Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates or repeats. Let us call that the “decimal characterization” of rationality. The proof of that characterization of rational numbers is obviously just as applicable to other bases as it is to base $10$.
$\mathscr Question:$ Are there any proofs of the irrationality of $\pi$ or $e$ or $\sqrt 2$ or $\log_2 3$ or any other naturally occurring number that use the decimal characterization, showing directly that the decimal expansion does not terminate or repeat, and that are at least as simple as any proof that uses the characterization that is conventionally taken to be the definition?