0.999... does not represent a function or a process or anything like that. It represents a fixed number: the limit of sum from k=1 to n of 9/10^k , as n approac

0.999... does not represent a function or a process or anything like that. It represents a fixed number: the limit of sum from k=1 to n of 9/10^k , as n approaches infinity. You're demonstrating why so many people get confused by the 0.999... notation: they (and you) are thinking of it as the sequence (0.9, 0.99, 0.999, ...) ; but actually it doesn't represent this sequence, or a process for producing this sequence; rather, it represents the limit of this sequence, which is a different type of object: a real number. > Mathematicians are using the symbol … ambiguously Sure, it can mean other things in other contexts (like in my description above of a sequence), but we're talking about what it means in the case 0.j_1 j_2 j_3 ... where the j_k's are base 10 digits. In that case it has a well-defined, non-ambiguous meaning: \lim_{n\to\infty}\sum_{k=1}^n \frac{j_k}{10^k} . > If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. OP is right. I have a degree in math and this is what mathematicians mean by 0.999... > What I'm saying is that the expression could be taken to be one of any number of things Yeah we could have had the notation mean something else. But we didn't. We have other notations for the things you're talking about. Who cares? > That you say the convention is that it most assuredly is 4 is not something I'm buying into. What would it take to make you believe that this is what mathematicians mean by this notation? I promise with all my heart that this is true. Don't take my word for it. Here's some random analysis book. Flip to page 31 (51/686 in the numbering of the PDF): http://www.matematica.net/portal/e-books/Apostol%20-%20CALCULUS%20-%20VOLUME%201%20-%20One-Variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra.pdf http://www.matematica.net/portal/e-books/Apostol%20-%20CALCU... > (I'd nearly go a step further and assert that it ought not be defined this way). Plenty of stuff in math has pedagogically dubious notation, it's not really novel or philosophically interesting to claim you dislike some piece of notation or think it could have been better. It's all just convention. > Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. No mathematician is saying that. The sequence (0.9, 0.99, 0.999...) quite clearly never reaches 1, and any mathematician will agree with that. But its limit is 1. > Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. Well, things don't have to "exist" for us to be able to study them in mathematics, so what's the point of claims like this?