# Thread: rational number well ordering principle.

1. ## rational number well ordering principle.

rational number well ordering principle. thanks.

2. For the first one, consider $\displaystyle u=\sqrt{2}^{\sqrt{2}}$. If $\displaystyle u$ is rational, we are done. Otherwise, what can you say about $\displaystyle u^{\sqrt{2}}$?

For the second one, where are you stuck?

3. For the first one, what about $\displaystyle e^{\ln(2)}$?

4. Sure, but assuming $\displaystyle e$ is irrational is assuming much more than assuming $\displaystyle \sqrt 2$ is. Moreover, while I'm sure $\displaystyle \ln 2$ is irrational, I wouldn't know how to prove it. (Though I can show $\displaystyle e$ is irrational!)

5. Originally Posted by Bruno J.
Sure, but assuming $\displaystyle e$ is irrational is assuming much more than assuming $\displaystyle \sqrt 2$ is. Moreover, while I'm sure $\displaystyle \ln 2$ is irrational, I wouldn't know how to prove it. (Though I can show $\displaystyle e$ is irrational!)
Suppose that $\displaystyle \ln(2)=\frac{p}{q}$ then $\displaystyle 2=e^{\frac{p}{q}}$. You said you can prove that $\displaystyle e^x$ is irrational for rational values, yes?

6. just have no clue to start it ?

7. Originally Posted by Drexel28
Suppose that $\displaystyle \ln(2)=\frac{p}{q}$ then $\displaystyle 2=e^{\frac{p}{q}}$. You said you can prove that $\displaystyle e^x$ is irrational for rational values, yes?
Yeah. I'm that tired!

But, in any case, the OP is certainly not expected to come up with the proof of $\displaystyle e$'s irrationality. The proof I outlined is perfectly good (and the archetype of a non-constructive proof)!

8. Originally Posted by lemon721
just have no clue to start it ?
Did you try what I suggested?

9. The first one I think u r right...
I mean the second one...cannot handle the well ordering principle..