Proof a number is irrational

1. May 2, 2009

Pjennings

I've been thinking about pi^e lately, and trying to prove that it is irrational. By rewriting e as 1+1+1/2+1/3!+...+1/n! I got it to pi^2*pi^(1/2)*pi^(1/3!)*...*pi^(1/n!), and proved that each of these terms is irrational. I'm stuck when it comes to showing that multiplied together these numbers are irrational. Any ideas?

2. May 2, 2009

Hurkyl

Staff Emeritus
Although we know a lot about certain forms, proving numbers irrational is generally a very, very hard thing to do. For example, I think we don't even know whether or not pi+e is irrational!

3. May 2, 2009

CRGreathouse

I don't think that is a workable approach, since it's possible to form a rational number as the series where the partial sums are all irrational.

4. May 2, 2009

Pjennings

No, we do not know if pi+e is irrational, but I don't think that one is very interesting. I think it would be interesting to know if γ(gamma) is irrational though.