Originally Posted by Claudius
It has do with the fact that the complex numbers isn't an ordered field. The real numbers is either constructed from the rational numbers as dedekind cuts (look this up), or as equivalence classes of Cauchy sequences of rational numbers. You make R into an ordered field by saying a < b for two dedekind cuts a and b, if a is contained in b. (dedekind cuts are sets).
If follows by the axioms of an ordered field, that for any element x in the field
x^2 > 0. Now , since i^2 = -1, you see why it can't be a real number.
The proof I know was less sophisticated but probably amounts to the same. If i
is real, it must be >, =, or < than 0 (your ordered field). Since i
^2 = -1, and using your axiom, none of the 3 possibilities can be true.