September 30th, 2016, 12:43 PM  #11 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,081 Thanks: 87 
The difference between .5000... and .4999... approaches 0 as n approaches infinity, but it never equals 0. So .49999999....... never equals .50000000..... But, you say, if .499999... doesn't equal .500000... in the limit as n approaches infinity, there must be a number between them. NO. because in the limit as n approaches infinity they are real numbers and the real numbers are complete: there is no real number between them. 
September 30th, 2016, 02:35 PM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra 
In which case they are the same real number because for any two real numbers $a\lt b$ there exists a third real number $\frac{a+b}{2}$ and $a \le \frac{a+b}{2} \le b$. What you are saying about the elements of the sequence is obvious. But you have now contradicted your original claim that all decimals represent different real numbers because you have said that the infinite decimals 0.4999... and 0.5000... are equal. Last edited by v8archie; September 30th, 2016 at 02:42 PM. 
October 1st, 2016, 05:43 AM  #13 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,081 Thanks: 87 
$\displaystyle \lim_{n\rightarrow \infty}$1/n=0 but 1/n never equals 0.

October 1st, 2016, 05:59 AM  #14 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra 
${1 \over n} \ne 0$ for finite $n$, correct. But that has nothing to do with infinite decimals.

October 1st, 2016, 09:05 PM  #15 
Global Moderator Joined: Dec 2006 Posts: 17,707 Thanks: 1356  
October 2nd, 2016, 02:54 AM  #16 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra 
Those two phrases are equal. The interpretation that "each decimal representation represents exactly one real number" has nothing whatever to do with the content of Zylo's posts and is anyway a trivial consequence of the definition of the reals as a limits of Cauchy sequences. Last edited by v8archie; October 2nd, 2016 at 03:09 AM. 
October 2nd, 2016, 03:38 AM  #17 
Global Moderator Joined: Dec 2006 Posts: 17,707 Thanks: 1356 
The phrases aren't equal  in the original post, it's plainly stated that the decimal representations referred to have n decimal places (which makes them unique), whereas you used the second phrase in a context that suggests that infinite decimal representations are included.

October 2nd, 2016, 04:37 AM  #18 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra 
Do you not think that there is a clear implication that this extends to "infinity". With that, the statement is one of complete banality isn't it?

October 2nd, 2016, 04:51 AM  #19 
Global Moderator Joined: Dec 2006 Posts: 17,707 Thanks: 1356 
It was indicated by zylo in a later post that uniqueness is maintained as n approaches infinity, which hints that infinite decimal representations would be considered next, but zylo never got that far. Unfortunately, the uniqueness of the finite representations doesn't imply uniqueness for infinite representations.

October 3rd, 2016, 01:34 AM  #20  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,037 Thanks: 674 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
Quote:
$\displaystyle 7 \times (0.125 \pm 0.001) = 0.875 \pm 0.007$ Because the error just scales with the multiplicative factor. There is a general technique for any sequence of operations. This hasn't really got anything to do with the decimal number system. Quote:
 

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