This is what Euclid would do. Say there are finitely many of them: . So that is rational. Then has property that is rational. However, because . A contradiction!
I am teaching a student proofs this semester. I came across a problem in the text that I cannot think of a simple solution for.
It is in the chapter on "Proof by Contradiction" (the text we are using, for those who are interested, is "Mathematical Proofs: A transition to advanced Mathematics" by Chartrand).
Anyway, here is the problem and here is my proof, which I am pretty sure is valid.
Result: Prove that there are infinitely many positive integers such that is irrational.
I will use the fact that if an integer is a perfect square, then it is equivalent to 0 mod 4 or 1 mod 4, as a lemma. (I would prove this separately, or better yet, ask the student to prove it.)
Proof:
Assume, to the contrary, that there are only finitely many integers such that is irrational. Then, there is a largest such integer, call it . By the above lemma, we have that or . Say the former, then that means also, and so is irrational as well, since is not a perfect square. But , and so we have found a larger integer than our largest, a contradiction. Assuming yields a similar contradiction.
QED
So here's the thing: I know it's a famous result, but the book has not mentioned the lemma I used. And it is not something I think a person who is a beginner when it comes to proofs (and is also a novice when it comes to set theory and number theory) can come up with. I can't really think of any simple ways to prove it though. Do you guys know any easier way?
Thanks
Yes, it should be "irrational", and that causes the proof to fail: and are distinct irrational numbers, yet their product is not.
In order to prove the theorem, I would give an easy and well-worth knowing proof that is irrational, and then say that is irrational for all (otherwise, would be rational). This is a constructive proof, which perhaps is not what you want, but after all the fact that is irrational is a must-know one (both historically and since proof is elementary).
Ah, thanks. that is a nice way.
whether the proof is constructive or not doesn't matter to me. as long as it is a simple one that a beginner can understand. if anything, i can offer hints along with the question so the student knows what direction i want them to think in
in light of what Laurent said (it had occured to me when i read the proof at first) how does this follow? since if is irrational and is irrational, it does not imply that is irrational.
i am thinking of doing a combination of the proofs you and Laurent posted, but that might be too complicated. a constructive proof is perhaps best for the level this student is at. i will be asking them to prove is irrational--it's a famous result, i have to make sure they can do it--so maybe using that somehow would be good.
EDIT: Ah, it seems TPH's proof was about rational square roots, not irrational ones.