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>