rationalize the denominator

Question

Is there any theorem saying that in below fraction we can't rationalize the denominator
$\frac{2}{\pi}$
I couldn't find any way and I don't think there is but I was wondering if this actually proved?
for example for $\frac{1}{\sqrt{2}}$ we say
$\frac{1}{\sqrt{2}}=\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}$
result have meaning if we look at fraction like Division

Answer 1

Are you speaking of rationalizing the denominator and the numerator at the same time? If not, a trivial solution would be $\frac{2/\pi}{1}$, but that doesn't seem very interesting. So perhaps you are speaking about some sort of manipulation that creates $\frac{2}{\pi}=\frac{p}{q}$, where both $p$ and $q$ are rational. But, I would find that greatly disturbing, because then it would be the case that $\frac{p}{q}$ is also rational... in which case $\frac{2}{\pi}$ would be rational as well.

EDIT: Your example with $\sqrt 2$ is similar to the trivial solution above, but, in the case of a radical, might have some more use.

So, yes, you can rationalize the denominator, if you find such action useful. But, would you, in the case of $\frac{2}{\pi}$?