I was reading that you can associate a measure to any given number giving you "how irrational" the given number is. I was wondering is there any irrationality measure that would tell you that the number under consideration is 100 percent irrational.
I guess what I am asking is: Can you establish that any number is irrational purely by looking at its associated irrationality measure?
Also, for numbers whose irrationality is unknown like Pi+e, is anything known about their irrationality measure?
From Wikipedia: Liouville_number, Irrationality_measure:
As a consequence of Dirichlet's approximation theorem every irrational number has irrationality measure at least 2.
and
Every rational number $\frac{p}{q}$ has an irrationality measure of exactly 1.
So
$x \in \mathbb{Q} \implies \mu(x) = 1$ and
$x \in \mathbb{R} \backslash \mathbb{Q} \implies \mu(x) \geq 2$.
Thus, these are not just implications, but also equivalences, because let's say we know the irrationality measure $\mu(x)$ of some number $x \in \mathbb{R}$. Note that 1 < $\mu(x)$ < 2 can't happen. So if $\mu(x) = 1$, we know that $x$ cannot be irrational, thus it must be rational. And if $\mu(x) \geq 2$, then $x$ cannot be rational, thus it must be irrational. In conclusion we assert that
$x \in \mathbb{Q} \iff \mu(x) = 1$ and
$x \in \mathbb{R} \backslash \mathbb{Q} \iff \mu(x) \geq 2$.
This answers your second question. If we know the interval wherein $\mu(x)$ is, then we directly know if $x$ is rational or irrational, so if we would know, hypothetically, e.g. that $2 \leq \mu(\pi+e) < 1000$, then this would directly tell us that $\pi+e$ is irrational.
