Can Complex Numbers be Compared Using Greater-Than and Less-Than Relations?

[quawa99]

Are the less than (<) and greater than(>) relations applicable among complex numbers?
By complex numbers I don't mean their modulus, I mean just the raw complex numbers.

 

[jbriggs444]

The short answer is "no". The greater-than and less-than relations do not apply.

A longer answer is that the complex numbers together with the standard operations of addition and multiplication form a "field". But there is no greater-than relation that can be used to make it an "ordered field". http://en.wikipedia.org/wiki/Ordered_field

The problem comes when you try to decide whether i is positive or negative. i is different from zero, so it has to be either positive or negative. If it is positive then i*i must be positive. But i*i=-1 and -1 is negative. If i is negative then -i must be positive. So -i*-i must be positive. But -i*-i=-1 and -1 is negative.