In mathematics, if a structure is a field and has an order , two additional axioms need to hold for it to be an ordered field. These axioms express the notion that the ordering is compatible with the field structure:

1) For any three elements, implies

2) For any two elements, and implies

We can show that such an order does not exist in the set of Complex number.

Let us restate the two axioms:

If , we write

.

By (2),

.

i.e.,

a contradiction.

Otherwise (), we have

.

By (1),

We write it as

By (2),

i.e.,

, a contradiction again.

Therefore, the complex numbers do not possess an order.