1. are we finished with f(a).b and when a is odd f(a)=-(a+1)/2 and when a is even f(a)=a/2 ?

2. Originally Posted by DrSteve
$\mathbb{N}\rightarrow \mathbb{N}\times \mathbb{N}\rightarrow \mathbb{Z}\times \mathbb{N}\rightarrow \mathbb{Q}$.

The first is the inverse of the pairing function. The second is the map $(a,b)\rightarrow (f(a),b)$ where f is the function we've previously discussed, and the third is the map $(a,b)\rightarrow a.b$.
For the function suggested by Plato, one can take $p_1(n)=\max\{k\in\mathbb{N}\mid\text{2^k divides n}\}+1$ and $p_2(n)=(n/p_1(n)+1)/2$. (One must check whether $p_1(n)$ and $p_2(n)$ need to assume 0 and whether they actually do.) Then $n\mapsto(p_1(n),p_2(n))$ is a bijection $\mathbb{Z}^+\to\mathbb{Z}^+\times\mathbb{Z}^+$, which is the first step in the chain at the top of this post.

3. Thanks emakarov - I didn't remember which direction was technically called the pairing function.

gutnedawg, you essentially got my idea right. The bijection $F:\mathbb{N}\rightarrow A$ is given by $F(n)=f(a).b$ where $f$ is the function you described and $\pi(a,b)=n$, where $\pi$ is the pairing function.

(Here $A$ is the set of rational numbers with terminating decimal expansions.)

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