after this the quotients 2, 1, 3, 1, 2, 8 recur; hence
It will be noticed that the quotients recur as soon as we come to a quotient which is double the first.
Explanation. We first find the greatest integer in 19; this is 4, and we write 19 =4+(19-4). We then express 19-4 as an equivalent fraction with a rational numerator. Thus
The work now stands
We begin the second line with the denominator of this complex fraction, which is itself a fraction with a rational denominator. The greatest integer in this fraction is 2, and we write
We then multiply numerator and denominator by the surd conjugate to 19 - 2, so that after inverting the result , we again begin a line with a rational denominator. The same series of operations is
performed in each of the following lines.
The first seven convergents formed as explained in Art. 502 are
