="Dan Asimov" <dasimov@earthlink.net> The only way to understand an infinite decimal representation ... ="Fred lunnon" <fred.lunnon@gmail.com> Nobody else has had the poor taste to muddy the waters further by introducing non-standard arithmetic.
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? Of COURSE when we START by implicitly binding the strings 0.9, 0.99, 0.999 and so on to a convergent sequence of real numbers then 1.0 is where our interpretation of the limit of the sequence of strings MUST inevitably lead. But strings and numbers are truly different things, and different routes to the infinite can and do afford divergent perspectives. For example we've discussed the mirror world of 9.0, 99.0, 999.0 and so on. We can also model these nicely using the SAME closed form for the finite sums of a geometric series. Except now, in passing to infinity, we get a sequence of INCREASING POSITIVE values whose "algebraic limit" is demonstrably -1! The quandary is, if we favor the algebra we break with STotRN convergence. But once string ordering no longer entails strong ordering there's no longer an air-tight argument that 0.999... HAS to NAME the same THING as 1.000... And of course when it doesn't we can get "perfectly good" but "non-standard" arithmetical systems that are internally consistent (in some symbolic sense) but royally suck at modeling our conventional magnitudinous quantities. I'm totally OK with explaining to folks why STotRN PLUS the usual implicit interpretation of numeral strings leads to the identification of 0.999... with 1.000... Just, please, at least, can we somehow cool it with glibly calling 0.999... a "number"?