User:Trovatore/Arithmetic and exponentiation in various numerical structures

Many times, usually in the context of discussions about the expression $0^{0}$, I have wanted to express certain ideas that I think are relevant, but it would be too long-winded inline, and also I didn't want to spend the time. So I'm hoping that pointers to this essay may prove useful.

Opening remarks

It is often claimed that the natural numbers $\mathbf {N}$ form a subset of the integers $\mathbf {Z}$, which in turn form a subset of the rational numbers $\mathbf {Q}$, which in their turn form a subset of the real numbers $\mathbf {R}$, and the reals a subset of the complex numbers $\mathbf {C}$.

For most purposes this is true.