If x is a positive rational number, show that x^{x} is irrational unless x is an integer.

This question is amenable to a reductio ad absurdum proof.

Assume x is rational but not an integer; that is, x can be written as a/b, irreducible, with b > 1. Assume (a/b)^{a/b} = c/d is irreducible.

Raising each side to the power b we get (a/b)^{a} = (c/d)^{b}.

Hence a^{a} d^{b} = c^{b} b^{a}.

Now we use the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be written uniquely as a product of finitely many prime numbers.

Since b > 1, it will have at least one prime factor, p > 1. Consider the number of times p occurs in the prime factorization of each of the above terms. Letting b = p^{r}u, where p and u are relatively prime, then:

- b
^{a}: ra times - a
^{a}: 0 times, since a and b are relatively prime - d
^{b}: sb times, where s is the number of times p occurs in d (since p occurs on the right-hand side, it must also occur on the left-hand side, so it must be a factor of d) - c
^{b}: 0 times, since c and d are relatively prime

By the Fundamental Theorem of Arithmetic, ra = sb. But since a and b are relatively prime, b must divide r.

That is, r b p^{r}, which is absurd for p > 1.

This completes the reductio ad absurdum proof.

Hence (a/b)^{a/b} is irrational.

Of course, if b = 1, then x is an integer for any integer a, and x^{x} is rational.

Therefore, if x is a positive rational number, x^{x} is irrational unless x is an integer.

The above result, in conjunction with the Gelfond-Schneider Theorem, can be used to show that the positive real root of x^{x} = 2 is a transcendental number.

Clearly x^{x} = 2 does not have an integer solution. Hence, by the above result, x cannot be rational.

Now, using the Gelfond-Schneider Theorem, if x is algebraic and irrational, x^{x} is transcendental, and so cannot be equal to 2.

Therefore the positive real root of x^{x} = 2 is transcendental. The root is approximately equal to 1.559610469462369349970388768765; see Sloane's A030798.

- What is a number?
- Rational Irrational Power
- How to discover a proof of the fundamental theorem of arithmetic

Source: Original