I think part of the reason you're having problems here is that this is indeed counterintuitive. The concept of infinity can get rather complex, especially because there are many different things that we call infinity. For example, there are infinitely many integers and there are infinitely many real numbers, but there are not the same number (for those who prefer being more technical, think about this in terms of cardinality) of integers are there are real numbers.
So, since you've asked this as an honest question, I'm going to give you a serious answer, even though it's probably not the one that you're looking for. This is a historically hard problem -- mathematicians spent many centuries grappling with Zeno's paradox, because in mathematics, you don't debate whether something like this is true, but rather you prove it. And proving it requires really going back and understanding in a formal way what every one of these things means, so that you can reason logically from definitions. Trying to properly understand the answer to this question was a key impetus for the discovery of calculus, and really properly proving the answer to your question requires giving you a rigorous definition of a limit, then using that definition to demonstrate that indeed .9 repeating is equal to 1.
I can give you a couple of good arguments that it should be equal to 1. For example, the most common one was mentioned earlier in this thread.
We can let x = 0.9999...
Then, 10x = 9.9999...., and so 10x - x = 9, thus x = 1.
It bothers me a bit that people in this thread are jumping all over somebody who simply hasn't seen the same amount of math as they have, though, so I think perhaps I should add that this isn't really a very good proof, because we haven't really properly defined what it means to take 10x and what it means for all of these things to converge. Let me give you an example of a similar argument that you might not like as much:
First, a quick aside: since y'all have had some calculus, take the Taylor series for 1/(1+x)^2, and you'll find that it is 1 - 2x + 3x^2 - 4x^3 + 5x^4 - 6x^5 + ... We'll come back to this in a moment.
What is the sum of the natural numbers?
Let x = 1 + 2 + 3 + 4 + ....
Which also equals 0 + 1 + 0 + 2 + 0 + 3 + 0 + 4 + ...
Then, 2x = 0 + 2 + 0 + 4 + 0 + 6 + 0 + 8 + ...
So, x - 2x - 2x = -3x = 1 - 2 + 3 - 4 + 5 - 6 + ...
But wait, I recognize that series! It's the power series I gave earlier, at x=1, isn't it?
So, -3x = 1/(1+1)^2, or x = -1/12. Thus, 1 + 2 + 3 + 4 + ... = -1/12.
Hmm, you might not like that answer nearly as much as .9 repeating = 1. But, explaining why one is a more valid argument than the other is a lot more complicated than y'all were making it out to be upthread.
Incidentally, if you have seen enough math to find a hole in the derivation I just gave, I'd be happy to point you to a more rigorous argument that comes out of a technique we use to evaluate path integrals. This is actually a surprisingly hard problem to get rid of, and really does require a lot of careful work to define how limits work and so forth -- it's honestly not a simple question, even though it's certainly rigorously true that .9 repeating is the same thing as 1. So, I'd hope you'll have a little patience for somebody who hasn't seen as much mathematics as you have, and maybe you'll learn a little something from trying to really come up with a rigorous argument that it's true.
Tell me a bed time story.
on here and I really don't get how that happens. There must be something of which I am unaware. I'm playing with other players of roughly my own level, which of course means I win some and I lose some. I really can't see so many people being so good that they always win so much. Are they not playing people at their level?
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