326. Corollary. The values of a', a", &c., in the case of a vulgar fraction, are evidently the quotients which would be obtained by the process of finding the greatest common divisor of the numerator and denominator of x'. The preceding process might therefore be performed as follows: 263 351|1 =α 263 88 263 2 87|88|1=a" 1|87|87a"" 87 0 327. Corollary. If a fraction or ratio is transformed into a continued fraction by the preceding process, the approximate values of this continued fraction are also approximate values of the given fraction or ratio, which are often of great practical use. Thus the approximate values of 8, are 2, 3,, ; of which the last differs from the true value by only T Approximate Values of Fraction or Ratio. 328. EXAMPLES. 1. Find approximate values of the fraction Ans. 1, 2, 48, and 114. 2. Find approximate values of the fraction. Ans. 20, 11, 307, 1376. 3150 3459, and 246 5035 3. Find approximate values of the fraction 3215763. 94218374 Ans. 29, 8%, 275, 293, 2879, &c. 4. Find approximate values of the fraction 0.245. Ans. 1 and 13. 5. Find approximate values of the fraction 1.27. Ans. 4, 1, 14, 28, and 47. 6. The lunar month consists of 27-321661 days. Find approximate values for this time. 28, Ans. 27, 8, 765, 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, 215, &c. 8. The sidereal revolution of Venus is 224.700817 days. Find approximate values for this time. Ans. 225, 64, 1573, 2247, 26299, &c. 9. The ratio of the circumference of a circle to its diameter is 3.1415926535. Find approximate values for this ratio. Ans. 3, 27, 188, 115, &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. The approximate values of x are, therefore, 2,21 = 2.5, 282-492, &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. 332. EXAMPLES. 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. |