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In this proof can I show independently that positive rationals and negative rationals are countable by using Cantor's zigzag and then say that the union of two countable sets is countable so therefore

positive and negative rationals are countable.

Is this correct ?

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irrationals are countable? when did this happen. – simplicity Feb 10 '12 at 2:42
- ( 1/3) , -(1/4) , -(3/7) = negative rational numbers – user602774 Feb 10 '12 at 2:45
The confusion arose as a result of a bad edit. I don't believe OP ever intended to claim the irrationals are countable.//To OP: What you say about the union of two countable sets is absolutely correct. The set of all rational numbers (positive or negative) is indeed countable. – Austin Mohr Feb 10 '12 at 2:46
Sorry guys my bad. The OP spelt rationals as "rrationals" so subconsciously in my mind the edit was to put an "i" there. – user38268 Feb 10 '12 at 2:54
up vote 2 down vote accepted

You said "irrationals" in your last sentence by mistake - but "rationals" in the body of the problem. Yes, the argument is correct. Here's an even more direct method (without splitting into two cases)

Order everything in the usual positive matrix, and put the negative number immediately after the corresponding positive number:

$ \\ \frac{1}{1}, -\frac{1}{1}, \frac{1}{2}, -\frac{1}{2}, \ldots \\ \frac{2}{1}, -\frac{2}{1}, \frac{2}{2}, -\frac{2}{2}, \ldots \\ \vdots \\ \frac{n}{1}, -\frac{n}{1}, \frac{n}{2}, -\frac{n}{2}, \ldots \\ \vdots $

or use your favourite matrix (this one is missing $0$, by the way).

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