Can one work with any classes of numbers in a proof of number theory? Can one work with any classes of numbers, like natural, integer, rational, real and complex, in a proof of number theory, as long as the result tells something about the integers ? Or should the result be proven using integer-operations only (we can only divide one integer by another if the quotient is an integer, because otherwise we fall out of the default domain) ?
To be more specific, should we stay inside the group of integers when proving results about the integers ? If not, we are proving the result in a larger group containing the integers as a subgroup ?

We need to introduce the natural numbers $\Bbb N=\{0,1,2,3,...\}$ and their basic properties axiomatically. But then the integers $\Bbb Z$, the rationals $\Bbb Q$, the reals $\Bbb R$, the complex $\Bbb C$, the $p$-adics $\Bbb Q_p$ and all other "numbers" we deal with are constructed by means of relatively simple set-theoretic or algebraic-topological procedures which ultimately rely just on $\Bbb N$. Thus, all "classes of numbers" you can think of are, at bottom level, just "organized natural numbers".
