Let $(p, q)$ be a rational coordinate in range $[0,1] \times [0 , 1]$. My goal is to show that for each $(p,q)$, I can find a unique natural number $n$.
My current approach is to find an injective function which takes $(p, q)$ to a natural number. Since $p$ and $q$ are rational, there exist unique representation by fractions. So that
$p = \frac{a}{b}$ and $q = \frac{c}{d}$ where the ordered set $(a,b, c,d)$ is unique for each $(p,q)$.
Then the natural number $n$ which is the concatenation of $(a,b, c,d)$ is also unique. For example, if $(a, b, ,c,d) = (3, 82, 4, 9)$, $n = 38249$. As the ordered set only contains natural numbers, the concatenation should be possible.
I don't know if this is a valid way to show the existence of the unique natural for each coordinate as I can not show that the concatenation of natural numbers are allowed mathematically. Is there any other way to show that?
Hint : Consider the map $f : (\mathbb{Q} \cap [0,1]) \times (\mathbb{Q} \cap [0,1]) \rightarrow \mathbb{N}$ defined by $$f \left(\frac{a}{b}, \frac{c}{d} \right) \mapsto 2^a3^b5^c7^d$$