To Eric Angelini: Besides what Whit and Michael said, The only way to understand an infinite decimal representation N.abcde... is to understand a) That it represents the limit of the sequence S = N.a, N.ab, N.abc, N.abcd, N.abcde, ... . b) What it means for a sequence of real numbers to have a limit, and that when this is true, the sequence has only one limit. c) That the sequence S must in fact have a limit. So e.g. 0.99999... means the limit of the sequence of real numbers S_1 = 0.9, 0.99, 0.999, 0.9999, 0.99999, ... . This must be 1, because no matter how close you may want the sequence to get to the limit (say within the number e > 0 of the limit), then after a certain number of terms of the sequence, the rest of the terms all lie within e of 1. (In other words, they are all between 1-e and 1+e.) Let's *check* this claim. If we pick a sequence of closenesses e_1 > e_2 > e_3 > ... > e_n > ... > 0 such that this sequence contains arbitrarily small numbers, and we verify the above sentence for e = e_n no matter what n we choose, then we have verified that 0.99999... = 1. So, let's choose e_n = 1/10^n for each n = 1,2,3,... . Clearly these get arbitrarily small. Then for any e_n = 1/10^n, we can look at all the terms of the sequence S_1 *after the nth term*. Now, the nth term has n 9's after the decimal point, so the terms *after that* have at least n+1 9's after the decimal point. It's easy to check that this means they all lie between 1-(1/10^n) and 1+(1/10^n). This proves that the limit of the sequence S_1 is 1. This means that 0.99999... = 1. --Dan On 2013-01-28, at 6:12 AM, Michael Kleber wrote:
On Mon, Jan 28, 2013 at 7:16 AM, Eric Angelini <Eric.Angelini@kntv.be>wrote:
And what about: 1/3 = 0.6666666666666666666666666666666... ... re-written as: 1/3 = 0.6599999999999999999999999999999... 1/3 = 0.6666659999999999999999999999999...
The only real numbers with two decimal representations are integers divided by powers of ten; those can be written with an infinite tail of 0s, as usual, or an infinite tail of 9's. Your choice of 2/3 isn't of this flavor; it can only be written one way. Your versions with an infinite tail of 9s are indeed one of two representations -- but not of 2/3! For example,
0.6666659999999999999999999999999... = 0.666666
--Michael
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun