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What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational?

I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?


marked as duplicate by apnorton, user147263, Mark Fantini, Claude Leibovici, Najib Idrissi Sep 27 '14 at 8:01

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    $\begingroup$ This question is hardly homework... $\endgroup$ – lhf Sep 27 '14 at 3:04
  • $\begingroup$ See math notation guide. $\endgroup$ – user147263 Sep 27 '14 at 3:19

It is not currently known whether or not $\pi+e$ is rational. It is also not known whether or not $\pi e$ is rational. However, we can say that $\pi+e$ and $\pi e$ cannot both be rational.

For suppose to the contrary that they are both rational. Then $(\pi+e)^2-4\pi e$ is rational. But this is $(\pi-e)^2$, so $\pi-e$ is algebraic. But then adding and subtracting $\pi+e$, we find that $\pi$ and $e$ are algebraic. But in fact both are transcendental.

The same proof shows that $\pi+e$ and $\pi e$ cannot be both algebraic.

Remark: The natural conjecture is that both $\pi+e$ and $\pi e$ are transcendental. After all, "most" real numbers are. However, settling the question of transcendentality, or even irrationality, of either would represent a major mathematical achievement.


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