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CONTINUED FRACTIONS.

307. A CONTINUED FRACTion is a fraction having for its numerator 1, and for its denominator a whole number plus a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction, and so on. Thus,

1
3+1
2+1
5+1

4+ , is a continued fraction. The partial fractions composing the parts of a continued fraction are called its terms. Thus, in the fraction given above, }, 3, , &c. are its terms.

308. Continued fractions are used in obtaining, in smaller numbers, the approximate values of fractions whose terms, when reduced to their simplest forms, are expressed in numbers inconveniently large.

309.° To transform a common fraction into a continued fraction, and to find, in smaller numbers, its approximate values.

Ex. 1. Transform : into a continued fraction, and find its several approximate values.

OPERATION.

19) 60(3

19 1 57

Hence,

60 3 + 1
3)1966

6+$
18
1

1
1)3(3

1st approx. value,
3
3'

Ans. 3 1 x 6

6

2d approx. value,
(3 X 6) +1 19'
(6 X 3) +1 19

the original value, (19 X 3) + 3 60' Dividing both terms of : by the numerator, which operation will not change the value expressed (Art. 217), the fraction becomes 1

; the denominator of which being between 3 and 4, the value of 31%

=

}, we have

60

the given fraction must be between 1 and 1; and neglecting the fraction 85, for the present, in the denominator, we have } for the first approximate value. This approximation, however, is greater than the true value, since the denominator, 3, is less than the true denominator 316. We therefore divide both terms of the if by its nu

1 merator, and it becomes which is between į and ļ. By neglecting

63' the fraction } in the denominator, and taking the į instead of the

1

1 186, we have 3+

en for the second approximate value of

31
the given fraction; which approximation is too small, since in the
denominator, instead of 18, we used }, which is greater than the ff.
If we now include in the calculation the remaining partial fraction

1
1

, the original fraction.
34 +1 3.18
By the processes of the operation it will be seen that the first ap-
proximate value sought was obtained by disregarding all the partial
fractions after the first, the second approximate value by disregard-
ing all the partial fractions after the second, &c.

RULE. Divide the greater term of the given fraction by the less, and the divisor by the remainder, and so on, as in finding the greatest common divisor. The quotients thus found will be the denominators of the several terms of the continued fraction, and the numerator of each will be 1.

For the first approximation, take the first terms of the continued fraction.

For the SECOND pproximation, multiply the terms of the first approximate fraction by the denominator of the second term of the continued fraction, adding 1 to the product of the denominators.

For each suCCEEDING approximation, multiply the terms of the approximation last found by the denominator of the next term of the continued fraction, and add the corresponding terms of the preceding approximation.

NOTE 1. - When the fraction given is improper, the true approximations will be the reciprocals of the fractions found by the rule.

NOTE 2. In a series of approximations the first is larger, the second smaller, and so on, every odd fraction being larger, and every even one smaller, than the given fraction. Each successive approximate fraction, however, approaches more nearly than the one preceding it to the value of the given fraction. When the continued fraction indicates many approximations, it is generally sufficient for ordinary purposes to find only from three to six of them.

NOTE 3. — A continued fraction may, for convenience, be expressed by writing its terms one directly after another, with the sign plus (+) between the denominators; thus, the continued fraction equivalent to lë may be written 3+ + 5.

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EXAMPLES

2. Transform 1$ into a continued fraction.

Ans. 1
2+1

3+ #. 3. Transform 21 into a continued fraction. 4. Find the approximate values of ?. Ans. J, , , , 4 5. Find the first five approximate values of 1331 6. Find the first three approximate values of $34.

Ans. f, }, }, or 2, 21, 23.

1 7. Find the first six approximate values of

Ans. 13, 14, 21, 55, 2, 3 8. What are the first four approximate values of 1.27 ?

Ans. }, }, {, tt, 28, or 1, 13, 15, 111, 136.

1336

44

RATIO.

310. Ratio is the relation, in respect to magnitude or value, which one quantity or number bears to another of the

same kind.

311. The comparison by ratio is made by considering how often one number contains, or is contained in, another. Thus, the ratio of 10 to 5 is expressed by 2, the quotient arising from the division of the first number by the second, or it may be expressed by so 1, the quotient arising from the division of the second number by the first, as the second or the first number shall be regarded as the unit or standard of comparison. In general, of the two met ds, the first is regarded as the more simple and philosophical, and therefore has the preference in this work.

NOTE. Which of the two methods is to be preferred, is not a question of so much importance as has been by some supposed, since the connection in which ratio is used is usually such as to readily determine its interpretation. 312. The two numbers necessary

form a ratio are called

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the terms of the ratio. The first term is called the antecedent, and the last, the consequent. The two terms taken together are called a couplet ; and the quotient of the two terms, the index or exponent of the ratio.

313. The ratio of one number to another may be expressed either by two dots (:) between the terms; or in the form of a fraction, by making the antecedent the numerator and the consequent the denominator. Thus, the ratio 6 miles to 2 miles may be expressed as 6:2, or as s.

314. The terms of a ratio must be of the same kind, or such as may be reduced to the same denomination. Thus, cents have a ratio to cents, and cents to dollars, &c. ; but cents have not a ratio to yards, nor yards to gallons.

315. A simple ratio is that of two whole numbers; as, 3 : 4, 8:16, 9:36, &c.

A complex ratio is that of two numbers, of which one

316.

or both are fractional; as, 6 : 4},5 1,44:2}, &c.

ex

5 317. A compound ratio is the product of two or more ratios. Thus, the ratio compounded of 4 : 2 and 6:3 is X

4, or 4 X 6:2 X 3 = 4. A compound ratio is generally expressed by writing the ra. tios composing it, in a column, with the antecedents in one

4:2 vertical line, and the consequents in another; thus,

6:3 presses a compound ratio.

NOTE. — If a ratio be compounded of two equal ratios, it is called a duplicate ratio; of three ratios, a triplicate ratio, &c.

318. A ratio is either direct or inverse. A direct ratio is the quotient of the antecedent by the consequent; an inverse ratio, or reciprocal ratio, as it is sometimes called, is the quotient of the consequent by the antecedent, or the reciprocal of the direct ratio. Thus the direct ratio of 6 to 2 is or 3; and the inverse or reciprocal ratio of 6 to 2 is or }, which is the same as the reciprocal of 3, the direct ratio of 6 to 2.

Note 1. – - One quantity is said to vary directly as another, when both increase or decrease together in the same ratio; one quantity is said to vary irer

versely as another, when the one increases in the same ratio as the other de

creases.

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NOTE 2. The word ratio, when used alone, means the direct ratio.

319. When the antecedent and consequent of a ratio are equal, the ratio equals 1, and is called that of equality. Thus, the ratio of 6:6 = 6 1, and the ratio of 6 X 4:8 X 3

= t = 1, are ratios of equality. But if the antecedent is larger than the consequent, the ratio is that of greater inequality, and if the antecedent is smaller than the consequent, the ratio is that of less inequality. Thus, the ratio of 15:5 = = 3, is a ratio of greater inequality; and the ratio of 7:14 = 1 = }, is a ratio of less inequality.

320. The ratio of two fractions having a common numerator is the same as the inverse ratio of their denominators. Thus, the ratio of i: 3 is j = 2, which is the inverse ratio of the denominator 4 to the denominator 8.

321. The ratio of two fractions having a common denominator is the same as the ratio of their numerators. Thus, the ratio of : 4 is : 2, which is the ratio of the numerator 6 to the numerator 3.

322. The inverse or reciprocal ratio of two numbers denotes what part or multiple the consequent is of the antecedent. Thus, inquiring what part of 4 is 3, or what part 3 is of 4, is the same as inquiring the inverse or reciprocal ratio of 4:3. The inverse ratio of 4: 3 is , and 3 is of 4.

323. In order to compare one number with another, by ratio, it is necessary that they should not only be of the same kind, but of the same denomination. Thus, to compare 2 days with 12 hours, it is necessary that the days be reduced to hours, before we can indicate the ratio, which is 48 hours : 12 hours.

324. If the antecedent of a ratio be multiplied, or the consequent divided, the ratio is multiplied. Thus, the ratio of 6:3 is 2, but 6 X 2:3 is 4; or 6:3 • 2 is 4.

325. If the antecedent of a ratio be divided, or the consequent multiplied, the ratio is divided. Thus, the ratio of 18:6 is 3, but 18 • 3:6 is 1; or 18 : 6 X 3

= 1.

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