CONTINUED FRACTIONS.

§ 334. A continued fraction is one whose numerator is unity, and denominator the sum of an integer and a fraction whose numerator is unity, and so on, as before.

The following is the usual method of expressing a continued fraction:

1

2+1

3+1

5+%.

The partial fractions §, §, &c.,—each succeeding one of which is to be added to the denominator of the preceding one,-may be called the terms of the continued fraction.

A continued fraction may also be expressed by placing its terms one directly after another, with the sign+between the denominators; thus

1 1 1 1
2+3+5+6

§ 335. Continued fractions are employed to find, in lower terms, successive approximations to the value of a fraction or ratio whose terms are large, and prime to each other.

For example, suppose we wish to find, in lower terms, approximate values of 111, whose exact value cannot be expressed in lower terms.

Dividing both the given terms by the less, we find

131 1

418 33

Disregarding the fraction in the denominator, we have

the first approximate value of the given fraction.

for

This first approximation is greater than the true value, because the denominator 3 is less than the true denominator 335. 25 But since this denominator is between 3 and 4, the true value is between and 4.

To find the second approximate value, we reduce the in the same manner as the given fraction; thus

Disregarding the last fraction in the denominator, we have

1 1

3+5=1+3=18, for the second approximation.

This second approximation is less than the true value, because, by rejecting the, the term becomes greater than the true value to be added to the denominator 3.

To find the third approximate value, we reduce the in the same manner as before; thus

Then

6 1 25 4

131 1 1 1

4183+5+41

Disregarding the last fraction in the denominator, we have 1 1 1 1

3+3+4=3+1÷2184, for the third approximation.

This third approximation is greater than the true value; because by rejecting the the becomes too much to add to the denominator 5; and thence results too small a value to add to the denominator 3.

We have thus obtained the three approximating fractions,, , and 24, which are alternately greater and less than the given fraction. If the remaining partial fraction be included in the valuation, the result will be the given fraction.

From the preceding example we may derive the following principles in relation to

Approximating Fractions.

§ 336. Any given proper fraction may be reduced to a con tinued fraction by dividing both its terms by the numerator,proceeding in like manner with the fraction formed of the re mainder and divisor,—and`so on; and connecting each succeed ing partial fraction, by the sign +, to the preceding denominator Then,

1. The first partial fraction will be the first approximation to the value of the given fraction.

Thus in the preceding example (§ 335) is the first approximation to the value of 131.

2. The second approximation will have for its numerator the denominator of the second partial fraction; and for its denominator the product of the denominators of the first and second partial fractions, plus 1.

Thus in, in the preceding example, the numerator is the second partial denominator; and the denominator 16 was found by multiplying 5 into 3, and adding 1.

3. The two terms of each succeeding approximation, will be found by multiplying the corresponding partial denominator into the two terms, respectively, of the preceding approximation, and adding the terms of the next preceding one.

Thus in the third approximation, 4, the numerator 21 was found by multiplying 4 into 5, and adding 1; and the denominator 67 by multiplying 3 into 21, and adding 4, which is equivalent to 4x16+3.

If we proceed in like manner to find the fourth approximation, we have 6×21+5=131, for the numerator; and 6×67+16=418, for the denominator; that is, we reproduce the given fraction 13.

We may also conclude from the same example, that

§ 337. Every odd aproximating fraction, as the first, third, and so on, will be greater than the given fraction; while every even one will be less than the given fraction.

Thus we found to be greater, to be less, and to be greater, than 1.

131

The value, therefore, of the given fraction lies between any two consecutive approximating fractions. But it is desirable to ascertain more definitely the accuracy of each successive approximation.

§ 338. Any one of the approximating fractions differs in value from the given fraction by less than a unit divided by the denominator of that approximation multiplied into the denominator of the next approximation.

To prove this we first observe that if two consecutive approximating fractions be reduced to a common denominator, the resulting numerators will always differ by a unit.

270

Thus taking the and 21, found in the example before given, and reducing them to a common denominator, the numerators will become

5X67 and 21X16.

Recollecting the composition of the 67 and 21, according to § 336, we have 5X67=5X(4X16+3)=5×4×16+5×3;

and 21×16=(4×5+1)×16=4×5×16+1×16.

墨

From the manner in which these numerators are formed, as just shown. they necessarily differ from each other by a unit; and this method of illustration will apply to all like cases.

1 16X67°

The difference between the and 31 will then be But we have already seen that the value of the given fraction lies between and 4, (§ 335,) and therefore differs from either of these less than they differ from each other.

Hence differs in value from 13 by less than 1÷(16×67).

418

From the preceding principle it also follows, that

339. Any particular approximation differs in value from the given fraction by less than a unit divided, by the square of the denominator of that approximation.

418

་

Thus differs in value from 131 by less than 1÷162, since 1-163 is less than 1÷(16×67).

The principles of § 336 may now be applied to the following

To obtain approximate values of a decimal fraction, the denominator of the fraction must be supplied.

Thus for .83 we would take 83
100.

Ans. 231

Ans. 21, 21, 2

5. Find approximate values of .329. 6. Find approximate values of 2.17. 7. Find three approximate values of ratio of the circumference to the diameter of a circle.

3.14159', which is the

Ans. 34, 3, 311

ARITHMETICAL PROGRESSION.

§ 340. An ARITHMETICAL PROGRESSION is a series of quantities which continually increase or decrease by a common difference.

Thus 1, 3, 5, 7, 9, is a progression which increases by the continual addition of the common difference 2;

And 15, 12, 9, 6, 3, is a progression which decreases by the continual subtraction of the common difference 3.

The first and last terms of a progression are called the two extremes, and all the intermediate terms the means.

The following are the most useful principles relating to Arithmetical Progression.

The Last Term.

341. The last term, of an increasing arithmetical progression, is equal to the first term plus the product of the common difference multiplied into the number of terms less one.

This is evident from considering that the 2d term is formed by adding the common difference to the 1st term, the 3d by adding the common difference to the 2d, and so on,-the number of these additions being always one less than the number of terms in the progression.

When the progression decreases, each succeeding term will be found by subtracting the common difference from the preceding one. Hence,

§ 342. In a decreasing arithmetical progression the last term is equal to the first term minus the product of the common difference multiplied into the number of terms less one.

Or, more generally,

§ 343. The greater of the two extremes of an arithmetical progression, is equal to the less plus the product of the common difference multiplied into the number of terms less one; or the less extreme is equal to the greater minus the same product.