Elements of Algebra

CHAP. I. ADDITIONS. 479 that there is another fraction, $m/n$, whose value falls between the values of those two fractions, and whose denominator $n$ is less than $c'$, or less than $D'$. Now, since $m/n$ is between $c/c'$ and $D/D'$, the difference of $m/n$ and $c/c'$, which is $(mc'-nc)/(nc')$, or $nc-mc'$, must be less than $1/(c'D')$, the difference between $D/D'$ and $c/c'$; but it is evident that the former cannot be less than $1/(nc')$; and therefore if $n \not< D'$, it will necessarily be greater than $1/(c'D')$. Also, as the difference between $m/n$ and $D/D'$ cannot be less than $1/(nD')$, it will necessarily be greater than $1/(c'D')$, if $n \not< c'$, whereas it must be less. 13. Let us now see how each fraction of the series $A/A'$, $B/B'$, \ldots will approximate towards the value of the quantity $a$. For this purpose, it may be observed that the formulæ of Article 10 give $a = (Ab+1)/(A'b)$ $a = (cd+B)/(c'd+B')$ $a = (Bc+A)/(B'c+A')$ $a = (De+C)/(D'e+c')$ and so on. Hence, if we would know how nearly the fraction $c/c'$, for example, approaches to the given quantity, we seek for the difference between $c/c'$ and $a$; taking for $a$ the quantity $(cd+B)/(c'd+B')$, we shall have $a - c/c' = (cd+B)/(c'd+B') - c/c' = (BC'-CB')/(c'(c'd+B')) = 1/(c'(c'd+B'))$ because $BC' - CB' = 1$, (Art. 12). Now, as we suppose the