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Number Theory pp 179-222 | Cite as

Continued Fractions and Their Uses

  • W. A. Coppel
Chapter
Part of the Universitext book series (UTX)

Abstract

Let \(\xi = \xi_0\) be an irrational real number. Then we can write
$$\xi_0 = {\rm a}_0 + \xi^{-1}_{1},$$
where \(a_0 = \lfloor \xi_0 \rfloor\) is the greatest integer \(\leq \xi_0\) and where \(\xi_1 > 1\) is again an irrational number. Hence the process can be repeated indefinitely:
$$\begin{array}{c} \xi_1 = {\rm a}_1 + \xi^{-1}_2, \quad ({\rm a}_1 = \lfloor \xi_1 \rfloor, \xi_2 > 1),\\ \xi_2 = {\rm a}_2 + \xi^{-1}_3, \quad ({\rm a}_2 = \lfloor \xi_2 \rfloor, \xi_3 > 1),\\ \ldots \end{array}$$

By construction, \({\rm a}_n \in \mathbb{Z}\) for all \(n \geq 0\) and \(a_n \geq 1 \,{\rm if}\, n \geq 1\). The uniquely determined infinite sequence \([{\rm a}_0, {\rm a}_1, {\rm a}_2, \ldots]\) is called the continued fraction expansion of \(\xi\). The continued fraction expansion of \(\xi_n {\rm is} [{\rm a}_n, {\rm a}_{n+1}, {\rm a}_{n+2}, \ldots]\).

Keywords

Continue Fraction Diophantine Equation Modular Group Irrational Number Diophantine Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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