>>12962917The naturals are well-ordered, and you can define a well-order on integers (which are pairs of naturals) and rationals (which are pairs of integers) by reducing them back to natural arithmetic. (Insert statement about muh AC and a well-ordering on the reals/complex numbers)
But an additional element of the algebra "i" destroys this, since there is no way to express the numbers purely in terms of naturals. There is always an additional element that makes numbers two-dimensional, and there is no "preferred" way to order them. How would you even go about ordering the points on a plane? Maybe with a spiral shape? That's an additional structure you choose to impose, and isn't unique like the number line.
On the other hand, in finite fields, i can be assigned a natural value. If we have a field with elements, then the number n in the field obeys similarly to i. Ordering a field of this type can be tricky though, because there's two sensible options: counting from 0 up to and returning to 0; or counting from up to and similarly looping. There's still some ambiguity because both -i and i when squared are equal to -1, and there's nothing stopping you from arguing any of:
0 < i < -i
0 < -i < i
-i < 0 < i
i < 0 < -i
