I'm looking for a pointer to common techniques to map tuples of natural numbers into real or rational numbers such that the ordering is preserved (I assume tuples are ordered lexicographically).
I'm particularly interested in embeddings that map tuples of the form $(n, 0, \dots, 0)$ to $n$.
As an example for pairs, if I have a pair $(n,m)$ I can define $f: (\mathbb{N},\mathbb{N}) \to \mathbb{Q}$ as:
$f(n,m) = n + \frac{m}{m+1}$
and because $\frac{m}{m+1} \in [0;1[$, I have the desired property that $(n_1,m_1) \leq (n_2,m_2) \iff f(n_1,m_1) \leq f(n_2,m_2)$. I also have that $\forall n: f(n,0) = n$.
How does this generalize to tuples of arbitrary (but fixed) arity? Do these embeddings or some instances of such embeddings have well-known names?
How about this: $$(a,b,c)\mapsto0.\underbrace{11\dots11}_{a\text{ 1s}}0\underbrace{11\dots11}_{b\text{ 1s}}0\underbrace{11\dots11}_{c\text{ 1s}}$$ For example, $(3,4,5)$ maps to $0.11101111011111$.
You can consider that number to be written in binary or decimal; changing the base does nothing to the order. However, if you consider it to be binary, we can write it in a much simpler way like this: $$(a,b,c)\mapsto1-2^{-a-1}-2^{-a-b-2}-2^{-a-b-c-2}$$ (If you don't care about your stuff getting mapped to negative numbers, the $1$ in front is unnecessary.)
Here is a picture of what the triples get mapped to. The $x$-axis of each line is the image of a triple. The heights of the lines are only for you to see better; each line has a height of $2^{-a-b-c-1}$.
If you want to have $(a,0,0)$ map to $a$, just make $(a,b,c)$ map to $a+0.\underbrace{1\dots1}_{b\text{ 1s}}0\underbrace{1\dots1}_{c\text{ 1s}}$, so that $(3,4,5)$ maps to $3.1111011111$. Another way to write it would be $(a,b,c)\mapsto a+1-2^{-b-1}-2^{-b-c-1}$.