Elements of Algebra

CHAP. I. ADDITIONS. 469 quantity $\infty$. Now, $\frac{1}{\infty} = 0$, wherefore we shall only have $a = \alpha + \frac{1}{\beta + \frac{1}{c}}$. This case will happen whenever the quantity $a$ is commensurable; that is to say, expressed by a rational fraction; but when $a$ is an irrational, or transcendental quantity, then the continued fraction will necessarily go on to infinity. 4. Suppose the quantity $a$ to be a vulgar fraction, $\frac{A}{B}$, $A$ and $B$ being given integer numbers; it is evident, that the integer number, $\alpha$, approaching nearest to $\frac{A}{B}$, will be the quotient of the division of $A$ by $B$; so that supposing the division performed in the usual manner, and calling $\alpha$ the quotient, and $C$ the remainder, we shall have $\frac{A}{B} - \alpha = \frac{C}{B}$; whence $b = \frac{B}{C}$. Also, in order to have the approximate integer value $\beta$ of the fraction $\frac{B}{C}$, we have only to divide $B$ by $c$, and take $\beta$ for the quotient of this division; then calling the remainder $D$, we shall have $b - \beta = \frac{D}{C}$, and $c = \frac{C}{D}$. We shall therefore continue to divide $c$ by $D$, and the quotient will be the value of the number $\gamma$, and so on; whence results the following very simple Rule for reducing Vulgar Fractions to Continued Fractions. RULE. First, divide the numerator of the given fraction by its denominator, and call the quotient $\alpha$; then divide the denominator by the remainder, and call the quotient $\beta$; then divide the first remainder by the second remainder, and let the quotient be $\gamma$. Continue thus, always dividing the last divisor by the last remainder, till you arrive at a division that is performed without any remainder, which must necessarily happen, when the remainders are all integer numbers that continually diminish; you will then have the continued fraction,