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CHAPTER XLV.

CONTINUED FRACTIONS.

6 467.

is called a An expression of the form at:

d c+

et.. continued fraction; here the letters a, b, c, any quantities whatever, but for the present we shall only con

1 sider the simpler form a,+

where aj, , Ag

may denote

are

Ag +

Azt..

positive integers. This will be usually written in the more compact form

1 1 a,+

A2+ Ag + 468. When the number of quotients az, Az, az,

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.

469. To convert a given fraction into a continued fraction. Let be the given fraction ; divide m by n, let

a,

be the quotient and p the remainder; thus

m

n

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n

divide n by p, let a, be the quotient and a the remainder; thus

9

1 -A2+ =a,+-; р P P

9 divide p by

9,
let

аз be the quotient and r the remainder; and so Thus

1

1 1
- Q,+

=a,+
-

Ag+ Az+

on.

m

n

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If m is less than n, the first quotient is zero, and we put

1

m

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and proceed as before.

It will be observed that the above process is the same as that of finding the greatest common measure of m and n; hence if m and n are commensurable we shall at length arrive at a stage where the division is exact and the process terminates. Thus every fraction whose numerator and denominator are positive integers can be converted into a terminating continued fraction.

832 Example. Reduce to a continued fraction.

159 Finding the greatest common measure of 832 and 159 by the usual process, thus :

159)832(5

795
37)159(4

148
11)37(3

33
4)11(2

8
3)4(1

3
1)3(3

3

0 We have the successive quotients 5, 4, 3, 2, 1, 3; hence

832

1 1 1 1 1
5+
159 4+ 3+ 2+ 1+ 3

...

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1 is too

470. The fractions obtained by stopping at the first, second, third, quotients of a continued fraction are called the first, second, third, convergents, because, as will be shown in Art. 476, each successive convergent is a nearer approximation to the true value of continued fraction than any of the preceding convergents.

471. To show that the convergents are alternately less and greater than the continued fraction.

1 1 Let the continued fraction be a,+

A2+ az + The first convergent is ay, and is too small because the part 1 1 is omitted. The second convergent is a, +.

1

and is 02+ Az+ too great because the denominator a, is too small. The third

1 1 convergent is a, + and is too small because an+ A2+ Az

аз great; and so on.

When the given fraction is a proper fraction a,=0; if in this case we agree to consider zero as the first convergent, we may enunciate the above results as follows:

The convergents of an odd order are all less, and the convergents of an even order are all greater, than the continued fraction.

472. To establish the law of formation of the successive convergents. Let the continued fraction be denoted by

1 1 1

.;

12+ Agt 24+ then the first three convergents are

a, azaz+1, az(QzQ2+1)+1;
1
а.

ag .ag+1 and we see that the numerator of the third convergent may be formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent; also that the denominator may be formed in a similar manner.

Suppose that the successive convergents are formed, in a similar way; let the numerators be denoted by P1, P2, P3, and the denominators by 91, 92, 939 -

Assume that the law of formation holds for the nth convergent; that is, suppose

Pn=anPn-1+Pn-2, In=anqn-1+ In–2.

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The (n+1)th convergent differs from the nth only in having

1 the quotient an+ in the place of an; hence the (n+1)th con

On+1 vergent

1 an+1

an+1(an Pr-1+Pn-2) +Pn-1

An+1(an In-1 + 2n-2) + In-1
In-1 + In-2

ant

-) Pn+1+Pn-2

ant

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Anti Pn+Pn-1, by supposition.

An+19n+qn-1
If therefore we put

Pn+1=An+1 Pn+Pn-1, 9n+1=(n+19n+qn-1, we see that the numerator and denominator of the (n+1)th convergent follow the law which was supposed to hold in the case of the nth. But the law does hold in the case of the third convergent, hence it holds for the fourth, and so on; therefore it holds universally.

674 Example. Reduce to a continued fraction and calculate

313 the successive convergents.

674

1 1 1 By Art. 469, =2+

1 1 313 6+ 1+ 1+ 11 + 2 The successive quotients are 2, 6, 1, 1, 11, 2.

2 13 15 28 323 674 The successive convergents are

1' 6'7' 13' 150' 313 [Explanation. With the first and second quotients take the first and second convergents which are readily determined. Thus, in this example, 2 is the first convergent, and 2+

1

6 13

the second convergent. The numerator of the third con6 vergent, 15, equals the numerator of the preceding convergent, 13, multiplied by 1, the third quotient, plus 2, the numerator of the convergent next preceding but one. The denominator is formed in a similar manner: thus 7=1x6+1.

11(28) +15 323 The fifth convergent

.]

11(13)+7 150 473. If the fraction is a proper fraction, we may consider zero as the first convergent, and proceed as follows:

or

84 Reduce to a continued fraction, and calculate the suc

227 cessive convergents. Proceeding as in Art. 469,

227) 8400

00
84)227(2

168
5984(1

59

25)59(...
1 1 1 1 1 1 1
0+

2 + 1+ 2+ 2+ 1+ 3+ 2

We obtain

The successive quotients are 0, 2, 1, 2, 2, 1, 3, 2.
Writing for the first convergent we have, [Art. 472],

0
1

0 1 1 3 7 10 37 84
1' 2 3 8 19 27 100 227

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474. It will be convenient to call an the nth partial quotient;

1 1 the complete quotient at this stage being an+

Antit An+2+ We shall usually denote the complete quotient at any stage by k. We have seen that

Pn_anPn-i+Pn-2

In Anqn-1 +9n-2 Let the continued fraction be denoted by x; then x differs from Pn only in taking the complete quotient k instead of the partial an quotient an; thus

kpn-1+Pn-2.

kqn-1 +4n-2 475. If Pn be the nth convergent to a continued fraction, then an

Pn qn-1-Pn--19.=(-1)".

X=

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