Could there be an irrational number $x$ such that the product of $x$ and $\pi$ are rational?

pi irrational-numbers products

If $r=x\pi$ where $r$ is rational, then we have $x=\frac r\pi$.

This means that if you have an $x$ such that $x\pi$ is rational, then $x$ is of the form $\frac k\pi$ for some rational $k$. This explains why the answer to your question should always be "derived from $\pi$".

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