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.
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