Addition to Lemma 81. In [1], § 4, there is a variant of the matrix formula (64) for the simple continued fraction of a real number. Given integers a0, a1, . . . with ai > 0 for i ≥ 1 and writing, for n ≥ 0, as usual, pn/qn = [a0, a1, . . . , an], one checks, by induction on n, the two formulae ( 1 a0 0 1 )( 1 0 a1 1 ) · · · ( 1 an 0 1 ) = ( pn−1 pn qn−1 qn ) if n is even ( 1 a0 0 1 )( 1 0 a1 1...