Consider a “pseudo-ordering” on the complex field defined by saying a complex number 
is positive if and only if 
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 
with 
either is positive or 
is positive, but not both. By our definition, positive implies 
. Therefore, since 
we have 
. Hence, is not positive.
The third axiom is satisfied since by our definition of positive if 
and 
, then is not positive.
Finally, the first axiom fails since if we take 
, then is positive, but
![\[ z^2 = (1+2i)(1+2i) = 1 - 4 + 4i = -3 + 4i \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e97c425c4bb6088cf2c8541240266391_l3.png "Rendered by QuickLaTeX.com")
is not positive.
The second axiom is NOT satisfied ,consider i, it’s not zero yet both i and -i are not “positive” since their real part is not positive.