Irrationals (1 Viewer)

[seanieg89]

Carrot, have you ever seen the really sexy nonconstructive proof that you can find irrationals a,b such that a^b is rational?

 

[Carrotsticks]

Re: The Nonsense Thread

Carrot, have you ever seen the really sexy nonconstructive proof that you can find irrationals a,b such that a^b is rational?

Yep I saw it floating somewhere in my Number Theory book =)

 

[RealiseNothing]

Carrot, have you ever seen the really sexy nonconstructive proof that you can find irrationals a,b such that a^b is rational?

When I tried it last night I was doing something very similar to the proof you posted, but gave up because I had to do English.

But I'd have to agree after seeing it, it is freakin' sexy.

 

[seanieg89]

When I tried it last night I was doing something very similar to the proof you posted, but gave up because I had to do English.

But I'd have to agree after seeing it, it is freakin' sexy.

Glad you like it .

Of course, its sexiness is slighly detracted from by the fact that it is very easy to EXPLICITLY construct a pair:



(Some mathematicians object to nonconstructive proofs.)

But I still think its a pretty striking argument.

 

[RealiseNothing]

Glad you like it .

Of course, its sexiness is slighly detracted from by the fact that it is very easy to EXPLICITLY construct a pair:



(Some mathematicians object to nonconstructive proofs.)

But I still think its a pretty striking argument.

I liked how it didn't matter whether or not the previous case was rational, because if it were then you've already proven it, and if it wasn't then your current case proves it. ie:



If this rational then it is proven, but if it's irrational then we can just say:



Which is rational. So whether or not the first case is rational has no impact whatsoever on the fact that a^b can be rational.