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electricpanda



Joined: 03 Sep 2008
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PostPosted: Wed Nov 26, 2008 7:48 am    Post subject: Reply with quote

But I thought that was because it was so close to one, you could essentially just call it that because the difference would be indistinguishable (and also hard to fit on a calculator)?
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Tamir



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PostPosted: Wed Nov 26, 2008 7:54 am    Post subject: Reply with quote

No, it really is one. =/

Here's a basic proof that you might not like due to infinities:

x = 0.9999999999....
10x = 9.999999999.....
10x - x = 9
9x = 9
x = 1

But it's true, honest. I can find you other proofs if you want. (I know another one off the top of my head, but it's a bit strange for people who haven't heard that sort of proof before.)
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electricpanda



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PostPosted: Wed Nov 26, 2008 7:58 am    Post subject: Reply with quote

No, you've frightened me enough for one day. I'll just give you the benefit of the doubt XD No point arguing what you're ill-equipped to argue with.
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Nem



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PostPosted: Wed Nov 26, 2008 8:12 am    Post subject: Reply with quote

See that looks just as wrong as the one the other person posted to me. You've moved everything up by a decimal point, in other words you've added a zero to the far end, but the sequence has no end to add that to. It's like trying to add something to itself.
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Tamir



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PostPosted: Wed Nov 26, 2008 8:44 am    Post subject: Reply with quote

The other one was problematic because if you want the following to be true,

(1+1+1+1+.....) - (1+1+1+1+.....) = 0

then the number of ones in each infinity must be equal. This is ridiculous. If you're claiming that the number of ones is equal in each one, than there's a finite amount of them, and you can number them, and it's all good. You can't equate infinities like that.

In my proof, no infinities got equated. I assumed the nines go on infinitely, and therefore even if I move the decimal number a bit, I still have an infinitude past the decimal point.
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Nem



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PostPosted: Wed Nov 26, 2008 9:05 am    Post subject: Reply with quote

Tamir wrote:
The other one was problematic because if you want the following to be true,

(1+1+1+1+.....) - (1+1+1+1+.....) = 0

then the number of ones in each infinity must be equal. This is ridiculous. If you're claiming that the number of ones is equal in each one, than there's a finite amount of them, and you can number them, and it's all good. You can't equate infinities like that.


If they're not equal then one has to be greater or less than the other, which is a function of finite numbers. Besides what if I take one infinity and times it by two? Don't I then have two infinities that are equal?

I mean Nem doesn't believe in infinities anyway so it's a bit weird but assuming you can play with them that way.

Tamir wrote:
In my proof, no infinities got equated. I assumed the nines go on infinitely, and therefore even if I move the decimal number a bit, I still have an infinitude past the decimal point.


You've added nine infinities of .9999.... to your one infinity of .9999.... If you can't assume they're equal then how can we move the decimal point that way?
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Tamir



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PostPosted: Wed Nov 26, 2008 11:07 am    Post subject: Reply with quote

Nem wrote:
If they're not equal then one has to be greater or less than the other, which is a function of finite numbers.

Is it now? You may be happy to hear that infinity is defined as:
For any natural number n, inf > n.
And there are different orders of infinity, for which you can say that one is decidedly bigger than the other.

Nem wrote:
Besides what if I take one infinity and times it by two? Don't I then have two infinities that are equal?

Umm. No? They're both the same order of infinity, but the difference between them is not necessarily zero.

Nem wrote:
I mean Nem doesn't believe in infinities anyway so it's a bit weird but assuming you can play with them that way.

What's that supposed to mean? You think infinity is a human invention, and doesn't actually exist?

Nem wrote:
You've added nine infinities of .9999.... to your one infinity of .9999.... If you can't assume they're equal then how can we move the decimal point that way?

That's not quite what I did, but regardless....would you like me to give you a different proof for 0.9999... = 1? One that doesn't involve infinities (much)?
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theBSDude



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PostPosted: Wed Nov 26, 2008 12:56 pm    Post subject: Reply with quote

Well, .999... is 1 minus an infinitely small number; basically .000...0001 or 1 divided by infinity, which is zero.
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Tyris



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PostPosted: Wed Nov 26, 2008 5:49 pm    Post subject: Reply with quote

Here's a proof that isn't even slightly mathematically rigorous, but sounds good:

The difference between 0.9 recurring and 1 is 0.(0 recurring)1. But because the tail of 0s is infinitely long, you never actually reach the 1... the 0s go on forever, and it's simply 0. And if the difference between two numbers is 0, then they must be equal.
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theBSDude



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PostPosted: Wed Nov 26, 2008 6:16 pm    Post subject: Reply with quote

That's what I said.

My Calc3 teacher (the one who told me this) used this "proof," which, in my opinion, is really just a mind-screw:

1/9 = .111 repeating
2/9 = .222 repeating
3/9 = .333 repeating
...and so on...
8/9 = .888 repeating
so 9/9 should = .999 repeating
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Tamir



Joined: 22 Mar 2008
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PostPosted: Wed Nov 26, 2008 7:46 pm    Post subject: Reply with quote

Sure, but for that you need to accept that 1/9 is 0.111....

*shrug*

Analysis style proof time! First, an axiom: there exists no positive number x such that for every natural number n, x < 1/n. Translation: there are no infinitesimal numbers.

Proof: For any natural number n, we wish to show that that (1 - 0.999...) < 1/n.
If that holds, then the difference between 1 and 0.999... must be zero, since no infinitesimal numbers exist, and therefore 0.999... = 1.
How do we show that (1 - 0.999...) < 1/n? Well, let's say we choose n=10000, so 1/n = 0.00001. Now, we know that (1 - 0.99999) = 0.00001, but 0.999.... > 0.99999 of course, so naturally (1 - 0.999...) < 0.00001. The same goes for any n you can choose.

QED?
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Nem



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PostPosted: Wed Nov 26, 2008 8:09 pm    Post subject: Reply with quote

Isn’t the whole 0.9... thing extending to an infinitesimally small number?

Tamir wrote:
Is it now? You may be happy to hear that infinity is defined as:
For any natural number n, inf > n.
And there are different orders of infinity, for which you can say that one is decidedly bigger than the other.


If you could make a list of all the possible numbers in the universe, both positive and negative, and it wouldn’t show up anywhere on that list then how can it be bigger or smaller than anything?

Tamir wrote:
Umm. No? They're both the same order of infinity, but the difference between them is not necessarily zero.


If I take 2 and times it by two I've got two equal values. Whatever the value of the infinity I’m multiplying is when I times it by two why don't I get two of whatever it was?

Tamir wrote:
What's that supposed to mean? You think infinity is a human invention, and doesn't actually exist?


If it's greater than any natural number then I don’t understand how it could exist as anything other than a human invention.

Tamir wrote:
That's not quite what I did, but regardless....would you like me to give you a different proof for 0.9999... = 1? One that doesn't involve infinities (much)?


I’m even more confused now; multiplication doesn’t work by repeated addition of equal values?

Heh I got a nose bleed reading all this, I think my brain's broken Sad
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Tamir



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PostPosted: Wed Nov 26, 2008 8:18 pm    Post subject: Reply with quote

Nem wrote:
Isnít the whole 0.9... thing extending to an infinitesimally small number?

No no no no no no no. It's extending to one, darnit. -_-

Nem wrote:
If you could make a list of all the possible numbers in the universe, both positive and negative, and it wouldnít show up anywhere on that list then how can it be bigger or smaller than anything?

You list all of the natural numbers, then say that if there's a number bigger than them all, you shall call it infinity. Mmkay?

Nem wrote:
If I take 2 and times it by two I've got two equal values. Whatever the value of the infinity Iím multiplying is when I times it by two why don't I get two of whatever it was?

You do! But, infinity doesn't have a value. It's just infinity. Stop treating it like a normal number when it isn't one. =(

Nem wrote:
If it's greater than any natural number then I donít understand how it could exist as anything other than a human invention.

Okay, then, riddle me this - does our universe have limits? For if infinity does not exist at all, then space cannot continue indefinitely, and must be bounded.

Nem wrote:
Iím even more confused now; multiplication doesnít work by repeated addition of equal values?

Amusingly enough, not necessarily. But that's already past our point.
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Nem



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PostPosted: Thu Nov 27, 2008 6:15 pm    Post subject: Reply with quote

Tamir wrote:
No no no no no no no. It's extending to one, darnit. -_-


By adding values after the dp in increasingly small increments, that get increasingly small all the way out to infiniity, if there's are infimitesionally small numbers I don't understand how that can be done. Whatever comes at the far end of .999...9 would have to be infimitesionally small.

Tamir wrote:
You list all of the natural numbers, then say that if there's a number bigger than them all, you shall call it infinity. Mmkay?


How much bigger? If I just add one to whatever's at the top of the list of natural numbers then I've still got a natural number, however much I add I'm still going to have a natural number.

Tamir wrote:
You do! But, infinity doesn't have a value. It's just infinity. Stop treating it like a normal number when it isn't one. =(


If it doesn't have a value then why are you allowed to do any mathematical operations with it? It would be like me saying 1 * cheese = 20. Mmm, cheese.

Tamir wrote:
Okay, then, riddle me this - does our universe have limits? For if infinity does not exist at all, then space cannot continue indefinitely, and must be bounded.


Well that's kind of a difficult question since we don't know what, if anything, space is. My personal view is that space is just the absence of anything existing, it's not infinite and it's not bound, there's simply nothing there to count or quantify. Even when we count distance we're expressing not the amount of something inbetween the two objects but a derivative expression of the speed and time that a thing takes to travel somewhere.
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Tamir



Joined: 22 Mar 2008
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PostPosted: Thu Nov 27, 2008 6:32 pm    Post subject: Reply with quote

Nem wrote:
Whatever comes at the far end of .999...9 would have to be infimitesionally small.

Nothing comes at the far end. There is no end. Every one of those nines represents a real number (though a small one), and the list doesn't end.

Nem wrote:
How much bigger? If I just add one to whatever's at the top of the list of natural numbers then I've still got a natural number, however much I add I'm still going to have a natural number.

I know. That's why people have such trouble imagining infinities. It's terribly counter-intuitive, but that doesn't make it impossible.

Nem wrote:
Tamir wrote:
You do! But, infinity doesn't have a value. It's just infinity. Stop treating it like a normal number when it isn't one. =(


If it doesn't have a value then why are you allowed to do any mathematical operations with it? It would be like me saying 1 * cheese = 20. Mmm, cheese.

You're right - the mathematical operations we can do on infinity are very limited. That's why your friend's proof was illegal, by the way, because inf - inf is undefined. However, there are a number of operations we allow....
inf + a = inf (a = real number)
inf * a = inf (a = positive number)
inf + inf = inf, inf * inf = inf
1/inf = 0
And a few similar ones. These stem from the mathematical definition of infinity. By which I mean that you don't need to believe in infinity, but if you assume it exists, things like the above follow.

Nem wrote:
Well that's kind of a difficult question since we don't know what, if anything, space is. My personal view is that space is just the absence of anything existing, it's not infinite and it's not bound, there's simply nothing there to count or quantify. Even when we count distance we're expressing not the amount of something inbetween the two objects but a derivative expression of the speed and time that a thing takes to travel somewhere.

Umm. Okay? So, if two objects are flying away from each other in space, is there a limit to how far away they can get?
What about time, too? Must time stop as some point, so that it doesn't continue infinitely? Is time also an illusion?
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