Dirichletscher approximation theorem

The Dirichlet's approximation , named after Peter Gustav Lejeune Dirichlet , is a mathematical theorem about the quality of approximation (approximation) of real numbers by rational numbers .

The sentence is: For each and every one exist one and one , so that ${\displaystyle \alpha \in \mathbb {R} }$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle q\in \mathbb {N} ,1\leq q\leq N}$ ${\displaystyle p\in \mathbb {Z} }$

${\displaystyle \left|q\alpha -p\right|\leq {\frac {1}{N+1}}.}$

This theorem can be proven using the drawer principle .

After dividing by and observing from the theorem, it follows that for every real one there are infinitely many pairs of positive integers that ${\displaystyle q}$ ${\displaystyle q<N+1}$ ${\displaystyle \alpha }$ ${\displaystyle (p,q)}$

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}}$

fulfill. For rational numbers almost all such approximations have the form , so the infinity proposition is only interesting for irrational numbers . The set of Hurwitz improved the inequality a factor . ${\displaystyle \alpha ={\tfrac {a}{b}}}$ ${\displaystyle p=ka,q=kb}$ ${\displaystyle {\sqrt {5}}}$

Example : Be and . Then, according to Dirichlet's approximation theorem, (at least) one of the numbers is at most removed from an integer. Indeed it is ${\displaystyle \alpha ={\sqrt {2}}}$ ${\displaystyle N=10}$ ${\displaystyle {\sqrt {2}},2{\sqrt {2}},\dotsc ,10{\sqrt {2}}}$ ${\displaystyle 1/11}$

${\displaystyle \left|5{\sqrt {2}}-7\right|=\left|7,07106\dotso -7\right|=0.07106\dotso \leq 0.090909\dotso ={\frac {1}{11}}.}$

literature

• Hans Rademacher, Otto Toeplitz: Of Numbers and Figures , Chapter 15: "Approaching irrational numbers by rationale", Springer 1930 and numerous new editions.