Hi,
I have been reading a lot on here about proving that a number is irrational, which has helped. But I have not seen anything when it is simply variables that are either said to be rational or irrational. Here is my question:
Prove that, if x and y are rational, x is not equal to 0, and z is irrational, y + xz is irrational, and that y +x/z is irrational.
I am not sure at all on how to approach, and unfortunately, I have been assigned this problem without it being addressed in lecture or the text. =/
Thank you
I understand all the steps your present and it makes sense. All I have left to show is that the RHS is rational, but how do i do that? Gah! The reasoning behind number theory confuses me. >.< But the whole idea behind the proof is to present a contradiction no? So by saying that z is rational when in the beginning it was irrational correct?
The thing is I do not know how to say this mess of variables that represent integers will always come out to a rational number. Do I have to repeat the process by setting it equal to some rational expression r/s where r and s are non-zero integers?
Sorry for my incompetence in the area.
Edit:
Cool, after looking at it long enough, it makes sense. Since z is equal to a ratio of integers, then that is enough to prove it is rational. But since in the beginning it was said that z is irrational, this is a contradiction. Therefore, y + xz is irrational. Sweet! Thanks Moo.
Exactly, it was a proof by contradiction, but I didn't say that lol
It is not a matter at all.The thing is I do not know how to say this mess of variables that represent integers will always come out to a rational number. Do I have to repeat the process by setting it equal to some rational expression r/s where r and s are non-zero integers?
Sorry for my incompetence in the area.
Here is the key : a rational number is in the form where a and b are integers.
Are the ugly stuff integers ? It doesn't matter what they represent, all you have to know is whether their sum & product are integers ! Do you kind of get it or do you need further explanation ?