I. The sum of any two diametrically opposite figures of the circle of decimals, will be 9. II. The sum of any two diametrically opposite terms in the circle of remainders, will make the denominator 29. III. If we subtract the right-hand figure of the denominator from 10, and multiply the remainder by any decimal figure of the inner circle, the right-hand figure of the product will be the same as the right-hand figure of the corresponding remainder of the outer circle. IV. Commencing the circle of decimals at any point, and counting completely round, it will be the perfect repetend of the vulgar fraction, whose denominator is the same as in the first case, but whose numerator is the remainder in the outer circle, standing one place to the left. V. If we divide the product of any two remainders by 29, what remains will be the remainder in the outer circle, corresponding with the place denoted by the sum of the places of the two numbers. From the IVth property, it follows that this same circle of decimals expresses the decimal value of all proper vulgar fractions, whose denominators are 29. The following figure, formed from the perfect repetend of the value of, possesses similar properties to those just explained. Similar circles may be formed for all perfect repe tends. =0·5294117647058823. 1=0'5882352941176470. =0'6470588235294117. 1=0'7058823529411764. 1=0.7647058823529411. +4=0-8235294117647058. =0'8823529411764705. +4=0·9411764705882352. We will arrange the complementary repetend arising from the vulgar fraction, in the form of a circle, as was done for perfect repetends, as follows: 150 23 19 190 1 10 100 156 83 197 260663507 55 128 14 140 134 74 107 15 It will be seen that a complementary repetend possesses all the properties ascribed to the perfect repetend, as given under Art. 52, except the IV. 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? 0625 100009 or 1825; this, reduced by Rule under Art. 17, gives; therefore, 0.0625=16. 34 67 500 2. What vulgar fraction is equivalent to the decimal 0.134 ? Ans. 10. 3. What vulgar fraction is equivalent to the decimal 0.00125? 125 Ans. To=500• 4. What vulgar fraction is equivalent to the decimal 0.0256 ? Ans. 256 10000 -- 16 5. What vulgar fraction is equivalent to the decimal 0.06248? 6. What vulgar fraction is equivalent to the decimal 0.001069? Ans. 1069 Case II. When the decimal is a simple repetend. Since 01, it follows that 0.2 must, 0.3=}, 0·4=¦, and so on; therefore, a simple repetend of one figure is equivalent to the vulgar fraction whose numerator is this figure, and whose denominator is 9. Again,=0·01; consequently, 0·07=77, 0·45=4§, and so on for other simple repetends of two places of figures. In a similar manner, we infer that 0.432 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 T 2. What vulgar fraction is equivalent to 0-123? 4 1 Ans. 333. 3. What vulgar fraction is equivalent to the repetend 0.027? Ans. 2=37. 4. What vulgar fraction is equivalent to the repetend 0.142857 ? Ans. 142857 14. 999999 |