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 .
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?
The definition says that a complex number is “positive” iff it’s modulus is positive, now for the first axiom, we have that the complex number (1,0) ,which we treat as being identical to the real number 1 , is “positive” because its modulus is sqrt(1^2+0^2) =1>0, similarly the complex number (-1,0) has modulus sqrt((-1)^2+0^2)=1>0 so it’s “positive”. Now we have established that 1 and -1 are “positive” but 1+(-1)=(1,0)+(-1,0)=(0,0)=0 which is not “positive” since its modulus is 0 which is not positive, this contradicts the first axiom since we found two complex numbers that are “positive” but their sum is not “positive”. The second axiom is not satisfied since we have found a complex number (1,0) which is not zero yet BOTH (1,0) and it’s additive inverse (aka, negative) (-1,0) are positive.
Tldr; The first axiom fails because the complex numbers 1 and -1 are “positive” but their sum isn’t. The second axiom fails too since the complex number 1≠0 yet both 1 and its negative are positive.