Rudin chapter 1 question 9. One condition for an ordered field is x > 0 and y> 0 then xy> 0, multiplication in complex field is defined as (suppose z = (a,b) w = (c,d)) zw = (ac - ba, ad + bc). Taking the case of z < w then a < c and both z and w greater than 0 in both real and imaginary part in the question, then what is preventing b from being a massive number that makes ac - ba less than 0 and therefore zw < 0 and violating the ordered field condition?
Hi!
I think that you are answering your own question here. Because of the argument you give (among other things) the relation described here does not give the complex numbers the structure of an ordered field. Why do you think this should be an ordered field?
P.S This might just be a typo but note that your definition of complex multiplication is incorrect. It should be
:)
I think that you are answering your own question here. Because of the argument you give (among other things) the relation described here does not give the complex numbers the structure of an ordered field. Why do you think this should be an ordered field?
P.S This might just be a typo but note that your definition of complex multiplication is incorrect. It should be
:)But the premise of the question is that we can turn the complex number into an ordered field if we define the specific conditions in the question (and then the subquestion is if this ordered complex field has the LUP property). And yeah, thats a typo from me but im still wondering about the case of a < c but then c <<< b and c <<< d, wouldnt this mean that the ac - bd is gonna turn less than 0 and violate the ordered field axiom under multiplication?
Hi!
You are correct but the question is about ordered sets, not ordered fields. The relation described in the exercise gives an order on the SET of complex numbers (showing this is part of what you need to prove in the exercise) but as you mention this does NOT give
the structure of an ordered field. One way to prove that this is not an ordered field is to use the exact same argument as you use here so this is a very good observation. In fact, exercise 8, chapter 1, in Rudin asks you to prove that there is no order on 
which gives the complex numbers the structure of an ordered field and this can be done using a special case of your argument here. Does this make things clearer?
You are correct but the question is about ordered sets, not ordered fields. The relation described in the exercise gives an order on the SET of complex numbers (showing this is part of what you need to prove in the exercise) but as you mention this does NOT give
the structure of an ordered field. One way to prove that this is not an ordered field is to use the exact same argument as you use here so this is a very good observation. In fact, exercise 8, chapter 1, in Rudin asks you to prove that there is no order on which gives the complex numbers the structure of an ordered field and this can be done using a special case of your argument here. Does this make things clearer?Thats quite an important detail I overlooked, thank you! If I still cant solve it then i'll post something here again, thank you for the help :)