Can I rationalize the denominator of $\frac{1}{\pi}$?

Question

Can I rationalize the denominator of $\frac{1}{\pi}$?

Answer 1

No, you cannot.

This is a consequence of the fact that $\pi$ is transcendental. This implies that there are no algebraic manipulations with rational numbers we can do (besides multiplying by $0$, which is obviously not allowed) which will yield a rational number in the denominator.

The reason we can rationalize the denominator in e.g. $\frac{1}{\sqrt{2}}$ is that $\sqrt{2}$ is algebraic (as opposed to $\pi$, which is transcendental).