1. |R is the only complete archimedian field. But any of its subfields is also archimedian.
2. The compatibility between the order and the multiplication operation:
-One has the notion of "positive real numbers" (those numbers which are > 0) -|R satisfies one further property: If a>b and c is positive, then ca>cb.
3. It is impossible to define an order in the field of complex numbers with all the mentioned properties.
(See Walter Rudin's book <<The Principles of Mathematical Analysis>> for reference.)(Hillgentleman)