Irrational Number - Irrational Powers

Irrational Powers

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

Indeed, if √2√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2√2 and b = √2. Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2, which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √2√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 ab 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, 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 hence hence hence, 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 can be written either as aa for some irrational number a or as nn for some natural number n. Similarly, every positive rational number can be written either as for some irrational number a or as for some natural number n.