> 0.9 is not equal to 1,
> 0.99 is not equal to 1,
> 0.999 is not equal to 1,
> 0.9999 is not equal to 1,
> 0.99999 is not equal to 1,
> 0.999999 is not equal to 1,
> and so on, ad infinitum.
You are correct about all of these, and all finite strings of the above form.
> Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.
I think it's because you, and a lot of other people in this thread, are turning the question on its head. The difficulty does not so much lie in figuring out whether 0.999… is equal to 1 or not, but rather in what we mean when we write 0.999….
I know I'm repeating myself from elsewhere in the thread, but I'll try again. Try to go through these step by step, and feel free to let me know where you lose the thread.
DEFINITION: A finite decimal representation of a real number is a finite string of the form `a_m a_{m-1} … a_0 . b_1 b_2 … b_n` where each `a_i` and each `b_i` is a natural number between 0 and 9 inclusive (a digit). We say that this finite decimal representation represents the real number
a_m*10^m + a_{m-1}*10^{m-1} + … + a_0 + b_1*10^{-1} + b_2*10^{-2} + … + b_n*10^{-n}.
Note:
The previous definition deals with
finite strings and
finite sums. I hope we can agree that these are well-defined and unambiguous concepts.
EXAMPLE: The string `12.98` has `m=1`, `n=2` with `a_1=1`, `a_0=2`, `b_1=9` and `b_2=8`. It therefore represents the real number
1*10^1 + 2*10^0 + 9*10^{-1} + 8*10^{-2}
(duh!).
Within this standard framework, there is no way to ask "what is 0.999…?. It is not yet defined, because we have only defined what finite strings mean. The standard definition for what one means by 0.999… follows. (One can obviously also define these things 0.888…, 1.999…, etc., but let's stick to one case here).
DEFINITION: Let `(c_n)_{n natural}` be a sequence of real numbers (let me know if you need a definition of sequences!). We say that the sequence has the limit x as n tends to infinity (these are words, you don't have to ascribe meaning to "infinity" in that sentence – it's just a word, like "gnarf"!) if, given any real eps>0, there exists an M such that for all m > M, |c_m - x| Definition (this is the definition you have to wrap your head around before continuing): Consider the sequence `(c_n)_{n natural}` where `c_n` is the finite sum
9*10^{-1} + 9*10^{-2} + … + 9*10^{-n}
The string `0.999…` (which we colloquially speak of as "zero point nine nine nine with nines repeating forever") denotes the limit of the sequence `(c_n)_{n natural}` as n tends to infinity (if it exists).
"THEOREM": The limit defining `0.999…` does exist. It is `1`.
PROOF: You can fill this in. If you can't, I'm happy to do it.
As you can see, at no point in the above did feelings or beliefs matter :-)