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## #951 2008-12-26 17:55:16

George,Y
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Registered: 2006-03-12
Posts: 1,306

### Re: 0.9999....(recurring) = 1?

BTW, the "proof" of ∞=∞+1 is fake, or two proofs, one is mapping, the other is stating larger than finite.

for the former one
0.999...->0.0999... every 9 has already done one'o'one mapping
and +0.9 strictly adds in one more 9 to the latter
the only reason now to equate the 0.999... and the new 0.9999...
is by their similarity or unexaughstive counting, by simply looking at
0.9 0.99 0.999 ...
0.9 0.99 0.999 ...
and say since first second and third 9's are the same, and the examine process can go on beyond our human's ability to count all over
the 9's in the two are the same amount.

such finite "examination" is plausible because this method itself is defected since it is not done at all. (Have it examined "all" digits for either 0.999...?) You cannot force us to testify someone innocent by stressing s/he was not seen in the crime scene. And another piece of evidence might be sufficient to nail down the case.

The only exhaustive method is by wholistic mapping
when you do the division by 10 to 0.999...
every digit is already assumed moving rightwards by one digit
(or if you try to move the 9's one by one, you never can finish the division in your life
and you should not claim you can do the division in your life if you are rigorous),
I want to mention here not a single one more 9 is created or less 9 is diminished in this division
otherwise it is not defined as "every" or "all" (however it can be relaxed if Ricky you want)
then add in one more 9
yes just the one more 9 to compose 0.9+0.999.../10
and now we have already created ∞+1>∞
and then you can use my proof to just locate where the one more 9 is
yes the one more 9 is 0.9
but taking it at a different angle, suppose the 9's representing the same 9*10^-r play the same role and be treated the same in both 0.999...'s and from previous knowledge we know there is one 9's difference,  one not and one has, the former can be filled with 0.
And this pair of 0 and 9 will not have a succeeding 9 'n' 9 pair to compose the paradox
...09
...99
Logically it follows this pair is on the right end. A 9 has no 9's on its right is clearly the right end of a bunch of 9's

Simple,huh? And I did not use countive mapping to try to conclude not enough evidence for the last 9. Additional evidence has already been provided to pin it down.

appendix
"I want to mention here not a single one more 9 is created or less 9 is diminished in this division
otherwise it is not defined as "every" or "all" (however it can be relaxed if Ricky you want)"
Now I relax it, but could anyone tell me which 9 can be diminished? The 9 at the right end? or the 9 without another 9 on its right? Still, the answer leads directly to the 9 at the end.

Last edited by George,Y (2008-12-26 17:56:19)

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## #952 2008-12-26 22:13:17

luca-deltodesco
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### Re: 0.9999....(recurring) = 1?

George, you simply cannot understand, you constantly go on the lines of; oh look there is a load more 9's WITH AN END, you keep constantly talking about the 'end' of the 9's. THERE ISN'T ONE. You can't try and prove disprove anything by talking about what happens with the end of the 9's because there ISN'T AN END. It's like trying to say that if you keep going round the earth you will eventually get to the end of the earth, but we as humans can't get to it because we only live a finite amount of time.

'
the only reason now to equate the 0.999... and the new 0.9999...
is by their similarity or unexaughstive counting, by simply looking at
0.9 0.99 0.999 ...
0.9 0.99 0.999 ...
and say since first second and third 9's are the same, and the examine process can go on beyond our human's ability to count all over the 9's in the two are the same amount.
'

There is no difference between 0.9... 0.99... 0.999.... 0.99999999999999999999999999999999... it's just a choice of how many you want to write down before saying that the 9's are recurring.

'
and the examine process can go on beyond our human's ability to count all over the 9's in the two are the same amount.
'

There is no 'amount' of 9's for which to count over, there isn't some unimaginable number of 9's there are an infinite amount of 9's there is no finite end to speak of to give an amount to consider and to then say, at the end of all these 9's there will be 1 more 9 on that number, because that end can never be reached, not because we as humans cannot consider the end, but because it simply does not exist, in the same way that 0.000....1 does not exist, where is the 1? at the end of the zero's ? but there isn't an end to the zeroes, if you place a 1 at any imaginable, or unimaginable point then you are restricting the number of zero's to some finite number, which means it is not 0.000....1 but some number with some very large number of zero's followed by a 1.

Last edited by luca-deltodesco (2008-12-26 22:14:12)

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## #953 2008-12-27 03:43:26

George,Y
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### Re: 0.9999....(recurring) = 1?

but we as humans can't get to it because we only live a finite amount of time.

Luca, you don't understand.
Because human cannot reach the end of 0.999... by counting one by one, then the end of 0.999... does not exist?
By the same reason, if humans cannot count 0.999... all over, does it mean 0.999... does not exist?
Because humans cannot finish the recursive division 1/9 under finite time, does it mean 0.111... does not exist?

Counting is only one method, and it is not sufficient at all even to rigorously show the very existence of infinitesimals. Since infinitesimals need imagination beyond empirical counting, it is very fair to examine it by logic beyond counting.

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## #954 2008-12-27 03:49:52

George,Y
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### Re: 0.9999....(recurring) = 1?

Historically, the reason to reject any infinite set is that it cannot be constructed by finite counting. And hence studying that is totally ungrounded. The very people holding this idea is called intuitionist (there are already a bunch in this post), often regarded as stubborn by mathematicians for set theory and real number definitions. However, the latter have little thinking superiority when it comes to prove various conclusions in infinite sets. When there is a fallacy? Cannot count to, does not exist.

Last edited by George,Y (2008-12-27 03:52:42)

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## #955 2008-12-27 04:07:46

luca-deltodesco
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### Re: 0.9999....(recurring) = 1?

(Referring to first) you are taking that quote out of context, that was referring to you; you were using that argument of not being able to count it as humans to argue your own point, i was rebuking it.

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## #956 2008-12-27 04:29:52

George,Y
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### Re: 0.9999....(recurring) = 1?

Yes luca, but that was previous me. Someone whose name begins with D reminded me of thinking all the decimals as a whole, altogether, and imagine it is reached infinite amount. Since then I adopted his idea and found when there is a whole, there is an infinitesimal or infinitesimals within it and it is at the right end and its backwards.

However, this is not only to you, luca. All set theorists and real number believers took one by one counting approach since Cantor named infinite set. And they are handicapped by counting method to discover the very logic flaw in infinite sets.

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## #957 2008-12-28 11:38:15

Ricky
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### Re: 0.9999....(recurring) = 1?

Luca, you are arguing with a person who doesn't believe that infinite exists in mathematics.  You can't argue over the definition of a concept that the other person believes doesn't exist.  It's like arguing how long God's beard is with an atheist.

George has taken a philosophical stance that precludes him from asking whether or not 0.999... = 1.  What George should be arguing is whether the notation "0.999..." makes sense.  In his philosophical universe, it doesn't.  And you know what?  That's OK.  Honestly and sincerely, it doesn't matter that most of George's philosophical positions were abandoned shortly after the time of Cauchy and Dedekind.  Philosophical views can be argued over logical consistency, rationality, and effectiveness.  From what I can remember, George's view is logically consistent.  It is basically the axioms of Zermelo minus the axiom of infinity.  And I don't think I can call George's philosophy irrational.  Indeed, I can't argue that infinity exists in this universe, though I would absolutely love to be shown otherwise.  The only thing that's left is the effectiveness.  What is it that George's views accomplish?  Well, as he has himself admitted, not much.  It precludes him from doing any traditional mathematics (Analysis, Algebra, Differential geometry, number theory, and so on) except for perhaps some subsets of discrete mathematics.

So two out of three ain't bad.  The third for me is reason enough to not accept his views, but perhaps if he works hard he can reinvent calculus a third time.  Of course I doubt that it's possible and would be willing to bet large sums of money on this, if I were a betting man.  Indeed by definition George's view on the nonexistence of infinity seems to preclude the idea of any sort of calculus.  I'm not even sure how George sleeps at night since the concept of a rational number is so awkwardly (not) defined.

But all of this is philosophy and not mathematics.  In fact, the only real error I see George making is conflating his philosophical views with mathematics.  Why he is arguing over 0.999... = 1 is beyond me.  He should start a different thread on the existence of infinity in mathematics.  At least that way we can debate over the real issue inside of the corollaries that come off it.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #958 2009-01-11 06:24:35

1q2w3e4
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### Re: 0.9999....(recurring) = 1?

this is nothing i heard that 1+1=1,but i agree.

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## #959 2009-01-11 06:32:05

1q2w3e4
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### Re: 0.9999....(recurring) = 1?

i can still prove that infinite isn't same as infinite+1.

infinite=1*x,
but infinite+1=1*(x+1),
so 0.999... has 1 more decimal than 9.999...=10*0.999...

Last edited by 1q2w3e4 (2009-01-11 06:34:34)

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## #960 2009-01-11 07:18:04

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

### Re: 0.9999....(recurring) = 1?

you're assuming ∞+1 =/= ∞, to show that ∞+1 =/= ∞, that is not allowed, i can equally show that pi = 1, by assuming pi = 1 to show that pi = 1, it shows nothing.

Last edited by luca-deltodesco (2009-01-11 07:19:39)

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## #961 2009-01-14 13:51:47

1q2w3e4
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Registered: 2009-01-11
Posts: 10

### Re: 0.9999....(recurring) = 1?

No,no,no!
I am saying that 9,999...(=10*0,999...) has one less decimal than 0.999...
Also (10*0,999...)-(1*0.999...)=(10-1)*0.999...=9*0.999...

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## #962 2009-02-10 12:21:53

George,Y
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Registered: 2006-03-12
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### Re: 0.9999....(recurring) = 1?

Just irrelevant. Where is the guy who posted this topic anyway?!!!

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## #963 2009-02-26 08:44:49

vbjanine11
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Registered: 2009-02-26
Posts: 1

### Re: 0.9999....(recurring) = 1?

I have spent a while reading some of these posts and I thought Id put in my 2 cents
I feel as though I need to remind some people that infinity is not a number (its a concept). Therefore, properties of real numbers do not apply to it. So you cant put infinity into a mathematical equations such as infinity=1*x and then add one to both sides. (You can only do that with numbers; and infinity is not a number.)
Also: as hard at it is to believe, I am convinced that .999999 does =1. (I dont particularly like it, but I know that it is true.)

I havent seen this posted yet, so I thought Id add this argument to the mix

1/9 = .1111
2/9 = .2222
3/9 = .3333
4/9 = .4444
5/9 = .5555
6/9 = .6666
7/9 = .7777
8/9 = .8888
9/9 = .9999

But 9/9 also = 1. So, (using substitution) 9/9 = .9999 becomes 1 = .9999

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## #964 2009-02-26 08:50:26

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

### Re: 0.9999....(recurring) = 1?

vbjanine11 wrote:

But 9/9 also = 1. So, (using substitution) 9/9 = .9999 becomes 1 = .9999

George's argument would be that none of these equalities are correct because neither 0.111... nor 0.333.. exist as they are continously growing. Yes, growing .

Anyways yeh, i have no idea what to add; my stomach hurts so i'm going to sleep lol.

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## #965 2009-03-03 00:14:17

vlgi
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Registered: 2009-03-02
Posts: 2

### Re: 0.9999....(recurring) = 1?

The 0.9 recurring issue is one that is an artefact of the base 10 system for representing numbers.

Its important to understand that all numbers are abstract representations of values, and an infinitely recurring number is just a more complex abstract representation of a value.

I find a good way to convince people that 0.9 recurring is in fact the same as 1 is to subtract 0.9 recurring from 1 and see what you are left with. The answer is 0.0 recurring. Its hard to argue that a zero with an infinite number of zeros after it is not zero thus 0.9 recurring must equal 1.

The problem with the concept of 0.9 recurring is that it involves infinity, and that is a concept that is really hard to get your head around. It helps to understand that infinity does not exist and the number 0.9 recurring does not exist, no person could ever write it down, no one will ever have 0.9 recurring of anything, just as no one could count to infinity, because it is not a number its an idea, its a symbolic representation of concept.

Some people find it difficult to understand that a number can have more than one representation in a number system, however few people have a problem with 1/3 and 2/6 representing the same value. Both numbers represent the same value, just as 0.9 recurring and 1 are the same value.

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## #966 2009-03-03 01:45:26

Ricky
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Registered: 2005-12-04
Posts: 3,791

### Re: 0.9999....(recurring) = 1?

It helps to understand that infinity does not exist and the number 0.9 recurring does not exist, no person could ever write it down, no one will ever have 0.9 recurring of anything, just as no one could count to infinity, because it is not a number its an idea, its a symbolic representation of concept.

When you say "exist", I take it to mean in the real world, not mathematics.  I see no reason to limit mathematics by what a person can or cannot do.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #967 2009-03-03 02:55:57

vlgi
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Registered: 2009-03-02
Posts: 2

### Re: 0.9999....(recurring) = 1?

Ricky wrote:

It helps to understand that infinity does not exist and the number 0.9 recurring does not exist, no person could ever write it down, no one will ever have 0.9 recurring of anything, just as no one could count to infinity, because it is not a number its an idea, its a symbolic representation of concept.

When you say "exist", I take it to mean in the real world, not mathematics.  I see no reason to limit mathematics by what a person can or cannot do.

Well its difficult to explain exactly what I mean, maths doesn't have a corporeal existence, there is no 1 that exists in reality. I have one glass of water, but if I split it into 2 glasses do I have 2 half glasses or 2 glasses of water, whether it is 1 or a half is entirely context sensitive. If send those 2 glasses of water to different places so other people have no knowledge that this glass of water is actually half a glass of water, would it be half a glass of water or just 1 glass of water. If they split the glass of water in half again are they quarters or halves? The answer is only relevant to the context. Maths is a model or a tool for explaining things. It can be used to explain reality, but it can explain things that don't exist in reality as well.

Things like infinity get a lot more difficult to explain because this is an abstraction that exists in Mathematics which itself is an abstraction. The idea of oneness can be applied to things in reality, but the idea of infinity cannot be correctly applied to things in reality. Even in mathematics it is a nebulous entity.

So going by that reasoning, the number 1 does not exist (nor does maths), neither does infinity, but infinity exists half as much than the number 1 because its an abstraction of an abstraction.

Its sounds very metaphysical but I think the important thing to understand that maths does not exist in the world as corporeal thing, it is entirely manufactured by the mind, there are many things that are not possible in reality but can exist in the mind. Maths is a model with corresponds very closely with reality within strict or perhaps not so strict boundaries, but it is not reality, reality isn't as neat as maths.

You can explain some maths with reality, and some reality with maths but the problem is if you start to think the two are the same thing, if you think reality is maths or maths is reality.

Its down to human perception (And by this I mean what we perceive with our senses) really when we perceive things we perceive them in a way which fits with how we understand things. We expect to see things in a specific way and so we make them fit that way. Look at the stars, do they make patterns? Do you see a plough or a bear or a dipper? You recognise Orion the Hunter. They are just dots, you brain interprets them as a pattern you have learnt to recognise.

In this way if we rely only on our perception then we are limited to it, the trick is to understand the limits of perception, and that there are some things you can't perceive naturally, that don't make sense in your natural understanding, because of the way you are wired, it doesn't mean they don't exist, or you can't accept them.

In the mind we are only limited to what we can imagine, its easy to think in the same way you perceive the world through your senses. But there are things beyond what we can perceive, atoms and molecules so small we can't see them, galaxies so far away we can't see them. And the mind is a playground we can move beyond our perceptions into a bigger world, a world that does exist but we cannot perceive directly, and into those than don't exist as well.

Anyway so I think an understanding of the abstract nature of maths is important, and one thing that is rarely taught to people. Without such understanding simple things are difficult to accept, like 1 and 0.9 recurring being equal, it doesn't make sense or is not correct if you do not apply the correct rules. If you use the rules of reality, of your perception of the world then it doesn't make sense.

And so I agree with you there is no reason to limit maths to reality, just as we don't limit our understanding to only what we can sense.

And so you can see clearly that 0.9 recurring = 1

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## #968 2009-03-03 11:34:07

Ricky
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### Re: 0.9999....(recurring) = 1?

Its sounds very metaphysical but I think the important thing to understand that maths does not exist in the world as corporeal thing, it is entirely manufactured by the mind, there are many things that are not possible in reality but can exist in the mind. Maths is a model with corresponds very closely with reality within strict or perhaps not so strict boundaries, but it is not reality, reality isn't as neat as maths.

This is a philosophical stance, yet you state it as fact.  I'd be interested in discussing such philosophy in another thread, but not here.  Feel free to start one if you are interested as well, either in Euler Avenue or Dark Discussion I think would be the most appropriate.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #969 2009-03-17 17:13:13

MrRHQ
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Registered: 2009-03-15
Posts: 8

### Re: 0.9999....(recurring) = 1?

I think when numbers are divided by x/9, there is an unprecedented number on the very end of the infinite range of decimal places where there is a number 1 greater than 9 to equal a value of 1:
Example:

7/9 = 0.7777777777777.....................8
8/9 = 0.8888888888888.....................9
9/9 = 1

This is proven by dividing by 9, this is actually NOT like solving for a infinite sum of 9 put to every decimal place. This does not have the number in the ellipsis to change it to 1. This would be as illogical as saying 999,999,999,999,999,999,999,999,999,999,999,999,999 = 1,000,000,000,000,000,000,000,000,000,000,000,000,000. I think decimal places have meaning as much as normal integral places, they are parts, and imprecise as you want it too, but clearly the equation above must be false even if there is one less.
To conclude this:

0.999999999999999999999999999 = 0.999999999999999999999999999
0.999999999999999999999999999.......................1 = 1

Or don't trust me and trust Wiki
http://en.wikipedia.org/wiki/0.999...

Last edited by MrRHQ (2009-03-17 17:16:48)

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## #970 2009-03-17 20:31:03

luca-deltodesco
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### Re: 0.9999....(recurring) = 1?

MrRHQ wrote:

7/9 = 0.7777777777777.....................8
8/9 = 0.8888888888888.....................9
9/9 = 1

How can you call this logical? the elipses are a notation used to denote that there are an infinite number of the preceeding digit in this case, you can not put anything after this: You say there is an 8 after the end of the infinite number of 7's? That makes no sense, because placing an 8 at any point along the length of 7's would be to give the number of 7's a finite value, and the number of 7's would no longer be infinite.

MrRHQ wrote:

This would be as illogical as saying 999,999,999,999,999,999,999,999,999,999,999,999,999 = 1,000,000,000,000,000,000,000,000,000,000,000,000,000.

No, the comparison you should be making if you want to make such a silly comparison; is that

999999999999...9999.0 = 10^infinity = infinity, aka an infinite number of 9's preceeding the decimal point is equal to infinity; which is by definition infinity anyways and therefore true, since that length of 9's is infinitely long, it is infinite big, and therefore equal to infinity.

Nothing in the wikipedia article agrees with you; beyond that 0.999... = 1, which is something we (bar george) agree with anyways.

Last edited by luca-deltodesco (2009-03-17 20:31:53)

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## #971 2009-03-24 16:33:42

George,Y
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### Re: 0.9999....(recurring) = 1?

The illogical thing is that you can only get a bunch of 9's growing by arithemetic, you claim you can get them all. Moreover, when I question the ending 9 in that "all", you go back to the statement that you are still growing the 9's.

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## #972 2009-03-24 23:29:34

luca-deltodesco
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Posts: 1,470

### Re: 0.9999....(recurring) = 1?

And once more, you insist on this 'growing' idea, which none of us have said, the 9's aren't growing, they are already there; we don't have to claim we can get them all, because we don't have anything to get; they already exist.

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## #973 2009-03-26 00:17:47

George,Y
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### Re: 0.9999....(recurring) = 1?

Fine, I like your confession. Go back to post 952. "they" include the ending 9.

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## #974 2009-05-28 00:01:09

MathGenius1
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Registered: 2009-05-27
Posts: 11

### Re: 0.9999....(recurring) = 1?

Hi Everyone!

You Cannot End Something...Or Say Something Is/Or Has Already Ended  When It Is Possible To Count What Exists!!

Example Infinite 0.999...It Is Possible To Count The .9's From .9 Onwards Which Equals A Count Of 1 Then .99 Equals A Count Of 2

Infinite 0.999... Has To Have A Continuous Count Value And The Count Value Will Always Be From 1 Onwards  Because The Count Will Always Exist!!

As Soon As The .9's Are/Or Become Equal To 1 Then The Continuous  Count Value Ends? Which Is Contradictory To The Infinite Count Value Representing The .9's

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## #975 2012-06-22 12:57:16

anonimnystefy
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### Re: 0.9999....(recurring) = 1?

I  have been looking at this thread for quite a while now. I would have to state my newly developed opinion and that is that 0.999... doesn't exist.

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