Liouville's theorem on Diophantine approximation

Statement

Let $\alpha$ be a real algebraic number of degree $d \ge 2$. Then there is a constant $c > 0$ such that $\left| \alpha - \frac{p}{q} \right| > \frac{c}{q^d}$ for all integers $p$ and $q$ with $q > 0$. As a consequence, each Liouville number is transcendental.

Remarks

The proof uses the minimal polynomial $f$ of $\alpha$: $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\alpha - p/q|$. For a Liouville number $x$, the approximations are better than $c / q^d$ for every $d$, so $x$ is not algebraic. Roth's theorem improves the exponent $d$ to $2 + \varepsilon$ for each $\varepsilon > 0$.

History

Joseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.

Explanations

L1 human

Algebraic numbers cannot be approximated by fractions too well: the error is at least a constant times $1/q^d$. A number with better approximations of every order cannot be algebraic. This gave the first explicit transcendental numbers.