Elements of Algebra

470 ADDITIONS. CHAP. I. $\alpha + \frac{1}{\beta + \frac{1}{\gamma + \frac{1}{\delta, \&c.}}}$ which will be equal to the given fraction. 5. Let it be proposed, for example, to reduce $\frac{1103}{887}$ to a continued fraction. First, we divide $1103$ by $887$, which gives the quotient $1$, and the remainder $216$; $887$ divided by $216$, gives the quotient $4$, and the remainder $23$; $216$ divided by $23$, gives the quotient $9$, and the remainder $9$; also dividing $23$ by $9$, we obtain the quotient $2$, and the remainder $5$; then $9$ by $5$, gives the quotient $1$, and the remainder $4$; $5$ by $4$, gives the quotient $1$, and the remainder $1$; lastly, dividing $4$ by $1$, we obtain the quotient $4$, and no remainder; so that the operation is finished: and, collecting all the quotients in order, we have this series $1, 4, 9, 2, 1, 1, 4$, whence we form the continued fraction $\frac{1103}{887} = 1+\frac{1}{4+\frac{1}{9+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4}}}}}$ 6. As, in the above division, we took for the quotient the integer number which was equal to, or less than, the fraction proposed, it follows that we shall only obtain from that method continued fractions, of which all the denominators will be positive numbers. But we may also assume for the quotient the integer number, which is immediately greater than the value of the fraction, when that fraction is not reducible to an integer, and, for this purpose, we have only to increase the value of the quotient found by unity in the usual manner; then the remainder will be negative, and the next quotient will necessarily be negative. So that we may, at pleasure, make the terms of the continued fraction positive, or negative. In the preceding example, instead of taking $1$ for the quotient of $1103$ divided by $887$, we may take $2$; in which case we have the negative remainder $-671$, by which we must now divide $887$; we therefore divide $887$ by $-671$, and obtain either the quotient $-1$, and the remainder $216$, or the quotient $-2$, and the remainder $-455$. Let us take the greater quotient $-1$: then divide the remainder $-671$ by $216$; whence we obtain either the quotient $-3$, and the remainder $-23$, or the quotient $-4$, and the remainder $193$. Continuing the division by adopting the greater quotient $-3$, we have to divide the remainder $216$ by the