Whether $\pi e$ and $\pi+e$ etc. are irrational or not are famously unsolved problems in math. Is the irrationality of $\frac{e}{\pi}$ or $\frac{\pi}{e}$ equally hard to prove or is it trivial? Haven't found anything about it online.
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1See Wikipedia page on transcendental numbers. – VTand Nov 08 '21 at 11:40
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1It is unknown whether $e$ and $\pi$ are algebraically independent over $\mathbb Q$. If $\frac{\pi}{e}$ were rational, this would not be the case. – Peter Nov 08 '21 at 11:42
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The linear independency has not been proven either. Hence it must be open whether $\frac{\pi}{e}$ is rational. – Peter Nov 08 '21 at 11:49
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1BTW if $q=\frac{e}{\pi}$ was proven to be rational, then $e+\pi=(q+1)\pi$ would be proven to be irrational. – Nov 08 '21 at 12:34