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 ?

$$\text{Yes, only natural numbers!}$$
Says Kronecker. 
