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Determine which order axioms are satisfied for a given “pseudo” ordering on the complex numbers

Consider a “pseudo-ordering” on the complex field defined by saying a complex number z = x +iy is positive if and only if |z| is positive. Which of the three order axioms (listed in the previous exercise) are satisfied.


The final axiom holds since |z| > 0 is true for all z \neq 0. Hence, 0 is not positive so the axiom holds.

The first axiom fails since under this ordering both 1 and -1 are positive, but 1 + (-1) = 0 is not positive.

The second axiom fails since for any complex number z we have |z| = |-z| which implies that if z is positive then so is -z.

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