# Is it possible to express $e$ in terms of $\pi$ algebraically and vice-versa?

Am I right in thinking this is not possible since both are known to be transcendental?

Also, $e^{i\pi}+1=0$ suggests this is not possible - we can not isolate $e$ or $\pi$ from this since it involves taking a log at some point, thus "cancelling" $e$.

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This is unknown at the moment. They both are transcendental, but it is unknown if they are algebraically independent. –  M Turgeon Aug 30 '12 at 11:50
But doesn't $e^{i\pi}+1=0$ show them to be algebraically independent? –  pbs Aug 30 '12 at 11:55
No. It simply shows that there exists a non-algebraic relation between them. –  M Turgeon Aug 30 '12 at 11:57
So there could be an algebraic relation that exists that we just don't know of. Ok. –  pbs Aug 30 '12 at 11:59
Note that $e$ and $2e$ are both known to be transcendental, and despite that it is possible to express each in terms of the other. –  Gerry Myerson Aug 30 '12 at 12:50