| I agree with you that the ... notation is ill-defined. It's common in mathematics to use conventions that are a little woolly, but only where everyone understands how to express the idea more correctly. Here's one way of presenting this formally. Define a 'decimal' to be an ordered sequence of integers (called 'digits') a_1, a_2, a_3, and so on. (By 'and so on', I formally mean that for each positive integer k we have a digit a_k at the kth position in the sequence). Let's say each a_k has to be between 0 and 9 inclusive. For each positive integer k, define the 'kth partial sum' of the sequence to be the sum from j=1 to j=k of (1/10^j) x a_j. I'll skip over what it means if we say that the partial sums converge as k->Infinity, because it sounds like you understand what limits are and how they work. If not, I'd be happy to explain. Now, if the partial sums converge to some value 'd', we say that the decimal has value equal to d. It can be shown that any decimal has at most one such d (which is good, because a decimal shouldn't have two values). Now, I think you'd be happy to say that when we write 0.9999999..., what we mean is the decimal where a_k=9 for all k. Given this definition, it follows that the value of the decimal is precisely 1, using properties of geometric sequences. It is up to you how you define '...', but all mathematicians would agree that 0.99999... should be interpreted as above if it is to have any meaning at all. If you really want a more precise eplanation of '...' at the end of a truncated decimal, I would provide the following: "Writing 0.abcdef... asserts that the digits abcdef of the truncation provided have a pattern which should be obvious to the reader. Assign the first few digits a_1, a_2 etc as per the part of the decimal that is explicitly given; and then assign all subsequent digits values according to said pattern." - it's not a formal notation, as I say, but rather a convenient shorthand that is understood by working mathematicians. It is always possible to be more precise if one has to be. |