edit: to explain myself a little better, no, i'm not saying f(x) = 2*x. In fact, I'm saying that's the problem-- OTHER people are saying that, but it's not true. f(x) is a function, that is f of x (f is a function of x), meaning the variable f is applied to the variable x. In our equation, we would apply 2 to 3, which in effect is multiplying it... but it's not the same as saying f(x) = f*x (though isolated, those equations are redundant). The problem is that USUALLY creating a function simply means multiplying it, so we've been trained to think that 2(3) = 2*3. But it isn't.
Um. f(x) is literally function of x - and does not mean the variable f is applied to x.
How about functions of the form -
where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T.
Quote:
f(x) is a function, that is f of x (f is a function of x), meaning the variable f is applied to the variable x.
No. In the above example, there is no variable f that is applied to variable x.
The f in f(x) is simply a placeholder - for the operations that are applied to x.
Quote:
we've been trained to think that 2(3) = 2*3. But it isn't.
We've been trained to think that way for a reason. Commutation - it comes from commutation notation and means 2*3.
Honest.
So, by direct evaluation, the equation must become 6/2*3 - as had been said before.
Last edited by EarlyMon; April 28th, 2011 at 07:45 PM.
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Um. f(x) is literally function of x - and does not mean the variable f is applied to x.
How about functions of the form -
where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T.No. In the above example, there is no variable f that is applied to variable x.
The f in f(x) is simply a placeholder - for the operations that are applied to x.
We've been trained to think that way for a reason. Commutation - it comes from commutation notation and means 2*3.
Honest.
So, by direct evaluation, the equation must become 6/2*3 - as had been said before.
This post makes me realize how much I dislike math. (That is math, right?)
It's a common approximation(*) of the Fourier transform, used to estimate frequency components in time series data.
It was an Easter egg for Master Po, who may have missed implied time when I said rhythm.
(*)The even more common one is to use Euler's and just directly evaluate from there, but it was just easier to copy from wikipedia than spell out the eq'n. A great deal of what I've done and published started with that eq'n. (Stated for Po's benefit, just to beat him to that punch. )
Ok Bob - once again 1,1,1,2,__ or 1,1,2,1,3,1,4,__
I guess time is running out.
Last edited by EarlyMon; April 28th, 2011 at 08:09 PM.
Um. f(x) is literally function of x - and does not mean the variable f is applied to x.
How about functions of the form -
where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T.No. In the above example, there is no variable f that is applied to variable x.
The f in f(x) is simply a placeholder - for the operations that are applied to x.
We've been trained to think that way for a reason. Commutation - it comes from commutation notation and means 2*3.
Honest.
So, by direct evaluation, the equation must become 6/2*3 - as had been said before.
well, fine. :-p
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It's funny how people used a mnemonic to justify an answer.
That's like saying Math is the way it is because I tied this string around my finger
Multiplication and division are equal im PEMDAS it's actually P E MD AS. It could have just as easily be written PEDMSA.
The only argument for the answer being 1 is if you interpret the 2 being a function of the parenthesis. Since it's genreally considered shorthand for multiplication and not a function the answer is 9. (Unless your talking computer science or a different type of math like calculus where x and * dont mean the same thing anymore). Basically what EarlyMon said.
Still, my answer is 42.
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Quote:
Originally Posted by Vihzel
Calculations:
6÷2(1+2) =
6÷2(3) =
3*3 = 9
This is right.
6÷2(1+2) =
6÷2(3) =
6÷6 = 1
Wrong. To "see" it correctly, the equation should be written as (6/1)*(1/2)*((1+2)/1), which equals (6/1)*(1/2)*(3/1), which equals 9 no matter the order of the three "terms".
6÷2(1+2) =
6÷2+4 =
3+4 = 7
Now that's just stupid.
There you go. It's been said before, but I tried to express it differently.
PS: I like to riddle my friends with this. I start with the first statement and proceed from there.
________a = b________________(Premise)
_____a^2 = ab________________(multiply each side by a)
a^2 - b^2 = ab - b^2__________(subtract b^2 from each side)
(a+b)(a-b) = b(a-b)____________(factorization)
____(a+b) = b_________________(simplification)
_____b+b = b_________________(according to the premise)
______2b = b_________________(addition...)
Last edited by DaSchmarotzer; April 28th, 2011 at 10:00 PM.
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Quote:
Originally Posted by EarlyMon
The simplification step required dividing both sides by zero.
Of course, but you'd be surprised at the number of educated people who will be stumped. Maybe not the ones used to work specifically with Mathematics though.
For example, the CEO of a consulting engineering firm that I know didn't find it.
Welcome to one of the easiest algebraic equations in the world!... or is it? MUAHAHAHAHA!
Can you solve this pesky little problem while also giving damning evidence that the other 2 can not possibly work?
YOU MUST CHOOSE ONE AND ONLY ONE!
Calculations:
6÷2(1+2) =
6÷2(3) =
3*3 = 9
6÷2(1+2) =
6÷2(3) =
6÷6 = 1
6÷2(1+2) =
6÷2+4 =
3+4 = 7
ADDED: Those calculations are simply thought processes for each possible answer. I am aware that there may or may not be something missing that could help clarify, but I am leaving it up to you to figure it out.
My Answer is nine, using the order of operations:
1: Perform any calculations inside parentheses.
2: Perform all multiplications and divisions, working from left to right.
3: Perform all additions and subtractions, working from left to right.
And since parentheses means multiplication it is done after the division.
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Of course, but you'd be surprised at the number of educated people who will be stumped. Maybe not the ones used to work specifically with Mathematics though.
For example, the CEO of a consulting engineering firm that I know didn't find it.
And you didn't take the opportunity to show him that dividing by zero will double a number and is therefore perfectly valid?!?
Get on his payroll - you'll be set for life in the first month!
It was a Malcolm Gladwell "Blink" answer. I didn't ponder my decision at all. Doesn't make my answer correct, but it does prevent me from getting sucked into a self debate. Here's my logic:
We make the equation a bit more algebraic without changing the operators and parens:
c/2(a+b) where a=1, b=2, and c=6
If you assume c/2 comes after evaluating (a+b), you are essentially putting (a+b) in the numerator of your division:
c/2 * (a+b)/1 --> c(a+b)/2
So if the equation was written as c(a+b)/2 ((a+b) is in the numerator), your final answer is 9. But the equation was definitely not written in this way. So 9 is wrong.
As the original equation was written, (a+b) is clearly in the denominator because I proved above it can't be in the numerator without a drastic re-write of the equation. In other words, c/2(a+b) can only imply that (a+b) is in the denominator without re-writing the equation. Therefore, as the original equation was written, the answer is the following:
eval parens first: (a+b) = 3
3 is part of the denominator, so 6/2(3) = 6/6 = 1
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you totally misread what I posted. Cause the part you quoted and said was incorrect is exactly how you would arrive at "9."
Ah yer right sorry, but it was a mistake of quoting the wrong part of your post, not misreading what you said.
Your proof is still wrong though because it rests on the "because it was written this way, I can perform the order of operations this way"
This is what I should have put in the second quote:
Quote:
c/2(a+b) can only imply that (a+b) is in the denominator without re-writing the equation.
That's where you make an assumption and rewrite the equation yourself.
You're basically saying because there are parenthesis, you can do the multiplication of 2 * (a+b) before c/2 even though that's not how the order of operations work. After the parenthesis, you're supposed to go back to division/multiplication and left->right. x(y) is just another way of writing x*y, it doesnt change the rules
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The more I read this thread....the more it looks like its time for me to go back to school...lol.
But it is an interesting read. If I didnt post in here, this was gonna be one of my "subscribe to and just read" threads.
And where in the world is Vihzel!?!?!?
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You're basically saying because there are parenthesis, you can do the multiplication of 2 * (a+b) before c/2 even though that's not how the order of operations work. After the parenthesis, you're supposed to go back to division/multiplication and left->right. x(y) is just another way of writing x*y, it doesnt change the rules
Well there's debate on how order of operations work as it pertains to M and D. But I was trying to eliminate that debate entirely by focusing on the OP's statement in one of his posts:
Quote:
You have to make a decision simply based on the equation given in the title
And in the title, it was written as: 6÷2(1+2)
And my point was: there are two popular answers: 9 and 1, depending on whether (1+2) was considered to be part of the numerator or denominator of the division operation. We all agree that (1+2) has to be evaluated first, but as written, does it belong on the top or bottom of the fraction?
And my argument was that if it belongs on the numerator, you get 9. But if it was INTENDED to be part of the numerator, the equation would have been written as 6(1+2)÷2.
If it was INTENDED to be part of the denominator, the equation would look pretty much as it was originally written, with the (1+2) associated with the 2 in a multiplication operation.
So, yes, I AM saying that the 2 multiplies (1+2) first because, as written, the equation implies this MORE than it implies "treat division and multiplication equal and just go left to right."
BTW, I can easily argue against this viewpoint, but this justification satisfies me more.
-edit-
I re-read your post and see what your argument is. You think my numerator trick is not the same as (6÷2) * (1+2). But it is, because multiplication is associative and distributive.
Last edited by novox77; April 29th, 2011 at 10:22 AM.
Oh I'm here. I am just enjoying reading all of these fine posts.
It was a great thread, no doubt.
However, I personally like the problems where there actually is a clear-cut correct answer and yet all the bigshot PhDs of the world can't agree on a right answer. Such was the case for:
The thing is... my answer is as good as anybody else's. I don't have like the FINAL ANSWER THAT MUST BE OBEYED!!!!!!!! lol
I personally believe it's 9 and that's the answer I would put on the math test to save my life. hehe
I interpret it as 6/2(1+2) is not the same as 6/[2(1+2)], so one answer would come up 9 and the other one as 1.
It's really amazing to see how even really intelligent people can argue for days on this and not come to an agreement on other equivalent problems to this.
Last edited by Vihzel; April 29th, 2011 at 10:22 AM.
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There exist differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −3^2 is interpreted to mean −(3^2) = −9, but in some applications and programming languages, notably the application Microsoft Office Excel and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages −3^2 will be interpreted as (−3)^2 = 9. [1]. In any case where there is a possibility that the notation might be misinterpreted, it is advisable to use brackets to clarify which interpretation is intended.
Similarly, care must be exercised when using the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x). Again, the use of brackets will clarify the meaning and should be used if there is any chance of misinterpretation.
The takeaways are highlighted.
RED: neither 9 or 1 is correct or incorrect. The person posing the problem (not you Vihzel) is WRONG for not being explicit.
GREEN: if there is the possibility of ambiguity of INTENT, use parentheses!
Last edited by novox77; April 29th, 2011 at 11:11 AM.
Reason: fixed the loss of superscripts representing exponents. carets added.
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Bob's sequence is how the second, minute, hour hands cross the numbers 1-12 on a clock.
Again a "Blink" answer. I haven't had time to even think if that makes sense or not. Gotta run to the doc's office. be back in a few hours with some more thought on it.
That answer is 9, could not be anything else. Regardless of how you read it.
I guess you want me to prove it right?
Ok, easy.
6/2 (1+2)= 6/2 is what you do with them. (1+2) is what they are, so....
You have to make a dog house, you have 3 (1 pine +2 fir), six foot long, boards to do it with. Plans say your dog house must be 2 feet tall. The boards are exactly 6 foot long and you must cut them at 2 feet. (6/2=3)
How many boards do you have after you finished cutting them?
1 six foot long board cut at every 2 feet, is exactly 3 boards
You have 3 six foot long boards, after you cut them all you have 9 exactly equal boards. 3x3=9
You can not take 2 feet out of 3 boards equally, which is what the multiplication is telling you, until you know the actually feet of the boards,which is what the division is telling you. The parenthesis make it clear what is the object and what is the operation. Because the operation tells you what to do with the object.
All you have to do is look at it as real things, not just numbers.
Allow me to take another stance...
If 6÷2(1+2)
Then (6)÷(2)(1+2)
So (6)(1)/(2)(1+2), because 6 is not divided by 2, but by 2(1+2) instead. Here the answer is 1.
My point is both 9 and 1 can be regarded as answers depending on what math rules you apply (left to right or Multiplication over Division or a combination of both).
Let's just add brackets to "clarify". I am not changing any operator or priority by doing so.
(6)÷(2)(1+2)
And since a division is in fact a multiplication of the inverse:
(6)(1/2)(1+2)
And that's 9 no matter the order of the terms.
I see what you are trying to say...
but because the operator of division was used in that location... that it is understood that it is also used to separates that equation. like so...
(6) / ( 2(1=2))
or
(6)
______
(2(1+2))
if the equation was ment to mean your interpretation..it would read like this
6(1+2) / 2 =
this would not doubt = 9
It's not operator location that implies precedence - it's only operator type and parens that set that.
Probably the only reason the confusion ever existed is because at some point around 20 years ago compilers popularly ended up whenever there was code like this and evaluating multiplications first, then divisions - leading to computation tools that enforced the thinking that leads to the wrong conclusion.
6
--------
2(1+2)
and
6(1+2)
--------
2
are both correctly hand-written notations of two different problems.
6/2(1+2) is machine-form (the typewriter) notation - and so only the conservative and strict rules may apply, to free from compiler confusions.
A * B * C = A * C * B
A / B * C = A * C / B
Therefore = 9.
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...dear lord of the butterflies...the answers have gotten much more confusing >_>;;; I think I got confused when letters started appearing in the equations.
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Well, if you don't find me expert enough (I'm peer-reviewed published in higher math for signal processing, nuclear reactor stuff, hydrodynamics, and radar among other things, and have lectured on and taught related stuff at the post-grad level), or DaS, another cool guy in the science biz - then just use the wiki info that novox 77 kindly copied:
Similarly, care must be exercised when using the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x).
Substitute (1+2) for x and it's case closed.
Last edited by EarlyMon; April 29th, 2011 at 02:33 PM.
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Quote:
Originally Posted by dan330
but the way it is represented... there is an implied reason for the location....
so there is an implied () around the first part = (6) ..
and an implied () around the second part = (2(1+2))..
you can not change what is inside or outside ().
NOTE: that is my interpretation of "implied"... i can be wrong.
is there an expert / professor here that can state the facts?
There is not such thing as "implied" when it comes to mathematics, since it would change the result. Rigor is important in sciences for that precise reason. Also, I consider EarlyMon to be an expert. Well, I also consider myself an expert for many things but that's a whole different story.
Also, both Mathematica (it's the same as Wolfram Alpha) and Maple say 9. Computers can't assume stuff (unless it's coded in ). If you ever start programming (maybe you do), you will truly understand what I mean by that.
PS: Roze, what a disappointment.
EDIT: Wow, I wrote that before EarlyMon had posted again.
EDIT2: Note to self: Using a lot of smiley faces sure doesn't help my credibility.
Last edited by DaSchmarotzer; April 29th, 2011 at 02:39 PM.
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Well, if you don't find me expert enough (I'm peer-reviewed published in higher math for signal processing, nuclear reactor stuff, hydrodynamics, and radar among other things, and have lectured on and taught related stuff at the post-grad level), or DaS, another cool guy in the science biz - then just use the wiki info that novox 77 kindly copied:
Similarly, care must be exercised when using the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x).
Substitute (1+2) for x and it's case closed.
ok.. i aint no expert. so i will bow to expert interpretation....
there is no implied () and
if in doubt...
left to right ...
EDIT: Wow, I wrote that before EarlyMon had posted again.
My reply was turning into another EarlyMon War+Peace special , so I deleted it, hoping you might post the clear path.
Glad I waited!!!
Quote:
EDIT2: Note to self: Using a lot of smiley faces sure doesn't help my credibility.
Doesn't hurt it any.
Often, information has to be expressed in multiple ways before clarity occurs.
This is true at all times and at all levels.
I once spent a decade solving a single equation - final proof took 30 pages of trig. After lecturing on the results and implications, and engaging in 5 years of feedback, I found while sitting in an airport, pretty smug over the day's lecture, that by considering all arguments for over a decade that the entire matter reduced to a simple progression of merely 4 clear and simple equations.
Illustrating for me for all time the positive need and positive benefit for us all to argue our way through things - you just never can be sure until the process completes how things might end up.
One of the reasons I like the people here.
Multiple people came to the same conclusion, all for the right reason.
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but the way it is represented... there is an implied reason for the location....
so there is an implied () around the first part = (6) ..
and an implied () around the second part = (2(1+2))..
you can not change what is inside or outside ().
NOTE: that is my interpretation of "implied"... i can be wrong.
is there an expert / professor here that can state the facts?
I'm no professor, but this is what I think it comes down to: To change the order of operations around the adition part (1+2) you need to add a parenthesis to make that go first, we all agree there.
But it shouldn't (and doesn't in my mind) make any sense that I should have to infer that a number that is near a parenthesis should have it's order of operation changed too. The parenthesis should change only what's inside it, with the innermost being evaluated first.
Basically some other problems to illustrate:
we clearly know that 6 * (1+2) = 18
And that 6 * 1 + 2 = 8
And that 6 / (2(1+2)) = 1 <- this is clear and obvious
^ So how is it we can infer 6 / (2(1+2)) from 6/2(1+2) ?
Unless there's some clear exception to the order of operations rule, we should not make assumtions that require additional parenthesis (imo)
Though my own argument can be used against me here too because the opposing argument could be how can you infer (6/2) * (1+2) ?
IMO,the reason you can infer the second (6/2) * (1+2) is because 6/2 doesn't need parenthesis normally, as it would natually come first in the order of operation.
6/2*3 is certainly clear enough I think. Once the parenthesis have been evaluated, they shouldn't affect the rest of the equation.
Also, I think It's pretty clear that:
EarlyMon/Rose(dan330+usta)=nova777 (imo)
And that's my final proof. Irrefutable! Just be sure to evaluate the (imo) first.
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where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T.
)
, so I deleted it, hoping you might post the clear path.
