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I was just thinking of something mentioned above...

Perhaps this demonstrates the clash between potential (or theoretical) infinity and actual (or realistic) infinity.

Theoretical infinity might suggest that 0.999... is infinitely a fraction short of 1 or that one might never indeed reach the Pepsi in Vivisekt's example. But in acutality, in reality, this infinity does not exist.

I guess what I'm trying to ask is: what does this little demonstration mean for the idea of infinity as a whole?

Vivisekt wrote:What you're thinking of is the philosophical concept of an infinite number - which, really, isn't math. Calculus doesn't work that way.

Perhaps this demonstrates the clash between potential (or theoretical) infinity and actual (or realistic) infinity.

Theoretical infinity might suggest that 0.999... is infinitely a fraction short of 1 or that one might never indeed reach the Pepsi in Vivisekt's example. But in acutality, in reality, this infinity does not exist.

I guess what I'm trying to ask is: what does this little demonstration mean for the idea of infinity as a whole?

Another one which is fun (if you don't want to see 1=0.999...) is

1-0.999...=?

I was just bored I guess.

That's equivalent to my problem "1-x=0.999999.... Solve for x".

I'm bored too.

"Politics is the art of looking for trouble, finding it everywhere, diagnosing it incorrectly and applying the wrong remedies." - Marx (Groucho)

The difference between .999... and 1 is infinitesimal, but not non-existant.

Care to write it down then ?

0.999... is an expression of 1. Mathematics has certain definitions and 0.999... and 1 are the same thing. No number can possibly exist between them.

The difference between .999... and 1 is infinitesimal, but not non-existant.

Wrong.

0.999... is an expression of 1. Mathematics has certain definitions and 0.999... and 1 are the same thing. No number can possibly exist between them.

This doesn't make sense to me; just because "mathematics" defines something does not mean it is true. You say that no number can possibly exist between them. Then you are assuming that because the "distance" between the numbers is essentially 0, that they are the same number.

However, intuitively, to me at least, they remain distinct numbers, with distinct values; they are simply on the smallest scale you can go to, so it appears that they are one and the same. Logically though, one comes "after" the other on the number line -- the other one is merely approaching one.

Or so it seems to me.

In mathematics, trying to use intuition and mystical 'feelings' instead of logic will not get you very far. It has been proven by luke_w among others that 0.99999...=1. If it is possible to take the limit of a convergent infinite sum (and it is), then 0.9999...=1.

"Politics is the art of looking for trouble, finding it everywhere, diagnosing it incorrectly and applying the wrong remedies." - Marx (Groucho)

This doesn't make sense to me; just because "mathematics" defines something does not mean it is true. You say that no number can possibly exist between them. Then you are assuming that because the "distance" between the numbers is essentially 0, that they are the same number.

You're correct, in a way, but still wrong.

For anyone in advanced mathematics, they should know this.

People are saying that 1 = .999...

I'd disagree, I will say.

1 :⇔ .999...

It is defined as the logical equivlant, though that is just semantics.

The difference that I have yet to see anyone say is that the difference between the two and the reason they aren't in reality equivlant, except by logical deduction, is that

The art of the infinite is hard for some to grasp because it deals in a realm that isn't naturally possible as far as we know.

I recomend the books:

By Robert and Ellen Kaplan

By Robert Kaplan

They both deal with the same type of things and are well put together.

The difference that I have yet to see anyone say is that the difference between the two and the reason they aren't in reality equivlant, except by logical deduction, is that one is a finite number and .999... is an infinite number.

I have to disagree with this. What are we writing when we write 0.9999...? We are not writing an 'infinite number', but rather a number expressed as an infinite sum. Luke_w expressed it correctly as:

0.9999... = Sum 9/10^n

(n=1 -> Infinity)

"Politics is the art of looking for trouble, finding it everywhere, diagnosing it incorrectly and applying the wrong remedies." - Marx (Groucho)

I have to disagree with this. What are we writing when we write 0.9999...? We are not writing an 'infinite number', but rather a number expressed as an infinite sum.

That's what I meant. Though it is a number expressed infinitely.

You can do .99999999999999999999999999999

9999999999999999999999999999999

9999999999999999999999999999999

99999999999999999999999999999999

99999999999999999999999999999999

9999999999999999999999999999999999

99999999999999.... and never reach the end.

Day Zero wrote:This doesn't make sense to me; just because "mathematics" defines something does not mean it is true.

Numbers don't exist. We created them - they do what we define them to do. There is no x={.99...} floating around in space somewhere. It's a concept.

Kitty wrote:The difference that I have yet to see anyone say is that the difference between the two and the reason they aren't in reality equivlant, except by logical deduction, is that one is a finite number and .999... is an infinite number.

{.99...} is not a sum in this instance. There is a vast difference between an infinate sum {.99...} and the series {.99...} towards 1.

Last edited by Vivisekt on 09 Jun 2005 14:33, edited 1 time in total.

That's what I meant. Though it is a number expressed infinitely.

You can do .99999999999999999999999999999

9999999999999999999999999999999

9999999999999999999999999999999

99999999999999999999999999999999

99999999999999999999999999999999

9999999999999999999999999999999999

99999999999999.... and never reach the end.

But you do reach the end. You seem to be thinking of this almost as a temporal infinity; that you add each term after a couple of seconds. Obviously, in that case you never would get to the end. But it's not a temporal process. We can imagine adding each new term at the same time. There's no reason why we can't add all the terms at once. This is equivalent to taking the limit of Sum 9/10^n (n=1 -> infinity) as n goes to infinity. That limit exists, and is finite. If you're not happy using the decimal notation, then just write 0.9999.... as Lim_n->inf(Sum 9/10^n). You can certainly write that down in a finite amount of time, and it's exactly equal to 1.

wierd. no wonder.

You are thinking that I am saying that it doesn't have a defined value, I know it does. That is why 1 :⇔ .999...

You are thinking that I am saying that it doesn't have a defined value, I know it does. That is why 1 :⇔ .999...

Potemkin wrote:You seem to be thinking of this almost as a temporal infinity; that you add each term after a couple of seconds. Obviously, in that case you never would get to the end. But it's not a temporal process. We can imagine adding each new term at the same time. There's no reason why we can't add all the terms at once.

Well said. Calculus regards it as a contained division, and that is exactly what it is - we only regard the infinity insofar as it divides the whole. It is composed of the whole, thereby being limited in value by the whole, and is thereby equivalent to the whole. If people are having trouble reconciling an infinity with a whole - which is what it sounds like - your example is perfect. One may consider 1 to be an 'end of time' or a spherical 'outline of existence' within which this infinity manifests. Thus, there is no conflict: {.99...} is a state of 1.

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