Irrational number power irrational number
Can an irrational number power of an irrational number be a rational number?
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2 solutions
Why is log(2) irrational? Is there a simple proof of that?
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If ln(2)=p/q(p and q are integers),then 2^q=e^p,but e^p can't be an integer
Now by eulers identity e^ipie +1=0.Transpose it to get e^ipie=-1.
Yes, since 2 is irrational. Consider (\sqrt{2}^\sqrt{2})^\sqrt{2}=2 . Either \sqrt{2}^\sqrt{2} is rational or 2 and \sqrt{2}^\sqrt{2} are irrational and one power another is rational.
Okay, EITHER 2 2 is rational (it isn't, it's actually a transcendental number), OR both 2 and 2 2 both are irrational (they both are). All right, no need to prove that 2 2 is irrational, the argument has been made.
e ln 2 = 2