The most usual form of a continued fraction, having a

fractions. When the number of component fractions is limited, the continued fraction may be called terminate; and when the number is indefinite, it may be said to be interminate."

(600.) Continued fractions may be generated in various ways. One of the most common is when the value of a quantity, which is not integral, is to be found by a series of approximations.

Let a' be such a quantity; and find the next lower integer to it, which denote by a; then a'—a—1; there>1, which quantity may be denoted by b'.

fore

1

a- -a

If b' is not integral, let b be the next lower integer to

If c' is not integral, let c be the next lower integer to it;

1

then c'—c≤1; therefore, which quantity may be denoted by d'.

Proceeding in this manner as far as necessary, then

Hence by substituting successively the values of b', c', d',...

(601.) Continued fractions may also be derived from common fractions, by performing upon their terms the process of finding their greatest common measure. Thus, if

A

B

be a fraction, and a, b, c, d, ... the quotients obtained by this operation, and C, D, E, F, ... the corresponding remainders, then it is evident from art. (103) that

AaB+C, B=bC+D, C=cD+E, ...

called complete quotients. The quantities a, b, c, ... here

correspond with a, b, c, ... in art. (600); also

...

A B C B'C' D'

correspond with a', b', c', in that article. If any remainder as E be = 0, then c is the complete quotient, and the continued fraction terminates with c.

(603.) It is usual to represent a continued fraction merely by stating the quotients; thus, if a' be the given fraction,

As this is a proper fraction, the first quotient is 0; the others are 1, 2, 12; hence

(604.) Continued fractions may be easily reconverted into common fractions. Thus, taking the first example, the

be proved to be correct in the same manner, which may be done by the student for exercise.

U
Ꮴ

(605.) Let be a fraction, and let the partial quotients

of the equivalent continued fraction be a, b, c, d, ... then

U

the true value of

Approximate values to the fraction

the whole continued fraction, may be found by converting, into a common fraction, the continued fraction carried out to one, two, three, or any number of terms. Let the approximate fraction, found by carrying it inclusively to

A B C A B

a, b, c, d, ... p, q, r, ... be respectively noted by C

C

for

C suggests the following theorem in reference to the formation of these approximate fractions, which are called convergent fractions :

(606.) The terms of any convergent fraction are equal to the product of the corresponding terms of the preceding convergent, by the partial quotient corresponding to the former, together with the corresponding terms of the next preceding convergent.'

For let the theorem be true for then

the two preceding it, it follows that

P Q

R
R" P"Q"

being

r, as is evident by writing down the continued carried out to s; hence

fraction