Simply put, no. The complex number set C cannot be an ordered field. If a >= 0, a^2 = a*a >= 0 follows. If a < 0, a^2 = a*a > 0, meaning a^2 > 0 for any a, therefore 1=1^2 > 0, and -1 < 0.
If C were an ordered field, then i^2 > 0 which means -1 > 0. which is untrue. C cannot therefore be an ordered field.
It is possible to order C without adhering to the field axioms, nevertheless. Dictionary ordering is one such method. If a > c, or if a = c and b > d, a + bi > c + di. Although this is in line with the sequence on R, there isn't much we can do with it. It doesn't follow AT ALL that if z > w and v > 0 such that zv < wv.
Alternately, partial orders are possible: z > w if abs(z) > abs(w), but this isn't total order. Since z < w, z > w, and z = w are neither mutually exclusive nor exhaustive, there may be circumstances in which none of the three apply.
Caption this Meme
