CHAPTER XLIII.
Continued Fractions.
497. An expression of the form a+ {b}{c+{d}{e+}} is called a continued fraction; here the letters a, b, c, may denote any quantities whatever, but for the present we shall only consider the simpler form a_ 1+ {1}{a_2+{1}{a_3+ }} where a_1, a_2, a_3, are positive integers. This will be usually written in the more compact form
a_1 + {1}{a_2 + } {1}{a_3 + }.
498. When the number of quotients a_1, a_2, a_3, is finite the continued fraction is said to be terminating; if the number of quotients is unlimited the fraction is called an infinite continued fraction.
It is possible to reduce every terminating continued fraction to an ordinary fraction by simplifying the fractions in succession beginning from the lowest.
499. To convert a given fraction into a continued fraction.
Let {m}{n} be the given fraction; divide m by n; let a_1 be the quotient and p the remainder; thus
{m}{n} = a_1 + {p}{n} = a_1+ {1}{{n}{p}};
