Wikipedia has an article entitled 0.999...
The article is quite good, and includes at least sketches of the various proofs that 0.999... = 1 and some interesting historical tidbits; I won't recapitulate those here.
The talk page and the arguments page are interesting, in a different way; one gets to watch people who don't really understand why this should be true, but want to understand it. (Of course there is the occasional crackpot. It might give you a headache, though. It's like sausage being made.
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4 comments:
If I were teaching this material I would write 1.000... and 0.999... In the representation of the reals we are talking about the repeating decimal 1.000... rather than the counting number 1 ("one") which seems to me part of where the confusion comes from in the heads (and hearts) of students.
REH
I love the proofs.
Now I have go and reread calculus proofs.
The article also lead me to Cantor Sets; they are very beautiful.
I wish I had stumbled on them 40 years ago.
Aha! It looks like the war over this non-issue is still raging! As stated, it looks to me like a purely notational problem with an obvious solution. If we know some number exactly and we know that it is 1, we simply don't write it as .(9), and if that number is approximate (a result of an experiment or an approximate calculation, for example), the infinite trailing sequeces of 9s never enter the picture.
On the other hand, the persistence of this discussion is an uncomfortable reminder of how subtle the notions of the real numbers (and there is nothing objectively real about them, as they exist only in our minds) and limits are, and how poorly they are understood (at least by the lay public). Fortunately these thorny issues are relevant only on the philosophical or purely mathematical level, while in practice we are only concerned about drawing approximate answers from approximate data.
looks like a great piece;
i'll look it over more carefully
sometime soon ... but meanwhile,
they appear to have left
these poymathematics threads
out of their external links
(so i add a link here).
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