CONTINUED FRACTIONS. 268. If we take any fraction in its lowest terms, as 13, and divide both terms by the numerator, we shall obtain a complex fraction, thus: Reducing, the fractional part of the denominator, in the same manner, we have, Expressions in this form are called continued fractions. Hence, 269. A Continued Fraction is a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction whose numerator is also 1, and whose denominator is a similar fraction, and so on. 270. The Terms of a continued fraction are the several simple fractions which form the parts of the continued fraction. Thus, the terms of the continued fraction given above are, , t, and. CASE I. 271. To reduce any fraction to a continued fraction. 1. Reduce 199 to a continued fraction. 339 OPERATION. 109 = 1 339. 3 + 1 9 +1 ANALYSIS. We divide the denominator, 339, by the numerator, 109, and obtain 3 for the denominator of the first term of the continued fraction. Then in the same manner we divide the last divisor, 109, by the remainder, 12, and obtain 9 for the denominator of the second term of the continued fraction. In like manner we obtain 12 for the denominator of the final term. Hence the following RULE. I. Divide the greater term by the less, and the last divisor by the last remainder, and so on, till there is no remainder, II. Write 1 for the numerator of each term of the continued fraction, and the quotients in their order for the respective denom 272. To find the several approximate values of a continued fraction. An Approximate Value of a continued fraction is the simple fraction obtained by reducing one, two, three, or more terms of the continued fraction. 38 163 273. 1. Reduce 3 to a continued fraction, and find its approximate values. ANALYSIS. We take 1, the first term of the continued fraction, for the first approximate value. Reducing the complex fraction formed by the first two terms of the continued fraction, we have 13 for the second approximate value. In like manner, reducing the first three terms, we have for the third approximate value. By examining this last process, we perceive that the third approximate value, , is obtained by multiplying the terms of the preceding approximation, 7, by the denominator of the third term of the continued fraction, 2, and adding the corresponding terms of the first approximate value. Taking advantage of this principle, we multiply the terms of by the 4th denominator, 5, in the continued fraction, and adding the corresponding terms of, obtain, the 4th approximate value, which is the same as the original fraction. Hence the following RULE. I. For the first approximate value, take the first term of the continued fraction. II. For the second approximate value, reduce the complex fraction formed by the first two terms of the continued fraction. III. For each succeeding approximate value, multiply both numerator and denominator of the last preceding approximatien by the next denominator in the continued fraction, and add to the corresponding products respectively the numerator and denominator of the preceding approximation. NOTES.1. When the given fraction is improper, invert it, and reduce this result to a continued fraction; then invert the approximate values obtained therefrom. 2. In a series of approximate values, the 1st, 3d, 5th, etc., are greater than the given fraction; and the 2d, 4th, 6th, etc., are less than the given fraction. 5 39 83 ' 17' 21' 164' 349* 3. What are the first three approximate values of 2831? 20357 21 5 Ans. 4, 36, 131′ 4. What are the first five approximate values of 23? 5. Reduce 33 to the form of a continued fraction, and find the value of each approximating fraction. COMPOUND NUMBERS. 274. A Compound Number is a concrete number expressed in two or more denominations, (10). 275. A Denominate Fraction is a concrete fraction whose integral unit is one of a denomination of some compound number. Thus, of a day is a denominate fraction, the integral unit being one day; so are of a bushel, of a mile, etc., denominate iractions. 276. In simple numbers and decimals the scale is uniform, and the law of increase and decrease is by 10. But in compound numbers the scale of increase and decrease from one denomination to another is varying, as will be seen in the Tables. MEASURES. 277. Measure is that by which extent, dimension, capacity or amount is ascertained, determined according to some fixed standard. NOTE. The process by which the extent, dimension, capacity, or amount is ascertained, is called Measuring; and consists in comparing the thing to be measured with some conventional standard. Measures are of seven kinds: 1. Length. 2. Surface or Area. 3. Solidity or Capacity. 4. Weight, or Force of Gravity. 5. Time. 6. Angles. 7. Money or Value. The first three kinds may be properly divided into two classes— Measures of Extension, and Measures of Capacity. MEASURES OF EXTENSION. 278. Extension has three dimensions-length, breadth, and thickness. A Line has only one dimension-length. A Surface or Area has two dimensions- -length and breadth, A Solid or Body has three dimensions-length, breadth, and thickness. I. LINEAR MEASURE. 279. Linear Measure, also called Long Measure, is used in measuring lines or distances. The unit of linear measure is the yard, and the table is made up of the divisors, (feet and inches,) and the multiples, (rods, furlongs, and miles,) of this unit. SCALE ascending, 12, 3, 53, 40, 8; descending, 8, 40, 51, 3, 12. } The following denominations are also in use: 3 barleycorns make 1 inch, used by shoemakers in measuring the length of the foot. used in measuring the height of horses directly over the fore feet. 1 sacred cubit. 1 pace. 1 fathom, used in measuring depths at sea. 1 geographic mile, { used in measuring dis 1 league. tances at sea. 1 degree of latitude on a meridian or of longitude on the equator. the circumference of the earth. NOTES.-1. For the purpose of measuring cloth and other goods sold by the yard, the yard is divided into halves, fourths, eighths, and sixteenths. The old table of cloth measure is practically obsolete. 2. A span is the distance that can be reached, spanned, or measured between the end of the middle finger and the end of the thumb. Among sailors & spaus are equal to 1. fathom. 3. The geographic mile is of 365 or 100 of the distance round the center of the earth. It is a small fraction more than 1.15 statute miles. |