The complex numbers cannot be made into an ordered field. There certainly exist "natural" total orders of the complex numbers (e.g. one can take a one to one correspondence between them and the real numbers, which *are* ordered, and copy that order; or one can take a lexicographic order on x+iy by mapping it to (x,y) or to (y,x)). But none of them respect the algebraic properties we'd want; the order doesn't "play nicely" with the field axioms.

Suppose P were the positive half of **C**, the complex numbers. Then `i`≠0, so either `i`∈P or `-i`∈P.

In either case, we have some `j` (either `i` or `-i`) in P such that `j`^{2}=-1. But this is impossible for a positive element.

Q.E.D.