Approximate Values of Fraction or Ratio. 328. EXAMPLES. 1. Find approximate values of the fraction Ans. 1, 4, 8, and 114. 07 2. Find approximate values of the fraction 8. 169 18564 Ans. 20, 1, 307, 176. 34, and 246 5035 3. Find approximate values of the fraction 3215763. 94218374' Ans. 2, 25, 243, 2875, &c. 4. Find approximate values of the fraction 0-245. Ans. and 1. 5. Find approximate values of the fraction 1.27. Ans. t, t, t, 28, and 17. 6. The lunar month consists of 27-321661 days. Find approximate values for this time. 82 76 Ans. 27, 8, 785, 3907, &c. days, which show that the moon revolves about 3 times in 82 days; or with greater accuracy, 28 times in 765 days; and with still more accuracy, 143 times in 3907 days. 7. The sidereal revolution of Mercury is 87-969255 days. Find approximate values for this time. Ans. 88, 281, &c. 8. The sidereal revolution of Venus is 224-700817 days. Find approximate values for this time. Ans. 225, 7, 1573, 2287, 2828o, &c. 117 9. The ratio of the circumference of a circle to its diameter is 3.1415926535. Find approximate values for this ratio. Ans. 3, 2, 188, fíž, &c. Approximate Roots of Equation. 329. Corollary. The process of art. 324 may be applied to finding the real roots of an equation, the approximate values of which, obtained by this process, can easily be reduced to decimals. in the given equation, we obtain 6x3-9x2-6x-1= 0, = 2,225, 283 2.492, &c. 2. Find the real root of the equation x3 12 x 28= 0. Ans. x 4.30213. 331. Corollary. If the given equation is a binomial one, as in art. 223, we can obtain, by this process, a root of any degree whatever. 1. Extract the square root of 5 by means of continued fractions. Solution. Representing this root by x, we have which, being precisely the same with the equation for x, we may conclude that 4 = a = a' a'' = a""' = &c. Approximate Roots of Equation. and the approximating values are 21, 247, 247, 272, &c.; and the value in decimals is 2.23606. 2. Extract the third root of 46 by means of continued fractions. Ans. 3.58305. 3. Extract the third root of 35 by means of continued fractions. Ans. 3.271. 4. Extract the square root of 2 by means of continued fractions. Ans. 1-4142136. |
