# Proving irrational

• Sep 25th 2008, 10:48 AM
Sinuyen
Proving irrational
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
• Sep 25th 2008, 11:37 AM
Moo
Hello !
Quote:

Originally Posted by Sinuyen
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

Here is my go...
Let $x=\frac{x'}{x''}$ and $y=\frac{y'}{y''}$ (x',x'',y',y'' are nonzero integers if there are, you can simply see if the results are rational or not)

See for $y+xz$. Suppose it is a rational, that is to say $y+xz=\frac pq$

$xz=\frac pq-\frac{y'}{y''}=\frac{py''-qy'}{qy''}$

$z=\frac{py''-qy'}{qy''} \cdot \frac{x''}{x'}=\frac{px''y''-qy'x''}{qx'y''}$

Is the RHS a rational ?
Is z a rational ?

Same reasoning for the second one :)
• Sep 25th 2008, 12:48 PM
Sinuyen
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.
• Sep 25th 2008, 01:07 PM
Moo
Quote:

Originally Posted by Sinuyen
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?

Exactly, it was a proof by contradiction, but I didn't say that lol

Quote:

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.
It is not a matter at all.

Here is the key : a rational number is in the form $\frac ab$ 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 ?
• Sep 25th 2008, 01:12 PM
Sinuyen
LOL it came to me just now and then I saw your post, so I can see that it is as i thought. Thank you very much.