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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 x is positive. Which of the three order axioms (listed in the previous exercise) are satisfied.


The second and third axioms are satisfied, but not the first.

The second axiom states that for every z with x \neq 0 either z is positive or -z is positive, but not both. By our definition, z positive implies x >0. Therefore, since -z = -x-iy we have -x < 0. Hence, -z is not positive.

The third axiom is satisfied since by our definition of positive if z = x+iy and x = 0, then z is not positive.

Finally, the first axiom fails since if we take z = 1 + 2i, then z is positive, but

    \[ z^2 = (1+2i)(1+2i) = 1 - 4 + 4i = -3 + 4i \]

is not positive.

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