For your first sentence, it really depends. If you start with the naturals, you can close it under addition using a Grothendieck-style construction: take pairs

For your first sentence, it really depends. If you start with the naturals, you can close it under addition using a Grothendieck-style construction: take pairs (n,m) of naturals, and say (n',m') is equivalent to it if n+m'=n'+m. There is a copy of the naturals embedded in these integers through n identified as (n,0). Similarly, we can take pairs of these integers to close under division to get the rationals. Let's denote such pairs n/m (m nonzero), so n'/m' is equivalent when nm'=n'm. The integers are represented as a pair by n/1. Next we can close the rationals under the property that Cauchy sequences converge: take the set of all Cauchy sequences of rationals, and say two such sequences are equivalent if the term-wise difference between the two converges to the rational 0/1. This gives us the reals, with a rational q being identified with the sequence (q,q,q,...) Finally, we can close the reals under being able to have polynomials have roots. It turns out adding the root of x^2+1 is all you need. The complex numbers then are the set of all polynomials with real coefficients, where two polynomials are equivalent if their difference is divisible by x^2+1. The reals can be identified as a complex number by thinking of r as a constant polynomial. All I'm saying is that, while we commonly think of Z as a subset of Q as a subset of R as a subset of C, this is only after a sequence of closures of different kinds, each time creating a completely new set which the first number system is not actually a subset of (though the first number system can be identified as some subset in the new system). With this in mind, it's completely reasonable to define exponentiation of something to the integer or to the rational as being a completely different operation from exponentiation to the real or to the complex. In fact, in Rudin, exponentiation to the real is defined as a limit of exponentiation to the rationals, which is defined as a positive root of real to the integer.