January 17th, 2019, 05:42 AM  #21 
Senior Member Joined: Oct 2009 Posts: 784 Thanks: 280  Yeah, so why don't you show a proof that 1 doesn't converge to 2. Or that 0.999... does converge to 1. Begin with defining what it means that a number converges to another.

January 17th, 2019, 05:52 AM  #22  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
There is nothing about convergence or evaluation. The only notion of limit is that some real numbers are an endless sequence (limit) of digits. For example Pi is the limit of a sequence of digits. You can think of my definition as a system of symbols that allow you to talk about and practically work with real numbers. You can use them and interpret them any way you like, just as you did in HS and Calculus. For example, you can associate a rational number, 1/3, with the real number consisting of the limit of an endless string of 3's, .333333333......, which you can interpret in the conventional sense as a sum of powers of 10 and then define a conventional {working) epsilon delta limit. Note that the conventional Lim .3333....p, as the number of 3's becomes endless and p is ANY fixed number, is 1/3.  
January 17th, 2019, 06:06 AM  #23  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 Math Focus: Mainly analysis and algebra  Quote:
Quote:
So you require a proof that $$\sum_{n=1}^\infty \frac9{10^{n}} = 9 \sum_{n=1}^\infty \left(\frac1{10}\right)^{n} = 1$$ which is a triviality for anyone that has studied infinite series. The fact that $$\frac1{1x} = 1 + x + x^2 + x^3 + \ldots = 1 + \sum_{n=1}^\infty x^{n} \qquad (x < 1)$$ allows us to put $x = \frac1{10}$ and get \begin{align}\sum_{n=1}^\infty \frac9{10^{n}} &= 9 \sum_{n=1}^\infty \left(\frac1{10}\right)^{n} \\ &= 9 \left(\frac1{1\frac1{10}}1\right) \\ &= 9\left( \frac1{\frac9{10}}1\right) \\ &= 9 \left( \frac{10}9  1 \right) \\ &= 9 \left( \frac19 \right) \\ &= 1 \end{align}  
January 17th, 2019, 06:14 AM  #24  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 Math Focus: Mainly analysis and algebra  Quote:
Alternatively, the number 10 comprises two symbols (0 and 1) and there relative positioning has a meaning. This is a definition that you are relying on but refusing to acknowledge. The symbol ten means $0 + 1(10^1)$ and so any definition of a natural number via decimal digits requires that the corresponding sum can be evaluated. This, again, is a concept that you refuse to admit  you have even taken to denying it on occasion.  
January 17th, 2019, 06:37 AM  #25 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
Evaluation is strictly a matter of convention. A computer programmer might decide to round off (evaluate) all numbers to ten decimal places, where the tenth place is the next higher or lower number (integer). A mathematician might decide to evaluate .33333.... (endless string of 3's) as the epsilon delta sum (\Sigma). An engineering lab might decide to evaluate all shock load data to the nearest integer. In this case ,9 = 1. 
January 17th, 2019, 07:49 AM  #26 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 Math Focus: Mainly analysis and algebra  What a load of tosh! Approximation is approximation. Error correction is error correction. Nobody evaluates numbers differently. To the extent that convention does apply, you should follow it or state that you are doing stuff that doesn't apply to the conventional system. You do neither. 
January 17th, 2019, 10:58 AM  #27  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
I have said all along that what zylo is doing is using a nonstandard definition of real numbers and then saying that he has a proof that applies to the standard definition. The exercise is beyond silly.  
January 17th, 2019, 01:13 PM  #28 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  
January 17th, 2019, 06:23 PM  #29 
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics 
How are the moderators still allowing this same nonsense to be spouted over dozens of threads, month after month after month? This is absurd. I understand not wanting to be as ruthlessly moderated as other sites such as stack exchange, but enough is enough. Kick this idiot off his podium already.

January 18th, 2019, 03:50 AM  #30 
Global Moderator Joined: Dec 2006 Posts: 20,644 Thanks: 2084  That is how you started this topic, but you haven't explained what you meant by "unique". If the sequence 010 were used, why would it be unique? How would the fraction 1/9, which is a real in [0, 1), be obtained in this "read them in reverse" way? 

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