# Thread: Complex Number Ordering.

1. ## Complex Number Ordering.

I have trouble solving this problem from my real analysis class. It would be wonderful if someone could help me out or give me a hint. Thanks very much in advance.

Suppose z=a+bi, w=c+di. Define z<w if a<c, and also if a=c but b<d. Prove that this turns the set of all complex numbers into an ordered set. Does this ordered set have the least-upper-bound property?

2. Originally Posted by EmmWalfer
Suppose z=a+bi, w=c+di. Define z<w if a<c, and also if a=c but b<d. Prove that this turns the set of all complex numbers into an ordered set. Does this ordered set have the least-upper-bound property?
Here is a web-site that may help you.
Note that even though the set may be totally ordered, the system of complex numbers is not an ordered field.
The operations can be ordered. Note $i^2=-1<0$.

P.S. Here is another good web reference.