In yesterday’s post, I demonstrated that there is no subset of the complex numbers which satisfies the following four axioms:

• If , then
• If , then .
• For every , either or , but not both.
• 

However, it’s instructive (and fun) to try to construct such a set. Yesterday I showed that the following subset satisfies three of the four axioms:

Apostol’s calculus suggests two other subsets to try:

and

Neither of these sets work either, but I won’t spoil the fun for you by giving you the proofs. I leave a thought bubble if you’d like to try to figure out which of the four axioms are satisfied by these two notions of “positive” complex numbers.