Historically, did mathematicians:
1) construct the set of real numbers using rationals, integers, and naturals?
Or
2) did they already know the set of reals existed and partitioned it into rationals, integers, and naturals?
I'm curious to know the timeline of events. If #1 above is true, then I am assuming mathematicians could only define functions from $\mathbb{Q},\mathbb{Z},\mathbb{N}$ such as $f: \mathbb{Q} \to\mathbb{N}$. Then later on, once $\mathbb{R}$ is constructed, they could define functions like $f: \mathbb{R} \to\mathbb{R}$.
Also, where does the Completness Axiom fit into the timeline?