Let $\alpha\not\in\mathbb{Q}$. Let

The irrationality measure of $\alpha$, denoted by $\mu(\alpha)$, is defined by

If $M(\alpha)=\emptyset$, we set $\mu(\alpha)=\infty$.

This definition is (loosely) a measure of the extent to which $\alpha$ can be approximated by rational numbers. Of course, by the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$, we can make arbitrarily good approximations to real numbers by rationals. Thus this definition was made to represent a stronger statement: it is the ability of rational numbers to approximate $\alpha$ given a fixed growth bound on the denominators of those rational numbers.

By the Dirichlet’s Lemma, $\mu(\alpha)\geq 2$. Roth [6, 7] proved in 1955 that $\mu(\alpha)=2$ for every algebraic real number. It is well known also that $\mu(e)=2$. For almost all real numbers the irrationality measure is 2. However, for special constants, only some upper bounds are known:

It is worth noting that the last column of the above table is simply a list of references, not a collection of discoverers. For example that fact that the irrationality measure of $e$ is 2 was known to Euler.