Diablo® III

48÷2(9+3) = ? cont.

In my first example of a/b(x) would be equal to (a/b)x. It just looks funny being around the (x). When you multiply something outside of the parentheses you multiply it by the entire value. So then (x)a/b would mean that you have an (x) that has to be multiplied by everything outside of it. It a sense it is used as a bigger multiplication symbol. So then (x)a/b would equal a/b(x). It doesn't matter if it is in front of or behind a/b it means the same thing, the meaning doesn't change by (x) being on one side of a fraction or another.
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[ Exponents still take precedence over addition, no matter how you denote those two operations.


I don't know what you mean by precedence here. I just showed you a notational system--one that's actually used! In fact I used it for the proof I wrote which allowed me to pass Modal Logic and that was just a few years ago--and in that notational system the operations are carried out in reverse of the order in which they are written. If that's not a system where "exponentiation has no precedence over addition" then you'll need to explain what you mean by "precedence."
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One statistician on my campus thinks that it is natural to interpret 2x/3y (with the division symbol or the slash) as having "understood parentheses" around the 2x and the 3y respectively.
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06/12/2012 04:16 PMPosted by Speusippus
One statistician on my campus thinks that it is natural to interpret 2x/3y (with the division symbol or the slash) as having "understood parentheses" around the 2x and the 3y respectively.
I would agree. I wonder if this arguement could be avoided by saying that a/b{x} where a/b is a fraction multiplied by (x). Brackets are not used in calculators and ( ) = { }
Edited by Windscar#1524 on 6/12/2012 4:22 PM PDT
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What is happening here is that people are not following the golden rules of Math!!!

1) There is only one correct answer per question. Implying has no place in Math
2) NEVER divide by zero
3) e^((pi)i)+(sin(x))^2+(cos(x))^2=0
4) Squareing numbers is like women; If they are under thirteen, just do them in your head.

Applying these rules, it is easy to see that:
48/(2(9+3))=2
or
48/2*(9+3)=288

Its so simple, why doesn't everyone agree?
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06/12/2012 04:56 PMPosted by Sinew
This can (and should) be rewritten as 1/b*(a*x) or a(x)/b


Yes it should. 48÷2(9+3) is ambiguous to a reader, but that doesn't change the fact that applying pure math and logic (if you knew about programming languages, calls, and stacks, this would make a lot more sense) that the answer is 288. Again from a technical standpoint.

Implying that the author meant 48/(2(9+3)) is an assumption that has no logical base.

4 ÷ (3x)2x^2 = 4 ÷ 2x^2(3x)

If we assume that implied multiplication doesn't come first than these statements are not equal, which contradicts the whole system.


Yet you fail to justify or tie together how those two being equal has anything to do with the "whole system." You state as if it's true (assumption) then say if it's wrong, the universe would implode.

Again, as I said in the other thread, your entire line of logic is based around the assumption that implied multiplication through parenthesis has special privileges, but when you have no logical ground to base that assumption, your whole argument falls apart.
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Yet again, you've proved nothing. Congratulations. You've just written down a bunch of examples that would get different answers depending on whether or not implied multiplication gets special privileges.

Oh wait a minute...you've been doing that the whole time!

If you want something to be included in the denominator, you write it in parenthesis with the rest of the denominator, not on its own. Parenthesis ensure clarity. Your method does not.

Hence:

1/2

if you want to multiply the denominator by something, you multiply it in parenthesis. This is basic calculator operation.

1/(2*4) = 1/8

Still waiting for justification on why implied multiplication should get special privileges.

And don't even try to tell me what you have is justification. You're basically trying to justify "no special treatment of implied multiplication" as wrong because it gets a different answer than what the problem would if implied multiplication comes first. Because your answer is supposedly right. Wait, how did you determine that "right" answer? Through implied multiplication. That's circular logic dude.

Either that, or you got these examples out of a book with an answer check in the back, in which case I can tell you with utmost certainty that you typed them wrong.

Eg.

1
________
3*4

is not written down as 1/3*4, that's a completely different fraction. It's 1/(3*4).

2/(x+0)(2+0) = 2, x=2 because (2 + 0) is not in the denominator. Your method treats it like 2/(x+0)(2+0) is actually 2/((x+0)(2+0)). Of course their answers are going to come out differently, they're completely different fractions. That's the same argument as saying 4 != 7. No duh. I can see how this makes sense from an user-input standpoint (as wolfram alpha demonstrates with 1/ab, with it being more likely that the user meant 1/(ab) and would have typed b/a otherwise) but that does not change the technical aspect of pure math, which is why if you type it in on a TI-89, you will get the same results I have demonstrated, because TI-89's require you to type things in correctly.

Try again.
Edited by SweetWilly#1217 on 6/13/2012 9:47 AM PDT
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Order of operations is arbitary. You can build a completely consistent and valid system of mathematical notation under which you do addition and subtraction first, then multiplication and division.

Take the NORMALLY notated equation

2 * x + 3 = 3

Under normal notation, this means "the result of multiplying x by two then adding three yields three." Under normal notation, this "x" must be 0.

Now, imagine a new kind of notation, where you do parentheses, then additions and subtractions, then multiplications and divisions. How would you say, under this new kind of notation, "the result of multiplying x by two then adding three yields three"? You'd say it like this:

(2 * x) + 3 = 3.

You have to use parentheses now, because you're supposed to do additions first unless they are superceded by parentheses--and what we're trying to say with this equation is supposed to involve doing a multiplication first. Hence, under our new notational convention, we find we must put parens around the multiplication.

We haven't broken any fundamental laws of mathematics. We've simply adopted a new notation. And it works--it gets you the right solution. For both the old notational convention and the new one, you can see, you must first subtract three from both sides, then divide both sides by two, leaving x = 0 as the answer.

Another example:

Ordinary notation: 2 * (x + 3) = 4
The same expression under the new notation: 2 * x + 3 = 4

Under both forms of notation, the solution is found by first dividing both sides by two then subtracting three from each side (because under both notations, following the rules of the respective notations, you can see that the addition is performed "first," the multiplication "second." Hence when solving, you divide both sides "first," then you will subtract "second.") In both cases, the correct solution is yielded: x = -1

Order of operations is not fundamental in mathematics. It is literally a mere notational convention and nothing else. You can adopt a different convention to do exactly the same math. You could use no notation at all and simply write everything out in English. (That's what Newton and his contemporaries did btw.) You could instead write it all out in French. (Descartes did some of that I believe.) Deciding between English and French is simply deciding between two notational conventionss, just as deciding whether to do addition first or multiplication is just a way to decide between two notational conventions, i.e., languages.

Who understood this? Who is like "what?!" Who is like "no way!!" I believe I am willing to stick with this conversation for quite a while til I've explained it adequately.
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^Once again, completely valid argument. Completely agree, they are arbitrary.

My previous argument pertains to the standard of Order of Operations set forth in calculators and programming languages everywhere, the technical aspect. The reason why it is important to put the denominator in parenthesis, for this specific OoO.
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^Once again, completely valid argument. Completely agree, they are arbitrary.

My previous argument pertains to the standard of Order of Operations set forth in calculators and programming languages everywhere, the technical aspect. The reason why it is important to put the denominator in parenthesis, for this specific OoO.


That's fine, I was just responding to some comments I've seen from others that using a different order of operations than the standard one is somehow to miss something fundamental about mathematics.
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88 Gnome Death Knight
5585
Everyone that speaks of "order of operation" and still come up with 288 are not following their own rules. 48÷2(9+3) First step in order of operations is to deal with the parenthesis.
48÷2(12) wait...this is where you 288ers mess up! You don't finish dealing with the parenthesis! You MUST completely remove the parenthesis, not just simplify the inside of them. In order to do that, you MUST multiply 2 into the parenthesis. 48÷24. Now your done with parenthesis, and can move on in the order of operations. The order of operations DOES NOT say to work the inside of the parenthesis and then leave the parenthesis...you MUST remove them before continuing the order of operations. The only way to remove the parenthesis in this problem is to multiply 2 into the parenthesis.
Edited by Batlet#1797 on 6/13/2012 3:46 PM PDT
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06/13/2012 03:44 PMPosted by Killelz
The only way to remove the parenthesis in this problem is to multiply 2 into the parenthesis.


Completely false.

Lets say this is an equation where the answer is x.

48/2(9+3)=x

yes, do the addition in the parenthesis first, everyone agrees here.

48/2(12)=x

Now lets divide both sides by 12!

48/2=x/12

Parenthesis are gone OMG!

24=x/12

288=x
Edited by Juggernaut#1876 on 6/13/2012 6:06 PM PDT
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88 Gnome Death Knight
5585
Why does everyone rewrite the problem? The way the problem is written, parenthesis have to be removed first...multiplying 2 into the parenthesis. I specifically said...in this particular problem you remove the parenthesis by multiplying 2 into the parenthesis. I did NOT say that's the only way to get rid of parenthesis. For instance...48÷2*(9+3)...in this rewritten problem...simply adding 9 and 3 will remove the parenthesis. In the original problem... 48÷2(9+3)...2 is attached to the parenthesis by implied multiplication. Therefore you have to multiply 2 into the parenthesis in order to remove the parenthesis.
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@Juggernaut
That isn't the correct way to solve that anywhere i've ever seen. Please solve the identity at the end with this method.


@sinew-
What do you mean 'solve the identity at the end with this method'?

Not sure what you are talking about.
Edited by Juggernaut#1876 on 6/14/2012 10:19 PM PDT
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