Comment by seanhunter

For people who read this parent comment and are tempted to say “well of course complex numbers can be ordered, I could just define an ordering like if I have two complex numbers z_1 and z_2 I just sort them by their modulus[1].”

The problem is that it’s not a strict total order so doesn’t order them “enough”. For a field F to be ordered it has to obey the “trichotomy” property, which is that if you have a and b in F, then exactly one of three things must be true: 1)a>b 2)b>a or 3)a = b.

If you define the ordering by modulus, then if you take, say z_1 = 1 and z_2 = i then |z_1| = |z_2| but none of the three statements in the trichotomy property are true.

[1] For a complex number z=a + b i, the modulus |z|= sqrt(a^2 + b^2). So it’s basically the distance from the origin in the complex plane.