is called a continued fraction, though the term is commonly restricted to a continued fraction that has 1 for each of its numerators, as

We shall consider in this chapter some of the elementary properties of such fractions.

398. Any proper fraction in its lowest terms may be converted into a terminated continued fraction.

b

Let be such a fraction; then, if p is the quotient and

if q is the quotient and d the remainder of b÷c,

The successive steps of the process are the same as the steps for finding the H. C. F. of a and b; and since a and b are prime to each other, a remainder, 1, will at length be reached, and the fraction terminates.

Observe that p, q, r, ....., are all positive integers.

399. Convergents. The fractions formed by taking one, two, three, of the quotients p, q, r, .............,

are

and are called the first, second, and third convergents, respectively.

400. The successive convergents are alternately greater than and less than the true value of the given fraction.

401. If

are any three consecutive convergents, V1 V2 V3

and if m1, m2, m, be the quotients that produced them, then

For, if the first three quotients are p, q, r, the first three convergents are (§ 399),

From (399) it is seen that the second convergent is 1 formed from the first by writing in it p+ for p; and the

1

r

third from the second by writing 9+ for q. In this way, any convergent may be formed from the preceding convergent.

In (1) it is seen that the third convergent has its numerator=rX (second numerator) + (first numerator); and its denominator = X (second denominator) + (first denomi

nator).

Assume that this law holds true for the third of the three consecutive convergents

Substitute u, and v, for their values mu1+u。 and mvivo; then

Therefore the law still holds true; and as it has been shown to be true for the third convergent, the law is general.

Let the sign ~ mean the difference between, and assume

(substituting for u, and v2 their values, mu1+u。 and

Hence, if the proposition be true for one pair of consecutive convergents, it will be true for the next pair; but it has been shown to be true for the first pair, therefore it is true for every pair.

Since by § 400 the true value of x lies between two con

И1

Any convergent,, is in its lowest terms; for, if 1⁄4, and

V1

v1 had any common factor, it would also be a factor of u12~uv1; that is, a factor of 1.