On quadratic numbers and forms, and Markoff theory (Q2039516)

Let \(\alpha\) be a root of quadratic polynomial with integer coefficients and \(\frac {p_n}{q_n}\) its partials. Then the author proves that \(\alpha - \frac {p_n}{q_n}=\frac 1{aq_n^2(1+\sqrt{1+\frac 1{bq_n^2}})}\), where \(a\) and \(b\) are constants depending on the character of the number \(\alpha\). The numbers \(a\) and \(b\) are described in detail. He applies this on the Markoff theory and obtains several interesting results.