# Trying to understand the irrationality criterion with examples.

If the irrationality criterion below is wrong feel free to edit this post.

The irrationality criterion, as far I understood is that if exists sequences $$p_n,q_n (n=1,2,3,...)$$ where $$p_n,q_n$$ are integers then the number $$\alpha \neq p_n/q_n$$ is irrational if \begin{align} \left| \alpha -\frac{p_n}{q_n}\right|<\frac{1}{q_n^{1+\delta}} \end{align} where $$\delta>0$$.

If we multiply everything by $$q_n$$ we have $$|q_n\alpha-p_n|<1/q_n^\delta$$ so assuming that $$\alpha=r/s$$ with both $$r,s$$ integers, we find a integer in the interval $$(0,1)$$ because $$\delta>0$$, which is a contradiction.

So we must find sequences that $$p_n,q_n$$ such that $$p_n/q_n$$ converges very quickly to $$\alpha$$. How does one create such sequences and how to find $$\delta$$ ? I guess they don't fall from the sky.

For example, how we prove that $$e$$ is irrational using this criterion?

$$\left| e -\frac{p_n}{q_n}\right|<\frac{1}{q_n^{1+\delta}}$$

$$e$$ is rather exceptional in that it has a closed-form continued fraction:

$$e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8,\ldots]$$

The corresponding convergents $$p_n/q_n$$ are $$2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, \ldots$$ which satisfy your inequality with $$\delta = 1$$.

Of course, if you have the continued fraction you really don't need your criterion: any non-terminating simple continued fraction represents an irrational number.