CHAPTER XXIII. APPROXIMATIONS. 424. What number exceeds its square root by 3? Make two suppositions, 5 and 6. By logarithms, 5— √5 = 2.764, an error of -.236; and 6 – √6= 3.551, an error of +.551. The difference of the assumed numbers is 1, and the difference of the resulting errors is .787. The errors of results are approximately in proportion to the errors of the assumed numbers. Therefore, in assuming 5, the error is to 1 as .236 is to .787; and in assuming 6 the error is to 1 as .551 is to .787. Hence, 5 is nearly .3 too small, and 6 is nearly .7 too large. By logarithms, 5.3 – √5.3 – 2.998, an error of −.002; and 5.4– √5.4 = 3.077, an error of +.077. The difference of the assumed numbers is .1, and the difference of the errors is .079. Therefore, the error of 5.3 is to .1 as .002 is to .079; or, error of 5.3.12: 79. From this proportion the error is found to be .0025. 5.3 +.0025 = 5.3025, and this result is the nearest approximation attainable by four-place logarithms. 425. Hence, in solving questions of this kind, Assume two answers, test each, and note the errors of the results. Calculate the error of either supposition by assuming that it is in the same ratio to the difference of the two suppositions as the error of its result is to the difference of the two results. EXERCISE LXXXIV. 1. What number is 3 less than its square ? Assume 2.3 and 2.4. 2. A flag-staff 50 ft. high broke, and the top falling over rested one end on the stump and the other 17 ft. from its base. How high was the stump? Assume 22 ft. and 23 ft. 3. What number added to eight times its reciprocal is equal to 8? Two answers are required: one between 1 and 2, the other between 6 and 7. 4. Find a number whose reciprocal is equal to 4 miņus the number. Two answers are required: one between 0 and 1, the other between 3 and 4. 5. What number is ten times its own logarithm? 6. What number is double its own cube-root? 7. What number exceeds its cube root by 6? 8. What is the number which added to its own square makes 11? 9. What is the number which multiplied by 10 makes 8 more than the square of the number? 10. A certain number is equal to the sum of its own cube plus its own square. What is the number? 11. What number is equal to its square minus three times its logarithm? Assume 1.1 and 1.2. 12. The sum of the square, and the square root of a number, being divided by 1 plus the number gives a quotient of 21. What is the number? CONTINUED FRACTIONS. 426. As in decimal fractions a result accurate to a given number of places is often required, so in common fractions it is often required to find the most accurate value of a ratio that can be given with denominators limited to a certain size. Ex. Find the most accurate ratios of a circumference to a diameter expressed by a fraction with a denominator under 10; with a denominator under 100; with a denominator under 1000. The ratio 3.1416 is true to the nearest ten-thousandth; and, therefore, it is required to find values of 16 within the prescribed limits. 10000 The first step is to reduce the fraction to its lowest terms. Then, divide the denominator by the numerator; the last divisor by the last remainder; and so on, as in finding the greatest common measure. 177) 1250 (7 11)177 (16 176 1)11 (11 11 If, therefore, both terms of the fraction be divided by the numerator, the result is 1 7147 and if the fraction in the denominator be omitted, the required ratio, with a denominator less than 10, is 34 = 27. and the fraction in the denominator be omitted, the ratio becomes a result which shows that 22 is the nearest ratio expressed by a fraction with a denominator under 100, and that 353 is the nearest ratio expressed by a fraction with a denominator under 1000. 427. After the quotients have been found, the results be written as follows: may 428. The successive approximate values of a continued fraction are found by beginning at the top and taking first one, then two, then three, and so on, of its parts. Thus: 429. In reducing the part of a continued fraction selected for an approximate value, begin with the last fraction. Ex. Find the value of the continued fraction EXERCISE LXXXV. 64 1. Convert, 18, 29, 135 into continued fractions. 2. Find the approximate values of 29; 4; 31. 734 3. Find common fractions approximating to .236; .2361; 1.609. 4. Find common fractions approximating to .382; 1.732; .6253. 613 937 5. Find approximate values of 177; 917; 711; 117. 6. Find the proper fraction that, when reduced to a continued fraction, will have 2, 3, 5, 6, 7 as quotients. 7. Find a series of fractions approximating to the ratio of the pound troy (5760 grs.) to the pound avoirdupois (7000 grs.). 8. Find a series of fractions approximating to the ratio of the side of a square to its diagonal; that ratio being 1:1.414214 nearly. 9. Find a series of fractions approximating to the ratio of the ar to the square chain, from the equality 10. Find a series of fractions approximating to the ratio of the 48-pound shot to the weight of the French shot of 24kg. 11. If the mean diameter of the Earth is reckoned at 7912 mi., and that of Mars 4189 mi., find a series of fractions approximating to the ratio of the mean diameters of these two planets. 12. Find a series of fractions approximating to the ratio of a cubic yard to a cubic meter from the equality 1 cu. yd. .76453 of a cubic meter. 13. Find a series of fractions approximating to the ratio of the kilometer to the mile, from the equality 1m 1.09362 yds. |