Continued fractions and irrational rotations

Let $$\alpha \in (0, 1)$$α∈(0,1) be an irrational number with continued fraction expansion $$\alpha =[0; a_1, a_2, \ldots ]$$α=[0;a1,a2,…] and let $$p_n/q_n= [0; a_1, \ldots , a_n]$$pn/qn=[0;a1,…,an] be the nth convergent to $$\alpha $$α. We prove a formula for $$p_nq_k-q_np_k$$pnqk-qnpk$$(k<n)$$(k<n) in terms of a Fibonacci type sequence $$Q_n$$Qn defined in terms of the $$a_n$$an and use it to provide an exact formula for $$\{n\alpha \}$${nα} for all n.