Excluding square/cubic etc roots of numbers, I've read proofs about the irrationality of $e,\pi,\ln 2,\zeta(2)$ and $\zeta(3)$. In all of them is: assume $x=p/q$, where $x$ is the number trying to proved irrational and $p,q$ integers. Long story short, by assuming $x=p/q$ we found an integer between $0$ and $1$.
Is no other way to prove a number is irrational? Maybe finding a combination of $p,q$ that is not integer. For example, assume $x=p/q$ then $p+q = \text{something not integer}$.
PS. The only proof that I've that does not end this way is the beuker's proof for $\pi$, where he find the estimation $1/p^n<1/n!$ which is not possible thus $\pi$ is irrational.