The notion of complex numbers How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will not in some way contradict some statement that is an outcome of the real numbers.
For example if I defined an operator x with the property that $x^{2n}=-1$, and $x^{2n+1}=1$, for all integers n, this operator is not consistent when used compatibly with properties of the real numbers, since I would have $x^2=-1$, $x^3=1$, thus $x^5=-1$, but I defined $x^5$ to be equal to 1.
How do I know I wont encounter such a contradiction based apon the axioms of the complex numbers.

A field is a generalization of the real number system. For a structure to be a field, it should fulfill the field axioms (http://en.wikipedia.org/wiki/Field_%28mathematics%29).
It is rather easy to see that the complex numbers are, indeed, a field. Proving that there isn't a paradox hiding in the complex-number theory is harder. What can be proved is this:
If number theory (natural numbers, that is) is consistent, then so is the complex number system. The main problem is that you can't prove the consistency of a theory without using a stronger theory. And then you have the problem of proving that the stronger theory is consistent, ad infinitum.
