Whoa... let me stop you RIGHT here. No one is saying that we sum up an infinite number of terms. What we mean by the limit in that situation above is "what's the least upper bound of the nondecreasing sequence of summations?" Or a bit more loose... what's the smallest number that the sequence gets close to? Draqon... what's the highest level of mathematics you have taken? You're arguing from a layman's point of view it seems. That's fine until you start making incorrect assumptions about notation and definition.
As has already been stated, it does equal 1, because there is no number that lies between 0.999... and 1. Your statement about "that cannot be summoned in the first place" doesn't make any sense. Yeah, obviously you couldn't really write out an infinitly long string of 9s. But when you write 0.999... you are talking about what the value would be if the 9s did actually extend out to infinity. Once again, if you don't think that they are equal, then tell us what number lies between them. If there isn't a number that lies between them, they are equal by definition.
I know for a fact klippymitch has seen me argue this before on another forum. It got to the point that I had to start getting very technical about the nature of the numbers. At some point, I brought up the uncountable set [0,1]. I forget how that happened.
So \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{9}{10^{i}}\) is not a number? And don't forget to address my previous reply to you.
There is no number between them. I'll give you an example for an .X... number. I cut a piece of wood .999 inches and the universe grows 1 square inch by 1 square inch by Time frame. Meaning Everything in the universe grows 1 square inch by 1 square inch per time frame. The piece of wood is always .999 for infinite Time frames even though it is growing it doesn't matter because everything else is growing at the same rate so it cancels out. note: This is just made up example. I have no idea what the rate of space expansion per square inch is by time frame.
I'm going to disregard your "example" because it doesn't make any sense. I'm sure others would agree. Now, let's talk STRICTLY about numbers. No physics BS or stupid "real life examples." Those are meaningless to the discuss because mathematics can deal with many things that cannot be represented in real life. Take for example Gabriel's Horn. Perhaps you have heard about it. Basically, it's a mathematically object that have a finite volume equal to \(\pi\). But, it has an infinite surface area. CLEARLY this doesn't exist in the real world.. nor could it ever. But we can invent one with mathematics. As far as your statement that there exists no number between 0.999... and 1, there must be a number next to 0.999... what number is this? How about the number next to that one? If you can answer those questions, I will detect "the pattern" and I will devise a counting function. I will explain everything else at that time. No, it's an expression. I see that you don't even use the proper terminology learned in high school algebra.
Oh really? Did you read what I said? I said the exact same thing. Now read EVERYTHING I posted because I expect answers to my questions. When did I say different? Fuck, all numbers are concepts. And I don't consider there to be just "infinity." There are actually many different "infinities." Some bigger than others... but none of them are numbers since they cannot be manipulated to behave like numbers.
.999... .999... = 123456789 01233456789 00123456789 000123456789 0000123456789 00000123456789 .... That's the best I can explain it.
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Heh...I have no idea. Apparently when Dragon says "0.999..." he means something different from the rest of the world, but I don't really understand what his unique definition is. Klippymitch's comments make me think that he is somehow imagining that the "..." is sort of an active process, with the series of 9s appearing one after another, with new ones appearing at every moment. So, although it might be arbitrarily long, it will never actually equal 1 because you would have to wait an infinitely long time for the numbers to all appear. Of course, that's not what "..." means. In 0.999... the 9s are already at an infinite length, so there isn't any sort of time issue. Of course, maybe I'm completely misunderstanding him.
Ok but your "time frame" example thingy (???) doesn't address my questions. If we're talking strictly about mathematics then it's no contest: you're wrong and I'm right. If you are making the claim that 0.999... does not exist in the real world... then we have some issues. 1) What is a number in the real world? 2) What does it mean for a number to exist in the real world? Trust me, you are. I have PLENTY of experience dealing with him... and even I have no fucking clue what he is talking about.
