=0.5294117647058823. 1=0.5882352941176470. 1=0.6470588235294117. 1=0-7058823529411764. =0-7647058823529411. =0-8235294117647058. 1=0.8823529411764705. 1=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: 10 100 156/83 197 71 10900473933649 09899097966068 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 이, or ᄒᄒ; this, reduced by Rule under Art. 625 100 17, gives; therefore, 0.0625=. Ans. 67 134 = 이이이이 에이 2. What vulgar fraction is equivalent to the decimal 0.134? 3. What vulgar fraction is equivalent to the decimal 0.00125? 125 Ans. 이이이이이=예아이. 4. What vulgar fraction is equivalent to the decimal 256 16 Ans. 이=이비에 0.0256? 5. What vulgar fraction is equivalent to the decimal 0.06248? 248 Ans. 하이 6. What vulgar fraction is equivalent to the decimal 0.001069? 1069 Ans. 디이이이이이이. Case II. When the decimal is a simple repetend. Since 01, it follows that 0.2 must =, 03, 04, 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, =00i; consequently, 0.07=7, 0.45=1, and so on for other simple repetends of two places of figures. 432 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? 8 ; this, reduced by Rule under Art. 17, becomes Ti 2. What vulgar fraction is equivalent to 0.123? 7. What vulgar fraction is equivalent to the repetend 0123321? Ans. 8. What is the value of 0.999 continued to infinity? Ans. =1. 9. What is the value of 0.987654320? Ans. 654320 [A very simple method of finding a vulgar fraction equivalent to any repe tend, may be found in my ELEMENTS OF ALGEBRA.] Case III. When the decimal is a compound repetend. RULE. I. Find the vulgar fraction which is equivalent to the decimal figures which precede those that circulate, by Rule under Case I. of this article. II. Find the vulgar fraction which is equivalent to the circulating part of the decimal, by Rule under Case II of this article; to the denominator of this fraction annex as many ciphers as there are decimals which precede the circulating part of the repetend; then add these two fractions together. EXAMPLES. 1. What vulgar fraction is equivalent to the compound repetend 0-343? 34 Ans. 이 이름이응응용. = 4. What vulgar fraction is equivalent to the compound repetend 0.03571428? 571428 Ans. T음이 이름이비에 5. What vulgar fraction is equivalent to the compound repetend 0.0714285? Ans. 714285 6. What vulgar fraction is equivalent to the compound repetend 0-123456? Ans. +이이이이이다. 123 456 41111 If we take the last example, which is 0.123456, and multiply it by 1000000, it will become 123456.456. Again, if we multiply 0-123456 by 1000, it will become 123-456. The difference of these two results is 123456-456-123-456=123333. Now, since 123456-456 was 1000000 times the decimal 0.123456, while 123-456 was 1000 times the same decimal, it follows that 123333 is (1000000-1000) times its value; that is, 123333 is 999000 times the value of 0.123456; hence, 0.123456= , the same as already found. A similar process may be employed for changing any repetend into an equivalent vulgar fraction. 123333 41111 |