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# It's not proved yet.

State whether $$(\pi)^e$$ is a rational number or an irrational number with proof.

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Note by Sandeep Bhardwaj
2 years ago

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It's transcendal number. · 3 months, 2 weeks ago

Proving that a number is an irrational is extremely difficult. As Shivang Jindal has mentioned, there a lot of "at least one of them is irrational" cases, but to prove which one is in the stars with respect to difficulty. Even the irrationality of $$\pi$$ is very difficult to prove. · 2 years ago

Even proving $e+\pi , e\pi$ is irrational/rational is unsolved,despite the fact that, at-least one of them has to be irrational![You can show this easily] . · 2 years ago

More strongly, at least one of $$e+\pi, e\pi$$ is transcendental. Proof here: $$x^2-(e+\pi)x+e\pi$$ has roots $$e,\pi$$. Assume for contradiction both $$e+\pi, e\pi$$ are algebraic. Then $$e,\pi$$ are algebraic (since they're roots of a polynomial with algebraic coefficients), contradiction, since $$e,\pi$$ are transcendental. · 1 year, 7 months ago