Monday, 8 July 2013

infinity - If 0.999cdots=1 Then Does frac110infty=0?




Recently I stumbled across a, to me, rather strange idea. I was messing around with the proof of 0.999...=1, when I figured that what 0.999... means is that those are all nines. That way I came upon a weird idea. Say a=0.999..., then a=1x. Yet, how do we define what x is? It should say x=0, but my theory did not. If 0.999... is just an endless sequence of nines, then why can't we say x just is an endless sequence of zeroes, ending with a 1, like 110?
If we take the equation n=0.9 for example, then, what would 1n be? Yes indeed, 0.1. Following that theory, can't we say that 0.999...=1110Now, if 0.999...=1 this would be impossible. Go figure. 1=1110 then 110=0 but that is impossible, because we cannot say 100=1When I came to this point, I really got stuck, because, in my head everything I did was right, however, it is impossible. Can somebody please explain to me what mistakes I may have made, and enlighten me about what else I did wrong?




Thanks in advance.
Sjoerd Dorrestijn.



EDT: I prefer 110 to use as an indication of 0.000...01, even though 10= in some way, I just seem to find this more clear.



EDT2: Just to be clear, I read a proof that 0.999...9=1 because it'd be 10.000...0=10=1. What I tried to prove here is that it is not equal to 1, because otherwise maths would collapse. My question was whether I am right or wrong. Since the original statement uses infinity (an infinite amount of nines) I think it is a must to use infinity as well. So, the question is if either the original statement is false, or if I made a mistake somewhere.


Answer



The (proven true) statement that 0.999=1 means that lim. That in turn is equivalent to \lim_{n\to\infty}\left(1-0.\underbrace{999\ldots9}_n\right)=0. Now note that 1-0.\underbrace{999\ldots9}_n=\frac{1}{10^n}, so we have \lim_{n\to\infty}\frac{1}{10^n}=0.




However, although the above reasoning works, it is easier and more straightforward to conclude that \lim_{n\to\infty}\frac{1}{10^n}=0 directly from the definition of the limit, without ever considering 0.999\ldots.



Finally, it is advisable to avoid the notation \frac{1}{10^\infty} altogether, as it is not defined given the usual definition of its constituent symbols.


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