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.

Let's say we have defined the complex numbers as the set $\Bbb R\times\Bbb R$ together with addition $$(a,b)+(c,d)=(a+c,b+d)$$ and multiplication $$(a,b)\cdot(c,d)=(ac-bd,ad+bc).$$It is then possible to show that this structure is a field, which basically means that all the usual algebraic laws hold.
We can then define a subset $R$ which contains exactly those complex numbers with zero imaginary part. The addition and multiplication, when restricted to this set, becomes $$(a,0)+(c,0)=(a+c,0)\quad\text{and}\quad(a,0)\cdot(c,0)=(ac,0).$$
It can again be proven that $R$ is a field, it's a subfield of the complex numbers. Since every element in this set is uniquely determined by the first component, we could just write $(x,0)=\bar x$. Now it holds that
$$\bar x+\bar y=(x,0)+(y,0)=(x+y,0)=\overline{x+y}\quad\text{and}\\ \bar x\cdot\bar y=(x,0)\cdot(y,0)=(xy,0)=\overline{xy}$$
where the addition in $\overline{x+y}$ is just addition of real numbers and similarly for multiplication. Thus, you can think of $\overline x$ as the real number $x$. There is no difference except for how you write it.
If we now define $i=(0,1)$ then we see that $$i^2=(0,1)\cdot(0,1)=(-1,0)=\overline{-1}$$ and so the axioms of the complex numbers are consistent (because we have a model for them). Now if you take any property which can be proven from the field axioms, for example $a^m\cdot a^n=a^{m+n}$ for integers $m$ and $n$, then this is valid for all complex numbers (because they form a field, too). Since the real numbers are just a special kind of complex numbers, we see that the complex numbers extend the reals in a way which is compatible with them in this sense.
