TITLE: Are there any natural proofs of irrationality using the decimal characterization?
QUESTION [6 upvotes]: 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 $10$.
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?

REPLY [5 votes]: I just learnt this, it is not about the decimal expansion, but I think it is clearly related to your question :
A simple proof that $e$ is irrational is using its factorial expansion $2+\sum_{n=2}^\infty \frac{1}{n!}$.
The theorem is that
$$\sum_{n=1}^\infty \frac{a_n}{n!}, \qquad a_1 \in \mathbb{Z},\quad a_n \in \{0, \ldots, n-1\}$$
is irrational if and only if $a_n  >0$ infinitely often and $a_n < n-1$ infinitely often. 
The proof is as follows : 

any real number can be written as 
$$\sum_{n=1}^\infty \frac{a_n}{n!}, \qquad a_1 \in \mathbb{Z},\quad  a_n \in \{0, \ldots, n-1\}$$
with the same algorithm as for the decimal expansion : $$a_n = \lfloor x_n n!\rfloor, \qquad x_{n+1} = x_n-  \frac{a_n}{n!}$$
with this algorithm, every rational number has a finite expansion, so if $x$  has a finite expansion then the algorithm finds it. More generally, if applied to $x = \sum_{n=1}^\infty \frac{c_n}{n!}, c_1\in \mathbb{Z},c_n \in \{0, \ldots, n-1\}$ where $c_n < n-1$ infinitely many times, then the algorithm finds back $a_n = c_n$.
when applied to $e-1$, the algorithm finds $a_n = 1$, thus $e-1$ doesn't have any such finite expansion, hence it is irrational.

The conclusion is that the factorial expansion (more generally the Cantor series expansion,  if someone knows a better reference ?) is much easier than the base $N$ expansion for proving the irrationality of a real number.