Can one work with any classes of numbers in a proof of number theory?

Question

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 ?

Answer 1

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

Answer 2

Why on earth would you impose such restrictions? There's only one thing you should require of a proof: that it be logically valid. The end.

Not only is there absolutely no reason to apply a restriction like that, but modern number theory very frequently steps outside the realm of ordinary integers. Analaytic number theory is roughly speaking the application of calculus to number theory, while algebraic number theory uses systems of numbers that can include complex numbers (for example, the Gaussian integers $a+bi$ where $a,b$ are integers). I've often heard of something called ergodic theory used in contemporary number theory, although I have no idea how or even what exactly that is.

Answer 3

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".