On Mar 28, 2010, at 3:09 PM, James Cloos wrote:
| Given A = a₁ + i·a₂ and B = b₁ + i·b₂, then:
|  A ≤ B if a₁ < b₁ || ( a₁ == b₁ && a₂ ≤ b₂ )
|  A ≥ B if a₁ < b₁ || ( a₁ == b₁ && a₂ ≥ b₂ )

Assuming that the last line should be "A ≥ B if a₁ > b₁ ...", this is called lexicographic ordering, and it's nothing new, especially not for complex numbers. (Someone might even have suggested ordering Complexes this way in the original discussion, but I can't find it right now.) The problem with this ordering and all other orderings of the complex plane is that it is impossible to make ℂ into an ordered field[1] that has all of the properties that we have come to expect from ordering the real numbers. Specifically, because -1 is a square in ℂ, ℂ being an ordered field would require that -1 > 0, which leads to a contradiction. I can give you a complete proof of this if you like.

-- Minimiscience

[1] <http://en.wikipedia.org/wiki/Ordered_field>

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