How to prove that the set of rational numbers are countable? Can any one tell me how to prove that the set of rational numbers are countable? Prove give me a prove?
Thanks.
 A: For an explicit enumeration of positive rationals, you can use the Calkin-Wilf sequence: $$q_{i+1}= \frac{1}{ \lfloor q_i \rfloor +1 - \{q_i\} }, \ q_0=1.$$
More details can be found in Proofs from THE BOOK.
A: Map each rational $\frac{a}{b}$ into the integer
$2^a 3^b$.
This shows that the number of rationals
is at most the number of integers.
If you want to handle the negative rationals,
map the sign ($-1$, $0$, or $+1$)
to $5^{\mathrm{sign}+1}$ and stick it on the end,
so the mapping is
$\mathrm{sign} \times \frac{a}{b} \to 2^a \, 3^b \,5^{\mathrm{sign}+1}$.
If you find this troubling, that's OK.
You are not the only one.
A: Here is another argument:
Consider the map $\varphi:\mathbb{Q}\rightarrow \mathbb{Z}\times\mathbb{N}$ which sends the rational number $\frac{a}{b}$ in lowest terms to the ordered pair $(a,b)$ where we take negative signs to always be in the numerator of the fraction.  This map is an injection into a countably infinite set (the cartesian product of countable sets is countable), so therefore $\mathbb{Q}$ is at most countable.  Since $\mathbb{Q}$ is not finite, it must be countably infinite.
