Elements of Algebra

CHAP. II. ADDITIONS. 529 $\checkmark$93 1 12 7 11 4 3 4 11 7 12 1 12 7 \ldots 9 1 1 1 4 6 4 1 1 1 18 1 1 \ldots $\checkmark$94 1 13 6 5 9 10 3 15 2 15 3 10 9 5 6 13 1 \ldots 9 1 2 3 1 1 5 1 8 1 5 1 1 3 2 1 18 \ldots $\checkmark$95 1 14 5 14 1 14 \ldots 9 1 2 1 18 1 \ldots $\checkmark$96 1 15 4 15 1 15 \ldots 9 1 3 1 18 1 \ldots $\checkmark$97 1 16 3 11 8 9 9 8 11 3 16 1 16 \ldots 9 1 5 1 1 1 1 1 1 5 1 18 1 \ldots $\checkmark$98 1 17 2 17 1 17 \ldots 9 1 8 1 18 1 \ldots $\checkmark$99 1 18 1 18 1 \ldots 9 1 18 1 18 \ldots Thus, for example, we shall have $\sqrt{2} = 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} +$, \ldots $\sqrt{3} = 1 + \frac{1}{1} + \frac{1}{2} \ldots$ and so of others. And, if we form the converging fractions, $\frac{p^o}{q^o}, \frac{p'}{q'}, \frac{p''}{q''}, \frac{p'''}{q'''}, \ldots$ according to each of these continued fractions, we shall have $(p^o)^2 - 2(q^o)^2 = 1, p'^2 - 2q'^2 = -1,$ $p''^2 - 2q''^2 = 1, \ldots$ and likewise, $(p^o)^2 - 3(q^o)^2 = 1, p'^2 - 3q'^2 = -2,$ $p''^2 - 3q''^2 = 1, \ldots$ M M