Subscribe to the weekly news from TrueShelf

## Dirichlet approximation

1. (Trivial approximation) For $x \in \mathbb{R}$ and $n \in \mathbb{Z}^+$, there is a rational number $\frac{p}{q}$, with $1 \leq q \leq n$, such that

$\left|x - \frac{p}{q}\right| < \frac{1}{n}$

2. (Dirichlet approximation) For $x \in \mathbb{R}$ and $n \in \mathbb{Z}^+$, there is a rational number $\frac{p}{q}$, with $1 \leq q \leq n$, such that

$\left|x - \frac{p}{q}\right| < \frac{1}{nq}$