how can i prove that if and are non-zero rational numbers, then
...using the fact that
Does this need proof? It should be obvious that a rational times an irrational is irrational, so is irrational.
The sum of a rational and an irrational is also irrational, so is irrational.
If it requires proof, then you probably need to use the fact that the set of rationals is closed under addition and multiplication...
But that rationals are closed under addition so:
is rational. Also the rationals are closed under division (by non-zero elements anyway) so:
is rational, which is a contradiction so our hypothesis that is rational fails, hence is irrational.