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
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
...<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 hand-write something like this: ½ with a little 'n' right beside the 2.
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:
Check out page 53 before and after "or more simply..."