Elements of Algebra

480 ADDITIONS. CHAP. I. approximate value of $d$, so that the difference between $d$ and $\delta$ is less than unity (Art. 3), it is evident that the value of $d$ will lie between the two numbers $\delta$ and $\delta \pm 1$, (the upper sign being for the case, in which the approximate value $\delta$ is less than the true one $d$, and the lower sign for the case, in which $\delta$ is greater than $d$), and, consequently, that the value of $C'd + B'$, will also be contained between these two, $C'\delta + B'$, and $C'(\delta \pm 1) + B'$; that is to say, between $D'$ and $D' \pm C'$; therefore the difference $a - \frac{C}{C'}$ will be contained between these two limits $\frac{1}{C'D'}$, $\frac{1}{C'(D' \pm C')}$; whence we may judge of the degree of approximation of the fraction $\frac{C}{C'}$. 14. In general, we shall have, $a = \frac{A}{A'} + \frac{1}{A'b}$ $a = \frac{B}{B'} - \frac{1}{B'(B'c + A')}$ $a = \frac{C}{C'} + \frac{1}{C'(C'd + B')}$ $a = \frac{D}{D'} - \frac{1}{D'(D'e + C')}$, and so on. Now, if we suppose that the approximate values, $\alpha, \beta, \gamma, \ldots$ are always taken less than the real values, these numbers will all be positive, as well as the quantities $b, c, d, \ldots$ (Art. 3.) and, consequently, the numbers $A', B', C', \ldots$ will be likewise all positive; whence it follows, that the differences between the quantity $a$, and the fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, \ldots will be alternately positive and negative; that is to say, those fractions will be alternately less and greater than the quantity $a$. Farther, as $b > \beta$, $c > \gamma$, $d > \delta$, \ldots by hypothesis, we have $b > B'$, $(B'c + A') > (B'\gamma + A')$, and also $> C'^*$, $(C'd + B') > (C'\delta + B')$, and therefore $> D'$, \ldots and as $b < (\beta + 1)$, $c < (\gamma + 1)$, $d < (\delta + 1)$, we have $b < (B' + 1)$, * For since $c > \gamma$, therefore $B'c > B'\gamma$; and, consequently, $(B'c + A') > (B'\gamma + A')$, which is $> C'$, because $B'\gamma + A' = C'$, page 476. And it is exactly the same with the other quantities. B.