Transformation of a Quantity to a Continued Fraction. quantity less than the fraction whose numerator is unity, and whose denominator is the square of the denominator of this approximate value. Demonstration. Let the denominator of the two successive approximate values be M' and N'; N' must, by art. 320, be larger than M'; and the difference between these two values must be 1 MN But, by the preceding article, the true value is contained between these two approximate values, and therefore differs from either of them by a quantity less than their difference. Now, since M' < N', we have M'?<M'N', and 1 M.MN M'N so that the true value must differ from the approximate value, whose denominator is M', by a quantity less than 1 that is, less than a fraction whose numerator is 1, and denominator M'2. 324. Problem. To transform any quantity into a continued fraction. Solution. Let X be the quantity to be transformed. Find the greatest integer contained in X, and denote it by A, and denote the excess of X above 1 A by the fraction; and we have From this value of x', find the greatest integer contained in x', and denote it by a, and the excess of x' 1 above a by ; whence 1 ac" x — a from which the greatest integer contained in x" is to be found, and so on; so that we have 1 X=A+ 1 at al +&c. 325. EXAMPLE. Transform it into a continued fraction. A = 2; 1 a = 2 = 263, a 263 Approximate Values of Fraction or Ratio. and the required continued fraction is 1 = 2+ 1 1+ 1 2+ 1 itor 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=a 263 176 87 0 = a 327. Corollary. If a fraction or ratio is transformed into a continued fraction by the preceding process, the approcimate 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 t, are 2, 3, , 4; of which the last differs from the true value by only Tatto Approximate Values of Fraction or Ratio. 328. EXAMPLES. 5035 32 1 57 63 94218374 87 1. Find approximate values of the fraction 44 Ans. $, %4, 47, and 144. 2. Find approximate values of the fraction ITT Ans. zo, tr, 36, 1376: 32639 and 246 3. Find approximate values of the fraction 3 Ans. zy, g'5, 275, 29°3, 2549, &c. 4. Find approximate values of the fraction 0.245. Ans. and Ms. 5. Find approximate values of the fraction 1:27. Ans. $, 1, 11, 13, and 47. 6. The lunar month consists of 27-321661 days. Find approximate values for this time. Ans. 27, 8, 9, 91, &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, 2835, &c. 8. The sidereal revolution of Venus is 224:700817 days. Find approximate values for this time. Ans. 225, 674, 173, 274, 26499, &c. 9. The ratio of the circumference of a circle to its diameter is 3:1415926535. Find approximate values for this ratio. Ans. 3, 4, 188, 11, &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. 330. EXAMPLES. 1. Find the real root of the equation 23 - 32 -8 = 0. Solution. We have, in this case, A= 2, and if we substitute x = 2 + =2+ x = 2 + in the given equation, we obtain 6 x 3-9x2 - 6x-1= 0, whence we have a=2; and the substitution of 1 al gives W3 — 30 212 - 27 x — 6=0; whence we have a' = 32, and so on. The approximate values of x are, therefore, 2, 2} = 2:5, 29% = 2.492, &c. . 2. Find the real root of the equation 23 28 = 0. Ans. x = 4.30213. 12 x |