May 27th, 2016, 09:32 AM  #21 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  
May 27th, 2016, 10:03 AM  #22 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,195 Thanks: 91 
Associate with each number to the right of the decimal point a number to the left of the decimal point. Now forget about the decimal point. These are natural numbers: 1,2,3,4,....... which include all repeating and nonrepeating sequences of digits. So the question is, can you associate any number in the sequence of natural numbers with its decimal number (not fraction) representation? Yes Can you associate any decimal number (not fraction) with the natural number it represents? Yes. V8archie and skipjack point of view: Pi is not a sequence of natural numbers, it is the limit of a sequence of natural numbers. I understand that. But are there enough natural numbers to uniquely define the sequence, ie, make the limit a countable number? 
May 27th, 2016, 10:07 AM  #23 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,195 Thanks: 91  
May 27th, 2016, 10:29 AM  #24  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra  Quote:
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Again, there is no limit here. Last edited by skipjack; May 27th, 2016 at 03:32 PM.  
May 27th, 2016, 11:10 AM  #25  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,195 Thanks: 91  Quote:
As for the rest: We are talking about decimal representation of numbers. By decimal fraction I mean .xxxx, by decimal number xxxx. I was trying to save on verbiage. .1845 <> 1845 Given any decimal representation of a natural number, say .14159... to n places, does it occur in the sequence of natural numbers for any n? I note the natural numbers have no limit. Last edited by skipjack; May 27th, 2016 at 03:44 PM.  
May 27th, 2016, 11:25 AM  #26 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra 
Every finite sequence of decimal digits appears in at least one natural number (and there is an obvious bijection of finite sequences of decimal digits with the natural numbers). But there is no infinite sequence of decimal digits in any natural number. $\pi$ itself is the limit of a sequence of rational numbers, but the digits of $\pi$ do not form a convergent sequence, even though the sequence is infinite. It is incorrect to talk of limits in this context. Limits are expressions of potential infinities. We are dealing with actual infinities. Last edited by skipjack; May 27th, 2016 at 03:45 PM. 
May 27th, 2016, 11:50 AM  #27 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,195 Thanks: 91 
Given pi = 3.1415926... , I am claiming that the natural number whose decimal representation is 1415926... to n places exists for every n.
Last edited by skipjack; May 27th, 2016 at 03:45 PM. 
May 27th, 2016, 12:16 PM  #28 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra 
Yes. Every finite $n$. But that's not the whole sequence because the sequence is infinite.

May 27th, 2016, 12:38 PM  #29 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,195 Thanks: 91 
f(n) =14159.. to n places. https://en.wikipedia.org/wiki/Sequence "Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first $n$ natural numbers (for a sequence of finite length $n$)." Last edited by skipjack; May 27th, 2016 at 03:54 PM. 
May 27th, 2016, 12:54 PM  #30 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra 
I know what a sequence is. Every term in your sequence is finite. It has $n$ digits where $n$ is a natural number, and therefore finite. So which term of this sequence of terms of finite length do you claim is equal to the infinite sequence of decimal numerals that represents the fractional part of $\pi$? Which natural number $n$ is infinite? There isn't one, from the definitions and proofs provided before now. 

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