# Math Help - explicit sequence

1. ## explicit sequence

Let $\bold{Z} \times (\bold{Z}- \{0 \}) = \{(a,b): a,b \in \bold{Z}, b \neq 0 \}$. This is a countable set. Now let us suppose that we have a function $f: \bold{Z} \times (\bold{Z}- \{0 \}) \to \bold{Q}$ defined by $f(a,b) := a/b$. Then $f(\bold{Z} \times (\bold{Z}- \{0 \})) = \bold{Q}$ which is at most countable. Since $\bold{Q}$ in infinite, then it must be countable. Then we can arrange the rational numbers in a sequence: $\bold{Q} = \{a_{0}, a_{1}, a_{2}, a_{3}, \ldots \}$.

How would you come up with a sequence, such that every element of the sequence is different from every other element, and the elements of the sequence exhaust $\bold{Q}$?

2. Is there an explicit formula?

3. Originally Posted by particlejohn
Is there an explicit formula?
Yes. It is not nice looking, one such formula can take advantage of unique factorization of the numbers into primes.