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

The final axiom holds since is true for all . Hence, is not positive so the axiom holds.

The first axiom fails since under this ordering both and are positive, but is not positive.

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

1. Anonymous says:

I don’t think the first axiom fails as the norm of z is always positive except when |z|=0. Hence, if |z1|>0 and |z2|> 0 then their sums and products are also greater than zero.

I don’t follow the logic in “1 + (-1) = 0 hence it doesn’t satisfy axiom 1”. The norm is never -1.

I think the second axiom is also satisfied since for |z| 0 we have |z|>0 hence -|z|<0. In fact |z| always belongs to the positives while -|z| to the negatives.

Another comment is that there is no complex number in the norm. But I am just self studying this (I have no math degree) so it would be nice if someone could confirm my reasoning or clarify RoRi's.

RoRi are you still answering this blog?

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