I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on why it doesn’t make sense to say one complex number is less than another complex number (using the usual definition of “less than”).
Part 1: A sketch of a direct proof based on the order axioms proving that is not an element of a number system that has the usual definition of inequality.
Part 2: An indirect proof.
Part 3: Discussion about the lexicographic ordering, which almost works.
Part 4: Two other partial orderings which almost work.