Thread: 6÷2(1+2) = ? View Single Post
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....
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