The Euclidean remainders

Abstract

The Euclidean algorithm applied to arbitrary real numbers \(\ r_{-1} > r_0 > 0\)  is closely related to the continued fraction expansion of     \(\ r_{-1}/r_0,\) but an explicit formula relating the remainders to the digits of the continued fraction is not found in the English language literature. (A German language reference for this is: Oskar Perron: Die Lehre von den Kettenbruechen Band I B.G. Teubner,
Stuttgart (1971)). In this note, we give a short and self-contained derivation of an explicit formula for the remainders \(\ r_n\) in terms of continuant polynomials, from which the well-know fact that \(\ r_n\) goes to zero at least as fast as  \(\phi^{-n}\) (where \(\phi\) is the golden ratio) follows immediately.