NOTE. The I., II., and III. of the above properties of perfect repe. tends have never been noticed by any author. I have given an algebraic demonstration of these properties, as well as several others, which was published in the American Journal of Science, Vol. 40, No. 81. 52. When many figures in the decimal are required, we may proceed as follows: Required the decimal value of. Operating by the rule under Art. 42, page 58, we get OPERATION. 29)100(0.03448 87 130 116 140 116 240 232 8 We have continued this process until we have found a remainder consisting of but one figure; placing this remainder, when divided by 29, at the right of the quotient, agreeably to the usual rules of division, we get, 29 I. =0.03448. Multiplying this by 8 we get = 0.27586. Substituting this value of in L, we get, II. 0.0344827586; this, multiplied by 6, gives -0.2068965517; which substituted in II. gives, III.=0.034482758620689655177. Again, multiplying 0.241379310344827586208. Substi. this by 7, we get tuting this in III. we get, IV.=0.0344827586206896551724137931034482758 62020. We shall gain nothing by continuing this process farther, since (by Art. 45) we know that the decimal figures can not extend beyond 28 places without repeating; in this case the number of places in the repetend is exactly 28: it is therefore a perfect repetend. 53. There is another way of decimating, which is as follows: Decimate. According to rule under Art. 42, we find 97)100(0.01 97 3 To continue the process we must add ciphers to this remainder, in the same way as we did to the numerator, 1. Now the remainder being 3 times as large as the first numerator, it follows that the next two decimal figures must be 3 times the two just obtained, that is 3×01-03; and for a similar reason we must multiply 03 by 3 to obtain the next two figures, and so on. Proceeding in this way we find 0.0103092781 243 729 2187 6561 &c. -0.010309278350515261 &c. Decimating by the above plan we get Decimating by this method we get =0.1248 16 32 64 128 256 &c. 0.1249999 &c. which will con stantly approximate towards 0.125. 54. To change a decimal fraction into an equivalent vulgar fraction. CASE I. When the number of places is finite, we can, from the definition of decimal fractions, Art, 34, deduce this RULE. Make the given decimal the numerator of the vulgar fraction, and for its denominator write 1, with as many ciphers annexed as there are decimal places. Examples. 1. What vulgar fraction is equivalent to the decimal 0.0625? 1 or 8; this reduced by rule under Art. 16, gives 0625 10000 625 10000 ; therefore 0.0625= 2. What vulgar fraction is equivalent to the decimal 0.134 ? 500 3. What vulgar fraction is equivalent to the decimal 0.00125? Ans. 67 Ans. Ans. 4. What vulgar fraction is equivalent to the decimal 0.0256 ? 5. What vulgar fraction is equivalent to the decimal 0.06248? 6. What vulgar fraction is equivalent to the decimal 0.001069? CASE II. When the decimal is a simple repetend. Since 0.1, it follows that 0.2 must=3, 0.3—3, 0.4— 4, and so on; therefore a simple repetend of one term, is equiv alent to the vulgar fraction whose numerator is this term, and whose denominator is 9. Again, '=0.01, consequently 0.07, 0.45—15, and so on for other simple repetends of two places of figures. In a similar manner we infer that 0.432–133. Therefore, we have the following RULE. Make the repetend the numerator, and for the denominator write as many nines as there are places of decimals. Examples. 1. What vulgar fraction is equivalent to 0.72? ; this, reduced by rule under Art. 17, becomes 2. What vulgar fraction is equivalent to 0.123? Ans. 3. What vulgar fraction is equivalent to the repetend 0.027 ? Ans. 4. What vulgar fraction is equivalent to the repetend 0.142857? Ans. 4. 5. What vulgar fraction is equivalent to 0.012345679? |