May 24th, 2016, 09:52 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 974 Thanks: 78  The Real Numbers are Countable
The Real Numbers are Countable The real numbers 0 $\displaystyle \leq$ x < 1 are countable because there is a unique countably infinite sequence of digits which identifies the real number and this same unique countably infinite sequence of digits is a natural number. Ex. Fractional part of pi .1415926........ equiv 1415926..... Note: .999999999999999999..... is not included. Use 1 Same applies for binary digits. 
May 24th, 2016, 10:00 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,555 Thanks: 2148 Math Focus: Mainly analysis and algebra 
This is nonsense... again. You have yourself stated in the past that all natural numbers are finite. Your last post correctly defined them as finite sequences of binary digits. This excludes $\pi$. Your exclusion of 0.99999... is somewhat pointless as it leaves many similar decimal representations. Last edited by skipjack; June 25th, 2016 at 08:15 PM. 
May 24th, 2016, 03:46 PM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 974 Thanks: 78 
.3333..... with a countably infinite (not finite) number of '3's defines 1/3. Is 33333.... with a countably infinite number of '3's a natural number? Can we count to it? Let's start counting: 1, 2, 3, 4, 5, ....... When we get to 333, do we have to stop counting? 3333333? 33333333333333333? The point is you can count to 33333..... (countably infinite number of '3's), i.e., it is a natural number. This illustrates the OP that the reals are countable. Countably infinite means COUNTABLE and NONFINITE. Any particular natural number is finite, but there are a countably infinite number of them. The above argument applies to any irrational number 0 $\displaystyle \leq$ x < 1. Last edited by skipjack; May 24th, 2016 at 09:20 PM. 
May 24th, 2016, 04:54 PM  #4  
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 492  Quote:
Last edited by skipjack; May 24th, 2016 at 09:21 PM.  
May 24th, 2016, 05:08 PM  #5  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,555 Thanks: 2148 Math Focus: Mainly analysis and algebra  No you can't. Such a process has to terminate. Since the infinite sequence of 3s does not, by definition, terminate, neither can your counting process. Furthermore, you defined your "countably infinite binary sequences" as terminating in an infinite string of zeros, which this clearly doesn't, so it's not in your enumeration. Quote:
We defined "countably infinite" only a few days ago. The definition was clear and unambiguous. "Countable" has not been defined. So this is an illformed definition. This is true! Unfortunately, you persist in guessing at the consequences instead of using any rigorous reasoning. Last edited by skipjack; May 24th, 2016 at 09:20 PM.  
May 24th, 2016, 10:19 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 16,788 Thanks: 1238 
Nonsignificant zeros are not included in the standard representations of natural numbers. According to zylo here, a binary [natural] number, $BN$, has the form $\displaystyle BN=p_{n}2^{n}+p_{n1}2^{n1}+...p_{0}2^{0}$, where the $p_i$ are the digits of the number. A decimal natural number, DN, is correspondingly $\displaystyle p_{n}10^{n}+p_{n1}10^{n1}+...p_{0}10^{0}$. In both definitions, $n$ is zero or a natural number (else the sums are divergent), so a natural number has a finite number of digits ($n+1$ digits). In zylo's words, "The definitions hold for every finite n, but not for n=∞ (undefined)." By zylo's earlier statement, which explicitly requires $n$ to be finite, this is not a natural number. I fail to see how zylo can maintain both posts are correct, as they directly contradict each other. 
May 25th, 2016, 05:55 PM  #7 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,555 Thanks: 597 Math Focus: Wibbly wobbly timeywimey stuff. 
@zylo: I haven't checked in here for well over a month and still you are here making the same complaints, confusion, and an almost obsessive level of insistence that all other Mathematicians don't know anything about this topic. I almost have to admire your tenacity. On the other hand your ship is not only sinking but it's become a playground for Spongebob and Patrick. Don't you think it might be time to admit you are in error? Dan 
May 27th, 2016, 03:41 AM  #8 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 974 Thanks: 78 
The real numbers are countable. Proof Every real number consists of a unique pair of natural numbers, one to the left of the decimal point, and one to the right of the decimal point. 
May 27th, 2016, 05:12 AM  #9  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,555 Thanks: 2148 Math Focus: Mainly analysis and algebra  Quote:
The statement is wrong anyway. As you have been shown many times, there are many real numbers that are infinite to the right of the decimal point. And an infinite string of digits is not a natural number because a natural number is finite.  
May 27th, 2016, 05:25 AM  #10 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 974 Thanks: 78 
Given: .1845 There is a 1:1 correspondence between .1845 and 1845: .1845>1845 1845>.1845 The decimal point is a convention of interpretation (the usual). 

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