Well, I just watched the video from the third post (or fourth or something).
And it really doesn't convince me.
Basically what the video says is: 1/2 + 1/4 + 1/8 + 1/16 + ad infinitum = 1
And it shows this by filling up a square.
The conclusion the video says is that the infinite divisions make up a finite answer, because the whole square is turned blue.
But that's not so. Yes... we can't see, eventually, the small piece that still isn't blue, but to randomly suggest maths has now proven that infinite divisions added together create a finite sum is... well... he could just as well have shouted: "BANANAS". Couldn't he?
This is doing my head in.
Originally posted by KazetNagorraWhy would you end up with a full pie? Surely there's always a wee piece missing?
Imagine a piece of pie that you divide into half. Now you keep adding half of the remaining piece to the pie. So you start with your half a pie, and add a quarter. Now you have three quarters of a pie. Add half of a quarter, and you get seven eights (you can visualize this quite easily with a drawing). At any point in the process, you will have some pie left, but keep doing it infinite times and you will end up with a full pie.
Originally posted by shavixmirNo, there cannot be a piece missing.
Why would you end up with a full pie? Surely there's always a wee piece missing?
Suppose there were a piece missing. Pick a position for the center point of that piece. Where ever that piece is in the pie, it is possible to find a finite number for which that center point is filled in. Therefore that piece cannot be missing.
Originally posted by wolfgang59This deduction assumes that the sum is convergent (in fact I think that it assumes that the sum is absolutely convergent (which it is) since you're making term by term matching) so in interest of completeness one must prove that the sum indeed is (absolutely) convergent.
Simply:
Let S= 1/2 + 1/4 + 1/8 + .....
Therefore 2S = 1 + 1/2 + 1/4 + ....
Now subtract the sequence S from the sequence 2S.
(by matching the terms)
Then you have 2S - S = 1
therefore S = 1
Originally posted by twhiteheadThere isn't a centre point though, because the size keeps shrinking. So as soon as you pinpoint the centre, it moves up by half a distance.
No, there cannot be a piece missing.
Suppose there were a piece missing. Pick a position for the center point of that piece. Where ever that piece is in the pie, it is possible to find a finite number for which that center point is filled in. Therefore that piece cannot be missing.
So, the piece that's missing gets smaller, but is still there.
I don't understand the S and 2S explanation either.
Why can you just scrap them away from each other?
Here's half a bottle of coke. And here's 2 half bottles of coke... Scratch them away from each other and you're left with 3/4 of a bottle coke or 1/2 a bottle of coke (depending if you added them together first), but certainly not a full bottle of coke.
Now, I hope you all don't think I'm trolling, but I genuinly don't get it.
But... I know thatthe glass breaks if I drop it. It's doing my head in!
Perhaps a practical example? I dunno...
Originally posted by shavixmirI’ve been following this thread and just wanted to say that I agree with you. All of these “this makes perfect sense” posts that I’ve seen in this thread... at best they sound like “just accept it ‘cause it’s true”.
There isn't a centre point though, because the size keeps shrinking. So as soon as you pinpoint the centre, it moves up by half a distance.
So, the piece that's missing gets smaller, but is still there.
I don't understand the S and 2S explanation either.
Why can you just scrap them away from each other?
Here's half a bottle of coke. And here's 2 ...[text shortened]... the glass breaks if I drop it. It's doing my head in!
Perhaps a practical example? I dunno...
In the pie example, the only thing that I can see happening is that at a certain point you reach either planck time or planck space where it’s physically impossible to divide the remaining pie in a smaller piece. So at best you get back to one pie. But not because you’ve been doing it infinitely, simply because there’s no more divisions possible.
So, you’re not the only stupid one here.
Originally posted by shavixmirI am talking about any given piece, not a sequence of pieces. If you claim that there exists a piece that is missing and it is of finite size, then it has a centre point. That centre point will have an exact position. That exact position cannot possibly be missing as I can always find a finite index in the sequence that has filled in that position. Therefore it is impossible for there to be a piece missing.
There isn't a centre point though, because the size keeps shrinking. So as soon as you pinpoint the centre, it moves up by half a distance.
Originally posted by Great King RatTomorrow I'll make a long post trying to shed more light in this. and if you two guys still don't understand it then I'll capitulate.
I’ve been following this thread and just wanted to say that I agree with you. All of these “this makes perfect sense” posts that I’ve seen in this thread... at best they sound like “just accept it ‘cause it’s true”.
In the pie example, the only thing that I can see happening is that at a certain point you reach either planck time or planck space w ...[text shortened]... , simply because there’s no more divisions possible.
So, you’re not the only stupid one here.
Originally posted by wolfgang59Why is 2S - S = 1???
Simply:
Let S= 1/2 + 1/4 + 1/8 + .....
Therefore 2S = 1 + 1/2 + 1/4 + ....
Now subtract the sequence S from the sequence 2S.
(by matching the terms)
Then you have 2S - S = 1
therefore S = 1
If I fill in S=3, then 2S - S = 2*3 - 3 = 3.
You filled in S= 1/2 + 1/4 + 1/8 + .....
Therefore 2S - S = 1/2 + 1/4 + 1/8 + .....
Right??
Originally posted by Great King RatWe have S = 1/2 + 1/4 + 1/8 + ...
Why is 2S - S = 1???
If I fill in S=3, then 2S - S = 2*3 - 3 = 3.
You filled in S= 1/2 + 1/4 + 1/8 + .....
Therefore 2S - S = 1/2 + 1/4 + 1/8 + .....
Right??
Therefore 2S = 1 + 1/2 + 1/4 + 1/8 + ...
Note how the terms after 1 + are again equal to S itself, so that:
2S = 1 + 1/2 + 1/4 + 1/8 + ... = 1 + S
Now we subtract S on both sides:
2S - S = 1
Therefore S = 1. (this is only an informal proof at best, though)
You can "prove" in a similar fashion that 0.999... (infinite sequence of 9s) = 1.