1. The complex number are not ordered. Which else number are not ordered?
The real numbers are ordered on a line; this is a familiar way to order the real numbers and is usually designated using the >,<,= operations. This is referred to as the "usual ordering." You can't apply this method to find an ordering of the complex numbers.
There are other ordering systems, though. For example, how do you order the points on the xy plane? Let me give you a quick demonstration, then we can use it for an ordering.
How do you arrange the words in the dictionary, such as "time" and "trinket?" Well, the first letters are the same so no information there. Let's look at the next letters... they are i and r respectively. Since i is "less than" than r we have time < trinket hence time is listed before trinket in the dictionary. Similarly private < privation. This is called, for obvious reasons, the dictionary order.
You can do this with the points on the plane. Given two points (a, b) and (c, d) we decide which one comes first. The rule is if a > c then (a, b) > (c, d). If a = c then if b > d then (a, b) > (c, d). So (3, 2) > (2, 6) and (3, 2) > (3, -2). (In fact you can use this as an ordering on the complex numbers as well.) Note, though, this is nothing at all like the "usual" ordering in the real numbers.. The points in the xy plane are not ordered in the "usual" ordering, either. If an ordering scheme is not given, assume you are using the "usual" ordering.
There's a whole raft of different orderings used in Topology.
-Dan
Addendum: This is probably too much information but there is a more general (in the sense of "less specific") kind of ordering called a "partially ordered set" (or poset.) Define an ordering on the complex numbers with the usual absolute value: \(\displaystyle |a + ib| = \sqrt{a^2 + b^2}\). In this ordering system we can have two different complex numbers with the same absolute value. For example \(\displaystyle \left | 1 + i \dfrac{1}{2} \right | = \left | 1 - i \dfrac{1}{2} \right | \). Since these two are not the same but equal this ordering is a poset.
