7. What vulgar fraction is equivalent to the repetend 0.123321 ? Ans. 12331. 999999 8. What is the value of 0.999 continued to infinity? 9. What is the value of 0.987654320 ? Ans. 1. Ans. 287654320 [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? Ans. +6=308=383. 2. What vulgar fraction is equivalent to the compound repetend 0.08783? Ans. +80=147. 8 100 13 3. What vulgar fraction is equivalent to 0.083? Ans. Toto='4. 4. What vulgar fraction is equivalent to the compound Ans. 99990 repetend 0.03571428? 571428 = 5. What vulgar fraction is equivalent to the compound repetend 0.0714285 ? Ans. 714285 9999990-14. 6. What vulgar fraction is equivalent to the compound repetend 0.123456? 123 1000 456 4 1 1 1 1 Ans. +0000=333000. 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= 123=2, the same as already found. A similar 123333 41111 333000, process may be employed for changing any repetend into an equivalent vulgar fraction. CHAPTER IV. CONTINUED FRACTIONS. 55. Ir we divide both numerator and denominator of the fraction 351 by the numerator, we obtain, 87; this value substituted in III., we finally obtain, In the last example, the parts, }, 1, &c., are called the first, second, third, &c., partial fractions. It has been proposed, by some authors, to write continued fractions in the following way, which is more. compact: Thus, the preceding fractions may be written, If we seek for the greatest common measure of the numerator and denominator of the first fraction 35, by Rule under Art. 9, we shall have the following Operation. 351)965(2 263)351(1 263 88)263(2 176 87)88(1 87 1)87(87 87 0 Here we discover that the successive quotients are the same as the successive denominators of the partial fractions which compose the continued fraction already drawn from 35. Hence, to convert a vulgar fraction into a continued fraction, we have this RULE. Seek, by Rule under Art. 9, the greatest common measure of the numerator and denominator of the given fraction; the reciprocals of the successive quotients will form the partial fractions which constitute the continued fraction required. EXAMPLES. 1. Convert into a continued fraction. |