Number Theory pp 179-222 | Cite as

Continued Fractions and Their Uses

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


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 \(\xi_n {\rm is} [{\rm a}_n, {\rm a}_{n+1}, {\rm a}_{n+2}, \ldots]\).


Continue Fraction Diophantine Equation Modular Group Irrational Number Diophantine Approximation 
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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

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

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