Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche

  title={Ueber die angen{\"a}herte Darstellung der Irrationalzahlen durch rationale Br{\"u}che},
  author={Adolf Hurwitz},
  journal={Mathematische Annalen},
  • A. Hurwitz
  • Published 1 June 1891
  • Mathematics
  • Mathematische Annalen
Man kann bekanntlich jede Irrationalzahl a durch eine unbegrenzte Reihe von rationalen Bruchen $$\frac{{{x_1}}}{{{y_1}}},\;\frac{{{x_2}}}{{{y_2}}},\;\frac{{{x_3}}}{{{y_3}}},\; \ldots $$ (1) derart annahern, dass, abgesehen vom Vorzeichen, $$\alpha- \frac{{{x_n}}}{{{y_n}}}< \frac{1}{{y_n^2}}\quad \left( {n = 1,\;2,\;3 \ldots } \right)$$ (2) ist. Dieser Satz ergibt sich unmittelbar aus der Lehre von den Kettenbruchen; er lasst sich aber auch durch andere sehr… 

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