I'm not sure on the following parts, I insert what I'm not sure on in () brackets.
(a) Let and be ratonal numbers satisfying . To which of the following three sets does the number belong:
(It seems fairly obvious that it's in set B but I'm not sure exactly why...)
(b) Let be a set and let be a total order on . We say that is dense if for any two with there exists such that
Consider the set
Is A dense?
(it looks as though it is dense because for every two elements you can incerease n so that you find an element between them but I wasn't 100% sure on this.)
(c) Prove that is the only positive rational number such that is an integer.
(I'm not sure what to do here, I tried to suppose that is an integer and that and then is an integer but that this only happens when , is any of that right?)
(d) How many positive real numbers x are there such that is an integer? Answer should be one of: 0,1,2,3,..., countable infinite, uncountably infinite.
(x=1 is the only rational from the previous part so its certainly not 0. But are there irrationals x such that is an integer? I don't know...)
Thanks for any help!
Suppose for some integer that:
then:
and is rational if and only if the discriminant of this quadraic is a perfect square, that is for some :
As the gap between consecutive squares is greater than when the lesser is the square of any number of absolute value greater than or equal , and the other cases are easily checked manualy the only solutions to this are
Now for irrational 's. Any such that:
but not a perfect square then is an irrational such that is an integer.
In particular let , then is irrational and a solution of:
.
But in part (c) I prove that:
(c) Prove that is the only positive rational number such that is an integer.
so x = 1 is the only rational such that is an integer. So there can't be an infinite number, there is just x=1.
otherwise I guess there are countable infinite irrationals.