1) should be written as 6 ÷ 2 * (2+1). Obviously the () gets done first, giving 6 ÷ 2 * 3. Now we're in the above situation where PEMDAS says division happens first because it's what occurs first in the problem due to equal precedence.
There was another Math debate thread a little while ago, would like to get you take on the .9999....... = 1 topic. That one was interesting to say the least....



January 16th, 2013, 07:59 PM

#320 (permalink)

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Quote:
Originally Posted by jhawkkw
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÷b = (a)*(1/b) where (1/b) is a fraction also known as the reciprocal of b. This is the common technique often taught for dividing fractions. (1/2) ÷ (2/3) = (1/2) * (3/2) = (3/4) is an example of this. So applying this property, 6÷2(2+1) = 6*(1/2)(2+1) = 6*.5(2+1). From here, it's much easier to see how one comes up with 9. Using the associative property, which states that you can change the parenthesis order for multiplication only and it doesn't change the outcome, one can come up with (6*.5)(2+1). Going backward across the definition of division, you would get back to (6/2)(2+1).

This I can follow, but the problem is that I see it that where
a÷b = (a)*(1/b), then a=6 and b=2(2+1). This is where the differing solutions diverge, because you see it as a=6 and b=2.
I agree that multiplying by the reciprocal of b is the way to solve the problem...
1 = 0.9999? bring it on!



January 16th, 2013, 08:49 PM

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Quote:
Originally Posted by jhawkkw
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÷b = (a)*(1/b) where (1/b) is a fraction also known as the reciprocal of b. This is the common technique often taught for dividing fractions. (1/2) ÷ (2/3) = (1/2) * (3/2) = (3/4) is an example of this. So applying this property, 6÷2(2+1) = 6*(1/2)(2+1) = 6*.5(2+1).....<snip>

Oh, I know the principles and laws of reciprocity, etc And to me, 6/2n = 6 * (2n)^1
Quote:
As for 1a/1a or 6 ÷ 3x, these are cases where the letter of the law completely contradicts logic. If you were to follow PEMDAS exactly, 1a/1a = a^2; just like 6 ÷ 3x = 2x. But just like calculators, the order of operations doesn't think logically and we as humans can see that the intent of the 1a/1a is suppose to be (1a)/(1a) = 1. That is the only way PEMDAS or a calculator or a computer is going to get the correct answer, just like 6 ÷ 3x should be written/typed 6 ÷ (3x). It's stupid, and I have been the victim of these stupid semantics like this typing up programs to solve problems in the past.

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 ÷ 1 mole = 24g ÷ 6.022x10^23
not (24 ÷ 6.022) * 10^23
ab/cd ? (ab)/(cd)
If I wanted abc/c, I write just that.
same for (1/2)x, is x/2
1/2x? 1 ÷ 2x
Just like (6/2)(2+1). I would write that, or i would say 6(2+1) ÷ 2
I apologize if it seems like a little rant
Quote:
...<snip> Therefore 6 ÷ 3x is actually equal to 6 ÷ 3 * x. It looks dumb and is not how it's normally written, but that's what it is really saying.

I think it is saying "six divided by 3 x's" and not "6 divided by 3, times x"
As I said, this was the intention of the meaning in 5 schools I went to.
6/3x is/was 2/x, otherwise you write 6x/3, or 2x
Once again a ÷ 1a = ? We both know it is one, but we have brains for a reason, and we know the 1a is a single unit, which translates to 6 ÷ 2n, where '2n' is a single unit, and n is 2+1.
I don't see how anyone can prove that a ÷ 1a = 1 and 6 ÷ 2a = 3a
NOW.... we didn't use calculators in most of my school. It was all on paper, and calculators were forbidden, so we weren't victims of having to input it into a computer. I can't remember when we used them. We had to "prove" all work. When you realize how a computer/calculator interprets things, usually incorrectly, you end up putting ( ) around EVERYthing when you input.
That said, it seems like our discussion is heading in the direction of, how would a computer interpret ..... ?? That is not where I wish it to go, as I care not what computer or program will come up with, or how they will handle the arguments entered.
I am talking about : "Here is a sheet of paper. Solve this problem showing all work" Having said that, I would never be limited to same size characters on one line across and would handwrite something like this: ½ with a little 'n' right beside the 2.
Quote:
There was another Math debate thread a little while ago, would like to get you take on the .9999....... = 1 topic. That one was interesting to say the least....

LOL !! Not another one !! I will pass on that one. I can tell you the first thing that comes to mind, but that is not my "final answer", Regis!
Initial response is "no". not equal. If you use a limit, somehow, with summation and limit of f(x) as x approaching ∞, then the limit could be 1, obviously if f(x) was set up right. been a while since I was in school, so I would have to refresh all areas of that, and I am really not interested right now... I am almost finished with this one, and it was exhausting, but fun to get 'back in the books'. the annoying part was everyone calling everyone names, etc. Rarely did anyone provide a single reference. Speaking of refs, here is one:
http://faculty.ksu.edu.sa/fawaz/481files/BartleIntroductiontoRealAnalysis.pdf
Check out page 53 before and after "or more simply..."
Regards!
MM



January 16th, 2013, 09:09 PM

#322 (permalink)

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by mathman26
Oh, I know the principles and laws of reciprocity, etc And to me, 6/2n = 6 * (2n)^1
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 ÷ 1 mole = 24g ÷ 6.022x10^23
not (24 ÷ 6.022) * 10^23
ab/cd ? (ab)/(cd)
If I wanted abc/c, I write just that.
same for (1/2)x, is x/2
1/2x? 1 ÷ 2x
Just like (6/2)(2+1). I would write that, or i would say 6(2+1) ÷ 2
I apologize if it seems like a little rant
I think it is saying "six divided by 3 x's" and not "6 divided by 3, times x"
As I said, this was the intention of the meaning in 5 schools I went to.
6/3x is/was 2/x, otherwise you write 6x/3, or 2x
Once again a ÷ 1a = ? We both know it is one, but we have brains for a reason, and we know the 1a is a single unit, which translates to 6 ÷ 2n, where '2n' is a single unit, and n is 2+1.
I don't see how anyone can prove that a ÷ 1a = 1 and 6 ÷ 2a = 3a
NOW.... we didn't use calculators in most of my school. It was all on paper, and calculators were forbidden, so we weren't victims of having to input it into a computer. I can't remember when we used them. We had to "prove" all work. When you realize how a computer/calculator interprets things, usually incorrectly, you end up putting ( ) around EVERYthing when you input.
That said, it seems like our discussion is heading in the direction of, how would a computer interpret ..... ?? That is not where I wish it to go, as I care not what computer or program will come up with, or how they will handle the arguments entered.
I am talking about : "Here is a sheet of paper. Solve this problem showing all work" Having said that, I would never be limited to same size characters on one line across and would handwrite something like this: ½ with a little 'n' right beside the 2.
LOL !! Not another one !! I will pass on that one. I can tell you the first thing that comes to mind, but that is not my "final answer", Regis!
Initial response is "no". not equal. If you use a limit, somehow, with summation and limit of f(x) as x approaching ∞, then the limit could be 1, obviously if f(x) was set up right. been a while since I was in school, so I would have to refresh all areas of that, and I am really not interested right now... I am almost finished with this one, and it was exhausting, but fun to get 'back in the books'. the annoying part was everyone calling everyone names, etc. Rarely did anyone provide a single reference. Speaking of refs, here is one:
http://faculty.ksu.edu.sa/fawaz/481files/BartleIntroductiontoRealAnalysis.pdf
Check out page 53 before and after "or more simply..."
Regards!
MM

Indeed. It more or less seems that our differing of opinions seems to result from my strict interpretation of the problem vs your logical interpretation of it combined with ambiguous notation. Since you noted that calculators were forbidden while learning this material (A stance I uphold in my own classroom), I have no doubt in my mind that if this was written in on paper that our opinions would result in the same answer. Indeed calculators are very helpful, but also very dumb at the same time. 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.
As for the .999.... = 1 question, the thread was quite frustrating because most of the proofs for it require a higher understanding of infinite limits/infinitesimals. The true answer is that it is true as long as the .9 is infinitely repeating forever, but the arguments I saw in that thread were less mathematically based an more on feeling. You mentioned one of the ways to prove it in your response, taking the limit of a convergent infinite geometric series.
My analysis book is old, http://www.amazon.com/PrinciplesMathematicalAnalysisInternationalMathematics/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1358393237&sr=81&keywords=mathematical+analysis+walter+rudin
Cheers


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January 16th, 2013, 09:13 PM

#323 (permalink)

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by davoid
This I can follow, but the problem is that I see it that where
a÷b = (a)*(1/b), then a=6 and b=2(2+1). This is where the differing solutions diverge, because you see it as a=6 and b=2.
I agree that multiplying by the reciprocal of b is the way to solve the problem...
1 = 0.9999? bring it on!

.9999...=1 Here you go, topic has died off it seems.



January 16th, 2013, 10:34 PM

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Quote:
Originally Posted by jhawkkw
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

Thanks pal. And you too! Nothing worse than "that's stupid, it's 9 you moron"
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...



January 17th, 2013, 04:43 AM

#325 (permalink)

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Why the continued fuss? The correct answer was given several posts ago.



January 17th, 2013, 08:42 AM

#326 (permalink)

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by CafeKampuchia
Why the continued fuss? The correct answer was given several posts ago.

What about 42? That should have been an option



January 17th, 2013, 08:46 AM

#327 (permalink)

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@ jhawkkw : Hey bud... I just spoke with a colleague of mine, who is a Space Science specializing in Physics graduate, with lots of defence research time. We chatted this morning over this equation, agreeing you can argue either way, and quickly ended up at the same place we did "It is a crap question with arguments either way.. you would need to know the flaws in the s/w program to avoid them... and on paper we would use proper notation". That aside, he did bring this up, and I would like your take on it.
The "9" people do this: 2n = 2 * n, so
6/2(2+1) = 6 / 2 * (2+1).
But then he said he would write 2n = (2 * n) and not just 2n = 2*n
I think he is more for '9', not that it matters, but that point did strike me as interesting... thoughts?
Quote:
Originally Posted by CafeKampuchia
Why the continued fuss? The correct answer was given several posts ago.

lol @ 'correct' answer.


Last edited by mathman26; January 17th, 2013 at 08:49 AM.


January 17th, 2013, 09:06 AM

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yep....Like I said...the correct answer is 9
pieces of bacon!
4 out of 3 people just don't get math!


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January 17th, 2013, 09:21 AM

#329 (permalink)

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Quote:
Originally Posted by huh
yep....Like I said...the correct answer is 9
pieces of bacon!

Perhaps, but like Jhawkkw said, 42 slices should be an option
Quote:
Originally Posted by huh
4 out of 3 people just don't get math!

Reminds me of another math joke:
There are 10 kinds of mathematicians. Those who can think in binary and those who can't.


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January 17th, 2013, 09:27 AM

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Re: 6÷2(1+2) = ?
Quote:
Originally Posted by mathman26
@ jhawkkw : Hey bud... I just spoke with a colleague of mine, who is a Space Science specializing in Physics graduate, with lots of defence research time. We chatted this morning over this equation, agreeing you can argue either way, and quickly ended up at the same place we did "It is a crap question with arguments either way.. you would need to know the flaws in the s/w program to avoid them... and on paper we would use proper notation". That aside, he did bring this up, and I would like your take on it.
The "9" people do this: 2n = 2 * n, so
6/2(2+1) = 6 / 2 * (2+1).
But then he said he would write 2n = (2 * n) and not just 2n = 2*n
I think he is more for '9', not that it matters, but that point did strike me as interesting... thoughts?
lol @ 'correct' answer.

Indeed, it really is a crap question with designed ambiguity meant to start debates/fights. The 2n = 2*n is the strict interpretation that I mentioned above. But indeed it may not make logical sense. I don't allow calculators in my class because it does 2 things:
1)Makes people lazy so that they don't know how to do basic arithmetic
2)Can actually make people get wrong answers that they would normally get correct on paper because of that strict vs logic entering of an equation.
Like I mentioned above, the answer should just be 42 since it's the answer to the meaning of life, the universe, and everything.



January 17th, 2013, 09:53 AM

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Quote:
Originally Posted by jhawkkw
Indeed, it really is a crap question with designed ambiguity meant to start debates/fights. The 2n = 2*n is the strict interpretation that I mentioned above. But indeed it may not make logical sense. I don't allow calculators in my class because it does 2 things:
1)Makes people lazy so that they don't know how to do basic arithmetic
2)Can actually make people get wrong answers that they would normally get correct on paper because of that strict vs logic entering of an equation.
Like I mentioned above, the answer should just be 42 since it's the answer to the meaning of life, the universe, and everything.

42 pieces of pie
oh.. and I always say 6.
6 is the best and safest answer to any question..i.e how many boyfriends did you have before me? 6
If your wife or GF asks you if you think that girl is pretty..you say I'd say she is a 6..
and so on
ok ok..I'll stop...
carry on..



January 17th, 2013, 11:48 PM

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What the world thinks today about this:
1 + 1 = 2
BUT in the REAL world it actually equals...
1 + 1 = 11
HA!
Yes man!
Like the ancient Babylonians who built The Tower of Babel to find God... so will I with the power of Mathematics to find God itself but I will not fail!
And I am the BEST at maths!
I am on a mission!
Wish me luck guys!
__________________
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Brb in 1 000 years



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January 18th, 2013, 02:00 AM

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Re: 6÷2(1+2) = ?
The answer is bacon.
After the Mayans.
With lettuce, and tomatoes.
The parentheses are the toast.
Stinky took my bacon.
Send more.



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January 18th, 2013, 03:21 AM

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Quote:
Originally Posted by EarlyMon
The answer is bacon.
After the Mayans.
With lettuce, and tomatoes.
The parentheses are the toast.
Stinky took my bacon.
Send more.

I don't like your tone LateMon!
...
AND aren't you LATE to reply to me myself and I!?



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January 18th, 2013, 07:07 AM

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Quote:
Originally Posted by Stinky Stinky
What the world thinks today about this:
1 + 1 = 2
BUT in the REAL world it actually equals...
1 + 1 = 11

This math problem has been answered long ago as well and found to be bacon:



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January 18th, 2013, 11:21 AM

#336 (permalink)

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Quote:
Originally Posted by mathman26
@ jhawkkw : Hey bud... I just spoke with a colleague of mine, who is a Space Science specializing in Physics graduate, with lots of defence research time. We chatted this morning over this equation, agreeing you can argue either way, and quickly ended up at the same place we did "It is a crap question with arguments either way.. you would need to know the flaws in the s/w program to avoid them... and on paper we would use proper notation". That aside, he did bring this up, and I would like your take on it.
The "9" people do this: 2n = 2 * n, so
6/2(2+1) = 6 / 2 * (2+1).
But then he said he would write 2n = (2 * n) and not just 2n = 2*n
I think he is more for '9', not that it matters, but that point did strike me as interesting... thoughts?

I claim a similar background and am on record from when this was first asked 
Quote:
Originally Posted by EarlyMon
Rules say the answer is 9, but some older compilers would evaluate that as 1.

Also on record for saying that the whole thing is sloppy, so I'm only going to defend this 
The parens are the toast for a BLT. The 1+2 signify that you take enough bacon for one BLT, then add twice that amount to make a proper BLT. Therefore, 2(1+2) says that two BLTs are required that way. 6÷2(1+2) means that each of the two sandwichs should be cut into thirds. (Yeah, I did that on purpose, just go with it.)
After all, you don't want to be a pig.



January 18th, 2013, 01:47 PM

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Quote:
Originally Posted by CafeKampuchia
This math problem has been answered long ago as well and found to be bacon:

So you think you have what it takes to take my place in the heavens and to be ... the one and only... "The Funniest Loser Alive" do you?
You think you have what it takes now do you?
Let me give you the greatest test of all timez to see if you are "The One"....
I don't know if you are the 1 but if you can guess what my favorite thing to do in the whole wide world is ... then I will step down from my throne... I will give you the crown... (sigh) and yes you will have a pony here and there to ride here and there blah blah blaaaah yakkitty smakitty and so on and so forth...
SO
What do I love to do more than anything in the whole wide world?
You have 1 day to answer me...
Time is ticking man!



January 19th, 2013, 06:42 AM

#338 (permalink)

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Don't worry, man. The crown is yours. I just like bacon!
But do tell and drown our curiosity!



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January 19th, 2013, 07:06 AM

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Quote:
Originally Posted by CafeKampuchia
Don't worry, man. The crown is yours. I just like bacon!
But do tell and drown our curiosity!

xD
Hahahahahaaa
Well since I cannot keep my pie hole shut for very long...
The one thing I Love more than anything in the whole wide world is to make people laugh or smile.



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January 19th, 2013, 07:14 AM

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Woops my trolling has forgotten the importance of this so very important thread of importnantness!
I trollishly change my mind and think the answer is  504193.
Yes!
Weird looking English guy from who wants to be a millionaire says:
Stinky Stinky... you could walk away with 1 million pounds... but if it is wrong you could walk away with less cash... Stinky Stinky... is this your final answer?
Stinky Stinky:
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January 25th, 2013, 12:11 PM

#341 (permalink)

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Division must be proven with Multiplication in reverse:
6 ÷ 2n = 3/n
3/n * 2n = 6
6 ÷ 2n = 3n ?? 3n * 2n = 6n^2
6n ÷ 2n =
(n+n+n+n+n+n) ÷ (n+n) =
[(n+n) (n+n) (n+n)] ÷ (n+n) =
3(n+n) ÷ (n+n) = 3
6n ÷ 2n = 6n ÷ (2n) = 3
obelus is a grouping symbol in itself.
Let n = 2+1. Ans: 1
I am having a hard time finding any references that say 6 / 2n = 3n ....



January 25th, 2013, 02:22 PM

#342 (permalink)

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Re: 6÷2(1+2) = ?
By the same token, you could build a case for
6/n(1+ 2) = 6/n3 != 6/3n due to nonstandard nomenclature
Stick with there not being enough parens because otherwise, it's compilerdependent.



January 25th, 2013, 02:39 PM

#343 (permalink)

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you are right. Machines are different than humans. Using mathematics though, it is 1.



January 25th, 2013, 02:47 PM

#344 (permalink)

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Re: 6÷2(1+2) = ?
Using mathematics, I'd claim that the sequence is expressed in an underconstrained fashion and send it back as not even wrong.
Which is where I thought you started.
To me this is right up there with calling y=sqrt(x) a function.





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