="Dan Asimov" <dasimov@earthlink.net> Mark, I respectfully could hardly disagree more strongly.
Yes, and also for my part. Please know any florid vehemence in this disputation in no way diminishes my affection or respect for anyone.
The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers.
[...] But when people want to know whether, and if so why, .99999... = 1, they are in almost all cases wanting to learn or be reminded of the conventional meaning of that statement.
And that requires understanding a) that an infinite decimal represents an infinite sequence, and b) that this sequence has a limit, and c) that having a limit has a very specific meaning that can be verified in this case.
I guess my main concern is that in our enthusiasm to get on with b) and c) we insufficiently acknowledge that step a) is where some key conventional choice-making comes in (from which the rest follows so agreeably). I feel a tug of intellectual obligation to examine and acknowledge that critical step (much as I'd feel some duty to somehow at some point distinguish a line drawn with pen and ink from an abstract geometrical line, or to mention there are non-Eucildean options, etc). Glossing too glibly that .999... "is" a real number leaves unexamined some curious topics which I tried to allude to before (no doubt unintelligibly). For example someone might wonder: if the half-infinite .999... "is" a number, then is the doubly infinite ...000.999... also a number? Surely they seem equal, and hence both equal to 1. But then if doubly infinite ...000.999... is a number, couldn't doubly infinite ...999.000... also be a number? And indeed one can engage in all sorts of arithmetic manipulations with these strings and very consistently find that ...999.000... = -1. So we have developed a kind of "numeral theory" which seems consistent with the intuitive ideas we have of numbers. But if we now try to incorporate that a)b)c) there's a dissonance to resolve, because the increasing sequence 9.0, 99.0, 999.0 clearly has limit -1, which is less than any of them. This line of reasoning seems to lead to a serviceable interpretation of repeating decimals which moreover covers a larger domain but which doesn't seem compatible with the usual ordered real line model. Now awareness of this approach doesn't mean I reject the utility of the STotRN nor that it's mandatory to drag in non-standard exotica, I'm just uncomfortable when presenting the conventional practice with implying it is the ONLY way to think on these matters, nor that it covers the whole topic.
="Allan Wechsler" <acwacw@gmail.com> I would love to hear of any theory in which it makes sense to say 0.999... != 1.
Um, I'd say "string theory" but that seems to be taken, so "numeral theory"?
="James Propp" <jamespropp@gmail.com> (Or, as I was taught to say forty or fifty years ago, these two _numerals_ represent the same _number_.)
Yes. Perhaps my mind was simply disfigured by "New Math", but thank you for acknowledging the distinction, and for the wealth of interesting commentary.