@abge: the pebbles are already paired. Theoretically before one pebble is crushed that would work, however, after it is crushed you will need to take one from somewhere in order for it to work. It would essentially be the question of what is infinity-(infinity-1) according to the commutative (or something like that) law of addition the order does not matter so you could just turn that in to infinity-infinity-1 which if infinity had a definite value would end up as -1. Even with the infinite number you had you would still have one left over after pairing the infinite amounts.

tvrocks, i bet you wouldn't agree that there are the same amount of even integers as there are positive integers. You can easily take both sets and lay them side by side like abge's pebbles and map every even x to x/2. So any positive number has a corresponding even number, and vice versa. If you start thinking "oh but the positive number set has 3 and that's not in the even numbers so there are more positive numbers", you're not understanding the concept of infinite. You just need a mapping.

So back to the proof, when you take .99999.... and multiply by 10 to get 9.99999....., you can easily map every 9 from .99999..... to a digit from 9.99999......., and that's all that is needed to show the sets have the same count, i.e. number of digits.

"the pebbles are already paired"

It's not illegal to reassign the pairing :)

"Even with the infinite number you had you would still have one left over after pairing the infinite amounts."

But this is where your argument falls apart because if there is an "after" it was, by definition, not infinite to begin with.

Don't think about it quite so hard. Crush one pebble. Now pair them up. Will you ever be left with one left over? No, because you will never stop pairing them. They are still both infinite.

tvrocks, yes, there are greater and lesser infinities, but that does not defeat the fact that infinity-1=infinity. It is an infinite amount. It cannot be compared. lim (x->infinity) x= infinity. lim (x-> infinity) x-1=infinity.

The pebble example can be taken to the extreme if you were to cut every green pebble in half. Are there more green pebbles than red pebbles now? The answer is still now because there is still one red pebble for every green pebble.

@y2k: post one about the number of even positive integers versus positive integers in general. The number of them would both be written as infinity though the even integers could just be paired with themselves such as 2 with 2, 4 with 4 leaving 1 and 3 unpaired at that time. Now you could pair them with the next two even numbers 6 and 8 though that would leave 5,6,7, and 8 unpaired. If the number of positive even integers was represented as infinity the number of positive integers in general would have to be represented as 2 infinity. Infinity is not a definite number though and as i explained earlier is defined as a number that is higher than one that can be assigned. Because it is saying a number it can be interpreted to mean that there are more than one number that are high enough to fit that qualification. Ot is all based off of the idea that some infinities are greater than others which should also be true as if you add 1 or multiply anyhing by 2 or something it will be bigger than it was before.

2*infinity=infinity...

well, 1/3 = .333....

if .333..... multiplied by 3 =.999.....

and 1/3 multiplied by 3 = 1.

Then I would guess .999.... = 1

But I am just the big stupid fighter, never took any math higher than Algebra 2 in community college so this sort of theory stuff is kinda beyond me.

That's because you're pairing them wrong. Pair them like this:

1-->2
2-->4
3-->6
4-->8

and you'll see they pair up perfectly. Nothing is missed.

@abge But this is where your argument falls apart because if there is an "after" it was, by definition, not infinite to begin with.

It is all theoretical as if there were a number that fulfilled the qualifications it would be that one. Though the entire idea of infinity is theoretical in the first place. I'm do e explaining for the night so later.

yeah maybe you're right.... really you're subtracting an infinite sum. and i guess i'm taking for granted that subtraction works on infinite sums like it does on typical numbers... maybe.

i don't know, i just don't like real numbers. this whole infinite-digit construction. i mean if you take all rational numbers and all (that exist in our universe) irrational numbers, add them together, you still have a countably infinite set. so the real number line is essentially all filled with numbers that do not and cannot exist in our universe.

My apologies, I didn't notice I was looking at the 3rd page of the topic. Prado basically said what I said already on page 1. Its past my bedtime. Good day.

cantor proves that there are different infinities, but if you can pair them up to a ratio (like even numbers to total numbers) the set has the same (infinity) number of numbers in it.

One quicke exception
@quickfighter: is .333... Actually 1/3 third or is it just infinitesimally close to 1/3? It's not 1/3 and is just cose.

I lied and am still here.

tvrocks, are you asking if I think they're the same, or telling me they're not, or both?

"i mean if you take all rational numbers and all (that exist in our universe) irrational numbers, add them together, you still have a countably infinite set. "

No, that is not a countably infinite set. Zultar, do you even vet your mods any more?

Jesus, I've been quoting bolshoi all thread and you still think I believe this stuff?

Hey, man, that's what you get for trying to front.

He made thug a mod after Bo was one. Obviously no vetting here

GHUG---fing auto correct

Hey, give Zultar a break! He's shown marked improvement recently.

I'd say it's a matter of definition. If you define the most accurate decimal number (real or not) as the fracture, sure, 1/3=0.333.... and 1=0.999....
If, however, you define 0.333.... as the limit towards 1/3, 0.999 is the limit towards one.

Pick your definition and move on I suppose?

Steephie, the difference between your definitions is that in the first the decimals are actually infinitely repeating and in the second they're not. They're limits as the sums approach infinity, so and your standard ellipsis or bar over your number signifies that the number itself is equivalent to the limit.

"Hey, give Zultar a break! He's shown marked improvement recently."

He gave you a cube, and youre not funny----hehe

He made thug a mod after Bo was one. Obviously no vetting here

Don't use the back button.

@tvrocks: You forget one thing, infinite series don't follow the same rules as finite numbers
(hence why 1+2+3+4+5...= -1/12, but that's another thread).

Vash, you can't manipulate every series into equality with whatever you want. That one is most definitely always infinity.