Is there an irrational number a such

Armeninilu

Armeninilu

Answered question

2022-06-27

Is there an irrational number a such that a a is rational?

Answer & Explanation

Judovh0

Judovh0

Beginner2022-06-28Added 16 answers

By continuity of the function x x , there is an a such that
a a = 2.
This number is irrational. Otherwise, let a be the irreducible fraction p / q and
( p q ) p / q = 2 , is equivalent to
p p = 2 q q p ,
which implies that p is even and q is even, a contradiction.

By the way, as
ln ( a a ) = a ln ( a ) = ln ( a ) e ln ( a ) ,
we have
a = e W ( ln ( 2 ) )
where W is the Lambert function.

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