Mark, I respectfully could hardly disagree more strongly. The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers. Now, I have nothing against exotic theories of numbers. (In fact, I love the surreal 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 have no idea what vertiginous or any other kind of semi-circular reasoning the word "only" obscures. But let me remind you that the fundamental group of a semi-circle is trivial, as compared to that of a circle, which is the integers. Of course, there's always this famous quote: << "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."
--Dan On 2013-01-28, at 11:42 PM, Marc LeBrun wrote: << "Dan Asimov" <dasimov@earthlink.net> << The only way to understand an infinite decimal representation . . .
Hear hear! And here we go 'round the infinite decimal discussion, again... Look, I've nothing 'gainst the Standard Theory of the Real Numbers (STotRN); an' I too'll testify t' y'all, it surely do come in plenty handy, of a time. So (pacem, please, concerns of arithmetical apostasy) what bugs me is that darn word "only", and the vertiginous semi-circular reasoning it obscures. Surely there are many ways to "understand" infinite decimal representations? . . .