Exploring the Complexity of 0: Real, Imaginary, or Both?

[pwsnafu]

The complex numbers are defined above the real numbers, as ordered pairs of real numbers, written in the form as z=x+yi, with addition and multiplication defined.

The construction of the complex numbers from the real numbers is via ordered pars of reals.
You can define the complex numbers without any reference to the reals.

You can not define real numbers with complex numbers.

From Wikipedia
Let K be a topological field, which contains a subset P, satisfying

1. P is closed under addition, multiplication and inverses.
2. If $x, y \in P$ and $x \neq y$ then $y-x \in P$ or $x-y \in P$, but not both.
3. For all non-empty $S \subset P$ there exists $x \in P$ such that $S+P = x+P$.

Further suppose K has a non-trivial involution such that for all non-zero $z \in K$ one has $zz^* \in P$.
Then K with the topology generated by $\{ y \, |\, p - (y-x)(y-x)^* \in P,\, x \in K, \, p \in P\}$ is isomorphic as topological fields to the complex numbers, and P is the set of positive numbers. Once you have the positive reals, you have all of the reals.

 

[micromass]

The construction of the complex numbers from the real numbers is via ordered pars of reals.

Also note that it is perfectly possible to construct the complex numbers in such a way that $\mathbb{R}$ actually is a subset of $\mathbb{C}$.