structural:scholar_edition
Irrational powers

Dov Jarden gave a simple non- constructive proof that there exist two irrational numbers a and b, such that a is rational:

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √ 2 is transcendental, hence irrational. This theorem states that if a and b are both algebraic numbers, and a is not equal to 0 or 1, and b is not a rational number, then any value of a is a transcendental number (there can be more than one value if complex number exponentiation is used).

An example that provides a simple constructive proof is

The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, log 2 3 {\displaystyle \log _{\sqrt {2}}3}, is irrational. This is so because, by the formula relating logarithms with different bases,

which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then log 2 3 = m / 2 n {\displaystyle \log _{2}3=m/2n} hence 2 log 2 3 = 2 m / 2 n {\displaystyle 2^{\log _{2}3}=2^{m/2n}} hence 3 = 2 m / 2 n {\displaystyle 3=2^{m/2n}} hence 3 2 n = 2 m {\displaystyle 3^{2n}=2^{m}}, which is a contradictory pair of prime factorizations and hence violates the fundamental theorem of arithmetic (unique prime factorization).

A stronger result is the following: Every rational number in the interval ((1 / e) 1 / e, ∞) {\displaystyle ((1/e)^{1/e},\infty)} can be written either as a for some irrational number a or as n for some natural number n. Similarly, every positive rational number can be written either as a a a {\displaystyle a^{a^{a}}} for some irrational number a or as n n n {\displaystyle n^{n^{n}}} for some natural number n.