$e+\pi$ and $e\cdot\pi$ Transcendental number or what?
I know what $e$ and $\pi$ transcrendental and $e\cdot\pi$ posible be rational and posible to be irrational, ($\sqrt2\cdot\sqrt2=rational$,$\sqrt2\cdot\sqrt3=irratinoal$)
Anyway, We can use this function $f_n=\dfrac{x^n(1-x)^n}{n!}$ in proof of $\pi$,$e$ which these irrational,
But in there, how we can prove these 2 equation ?
As of today, we don't know if $e+\pi$, $e\cdot\pi$ are irrational numbers.
This is an open problem.
From the expansion $$(x-\pi)(x-e) = x^2 - (e + \pi)x + e\pi$$ one sees that the above polynomial can't have rational coefficients.
Thus at least one of $e + \pi$ and $e\pi$ is irrational. As @Henning Makholm pointed out, actually at least one of them is not just irrational but transcendental (since the roots of a polynomial with algebraic coefficients are algebraic).
We are inclined to think both are transcendental numbers.
A similar question.