 The Old 0.99...=1 - Page 2 - Politics Forum.org | PoFo

Wandering the information superhighway, he came upon the last refuge of civilization, PoFo, the only forum on the internet ...

### The Old 0.99...=1

For the discussion of Philosophy. Discuss thought from Socrates to the Enlightenment and beyond!

Moderator: PoFo Agora Mods

Forum rules: No one line posts please. Religious topics may be debated in this forum, but those of religious belief who specifically wish to avoid threads being derailed by atheist arguments might prefer to use the Spirituality forum.
#656724
So in this case its just better to look at the whole thing in fractions. (or 1 as simpley being 0.999...)
#656725
Yes, exactly.
#656728
I was just thinking of something mentioned above...

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?
#656932
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.
#656942
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. #657387
T his must be what the Beatles song "Number Nine" was alluding to. That Yoko. What a genius.

*Sound of bong churning out another hit of deepness"
#657698
0.9999... = Sum 9/10^n
(n=1 -> Infinity)

= lim sum 9/10^n
(m -> Infinity) (n=1 -> m)

= lim .9(1-10^-(m+1))/(1-1/10)
(m -> Infinity)

= lim .9(1-10^-(m+1))/(9/10)
(m -> Infinity)

= .9/(9/10)

= 1
#657757
The difference between .999... and 1 is infinitesimal, but not non-existant.
#657776
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.
#657788
Apollos: if they are different, what is 1 - .999...? 0.000...?
#657846
The difference between .999... and 1 is infinitesimal, but not non-existant.

Wrong.
#657869
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.
#657911
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.
#657938
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 one is a finite number and .999... is an infinite number.

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:
The Art of the Infinite: The Pleasures of Mathematics
By Robert and Ellen Kaplan
The Nothing That Is: A Natural History of Zero
By Robert Kaplan

They both deal with the same type of things and are well put together.
#657987
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)
That is what we mean when we write 0.9999...., that and nothing else. The limit of that convergent infinite sum as n goes to infinity is 1. Therefore 0.9999...=1. QED.
#657992
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.
#658004
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.
#658009
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.
#658028
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...
#658031
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.
###### The Evolution Fraud

This is actually opposite of evolution. Wrong! […]

###### Trump's Dumb Economics

Trump is sinking that ship. The U.S. and world ec[…]

###### How ISIS replenishes its ranks

They aren't apostates, they're reformists. Don't[…]

###### Al-Aqsa Mosque Issue in Palestine May Be More Dangerous Than a Nuclear Weapon

If they want to pay for a walled corodor all the w[…]