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. 
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. 
.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. 
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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)." Quote:

@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 
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. 
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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. 
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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