and thus continue, at each successive figure of the quotient, to omit the right-hand figure of the divisor, until there is but one figure in the remainder. NOTE.-If we regard the dividend as the numerator of a fraction whose denominator is the divisor, the quotient will be the value of such fraction. Annexing a cipher to the numerator of a fraction is equivalent to multiplying its value by 10, and omitting the righthand figure of the denominator is also equivalent to multiplying the value of the fraction by 10. Hence, in the operation of division of decimals, instead of annexing a cipher to the dividend, as in the ordinary rules, we may, instead thereof, omit the right-hand figure of the divisor, as in the foregoing rule. EXAMPLES. 1. What is the quotient of 365 424907 divided by 0.263803? Operation. 0.263803)365 424907(1385.21892 263 S03 101 6219 79 1409 22 48100 21 10424 1 376767 1319015 57752 52761 4991 2638 2353 2110 243 237 6 5 1 2. What is the quotient of 0123456 divided by 1.912478? Operation. 1.912478)0 123456(0.064552 114748 8708 7650 1058 956 102 96 6 4 2 3. What is the quotient of 0'52600000 divided by 0.5260202? 5. What is the quotient of 7632038 divided by 3.716048? Ans. 2-053805. 6. What is the quotient of 2 divided by 15.314865? Ans. 0 13059207. 7. What is the quotient of 0.926954 divided by 0.3547898? Ans. 2612685. 8. What is the quotient of 13-75892 divided by 6.76897 ? Ans. 203264. 42. To change a vulgar fraction into an equivalent decimal fraction. It is obvious that the rule under Art. 33 will apply to this case by considering all the denominate values as decreasing regularly in a ten-fold ratio. Hence, this RULE. Annex a cipher to the numerator, and then divide by the denominator; to the remainder annex another cipher, and again divide by the denominator, and so continue, until there is no remainder, or until we have obtained as many decimal figures as may be desired. The successive quotients will be the successive decimal figures required. EXAMPLES. 1. What decimal fraction is equivalent to? Operation. 96 40 32 80 810 2. What decimal is equivalent to? Ans. 005555, &c. Ans. 004. 3. What decimal is equivalent to? Ans. 0.05. 4. What decimal is equivalent to? 5. What decimal is equivalent to ? Ans. 0.3333, &c. 6. What decimal is equivalent to T? 7. What decimal is equivalent to? 8. What decimal is equivalent to ?? Ans. 0.048. Ans. 0-875. Ans. 0.75. Since, in the above process of decimating a vulgar fraction, each successive dividend terminates with a zero, it follows that the right-hand figure of the remainder may be found by multiplying the right-hand figure of the denominator of the vulgar fraction by the quotient figure, and subtracting the right-hand figure of the product from 10; or, which is the same thing, if we subtract the righthand figure of the denominator from 10, and multiply the remainder by any decimal figure, the right-hand figure of the product will be the same as the right-hand figure of the remainder. 43. It will often happen, as in examples 2 and 5, of the last article, that the process will never terminate, in which case there is no decimal value which is accurately equal to the vulgar fraction. Since we constantly multiplied the remainders by 10, it follows that whenever the denominator of the vulgar fraction contains no prime factors different from those which compose 10, viz., 2 and 5, then the decimal value will terminate. But, in all other cases, the decimal expression must consist of an infinite number of figures. Hence, to determine whether a given vulgar fraction can be accurately expressed in decimals, we have this RULE. Decompose the denominator of the vulgar fraction, when reduced to its lowest terms, into its prime factors, (by Rule under Art.7,) then, if there are no prime factors different from 2 and 5, the vulgar fraction can be accurately expressed by decimals; but if it contain dif ferent factors, it cannot be expressed in decimals. EXAMPLES. 3 1. Can the vulgar fraction be accurately expressed in decimals? In this example, we find that 386=2×193; so that the denominator contains the prime factor, 193, which is different from 2 or 5; consequently, accurately expressed in decimals. cannot be 2. Can the vulgar fraction be accurately expressed in decimals? Ans. It cannot. 3. Can the vulgar fractions, having for denominators 640, be expressed in decimals accurately? Ans. They can. 44. When a vulgar fraction can be accurately expressed in decimals, we may determine the number of decimal places by the following |
