LESSONS IN A l: HUM OTIC. 



247 



and therefore by 



10 X 10 X 10, or 1000. 



by 5 x 5 x 5, we make it 



Hence } = -j- 



3 x 5 x 5x5 

 10 x 10 x 10 



= = '375. 



17. We Bee from the preceding remarks tho truth of the 

 following 



; wltetiur a yiiwi Vulgar Fraction will 

 l\rnrinating Dri-inml. 



I:. -i luce tho given fraction to its lowest terms, and eplit tho 

 denominator into its prime I'. 



It' the denominator h factors 2' or 5's, or both, 



unit no other factors, tho fraction will give a terminating 

 decimal, but not otherwise. 



18. 1 tins case tli* Decimal witlxnit actually 

 "K7- 



It c.i 10 of the factors 2 and 5 occur fewer times than the 



otluT, multiply numerator and denominator of the fraction by 



that power of the factor which occurs the fewest times in the 



denominator, which will make the number of times it occurs 



; 1 to tho number of times the other occurs. 



Thus, in the instance already given, 250 is made up of three 

 5's and one 2 as factors. We therefore multiply numerator and 

 denominator by the second power of 2. 



Similarly, in j, 8 being the third power of 2, we multiply 

 numerator and denominator by the third power of 5. 



06s. It will be perceived that the number of decimal places 

 in the terminating decimal which is equivalent to a vulgar frac- 

 tion, will be tho samo as the greatest number of times that either 

 of the factors 2 or 5 is repeated in its denominator, when the 

 fraction is reduced to its lowest terms. 



EXAMPLE. Determine the decimal which is equal to ffgj. 



JU*> *)&2 



Reduced to its lowest terms this is -j^r or jr. Multiplying 

 numerator and denominator by 2 3 , or 8, the fraction becomes 

 Sgj, or 3-056. 



19. Circulating or Recurring Decimals. 



Decimals in which the same series of figures is repeated in- 

 definitely, are called circulating or recurring decimals ; and the 

 series of figures thus repeated is called the period. 



Thus, 3-21737373, etc., . . . ., where 73 is continually repeated 

 "'' i i ijiii Hum, is a circulating decimal. 



Similarly, -01342342342 . . . ., and '6666 . . ., are recurring 

 decimals. 



A recurring decimal is indicated by writing a dot over each 

 figure of the period, or, sometimes, where the period is long, by 

 writing a dot over the first and last figures only of the period. 

 Thus, the decimals we have given before &s examples would be 

 written 



3-2173, -01312, or 'OlSt', and "6. 



Decimals in which the period commences immediately after 

 the decimal point, are sometimes called pure circulating or 

 recurring decimals ; others being entitled mixed circulating 

 decimals. 



Thus above -6 is a pure, while the other two are mixed circu- 

 lating decimals. 



20. Fractions producing Circulating Decimals. 



Wo have seen that all vulgar fractions in their lowest terms, 

 which have any other factors besides 2 and 5 in their denomina- 

 tors, will not produce terminating decimals ; that is to say, in 

 performing the division wo shall never arrive at a remainder 

 which is zero. Wo shall, however, arrive at a remainder which 

 is the same as one of the remainders which has already occurred. 



This is evident from the following considerations : 



The largest possible remainder in any division is the divisor 

 diminished by unity, and therefore there cannot possibly be more 

 than this number of different remainders. Hence, at the very 

 farthest, after this number of remainders have occurred, a re- 

 mainder will occur which is the same as one of the preceding re- 

 mainders. Now it is plain that when this is the case, the whole 

 of the operation which has been performed since that remainder 

 last occurred will be repeated, and that the same remainder will 

 occur again after exactly the same interval, and so on ad injini- 

 Xow to every remainder there will correspond a figure iu 

 the quotient, and therefore the figures in the quotient corres- 

 ponding to the interval between two remainders which are the 

 same will continually recur. 



Jl. This will be 

 Bedooe | to decimal. 



14 





 M 



40 



as 



BO 



U 



H 



Here it will be seen that at the point indicated by the star 

 the remainder 2 occurs, and therefore the division will, after 

 this point, be identical in every respect with that already per- 

 formed. Hence the figures in the quotient, 285714, will con- 

 tinually recur, or the quotient is the pure circulating A**l~*&^ 

 285714. 



It will be observed that the period here is as large as it could 

 possibly be, for the greatest possible remainder is 6, and all the 

 remainders from 1 up to 6 inclusive occur. 



[The process has been exhibited in the form of Long Division, 

 to allow of the remainders appearing in the operation.] 



22. Keduce ^ to a decimal. 



We see at once that the quotient will be a circulating decimal, 

 since being in its lowest terms, 3 is a factor of the denominator. 



30) 17-0000 (-566 .... 

 150 



200 

 180 



Here the remainder 20 is at once repeated, and therefore th 

 quotient after the first figure 5 will consist of 6 continually re- 

 peated, or it will be the mixed circulating '56. 



23. Reduce '$ to a decimal. 



55)129-00(2-315 Answer. 

 110 



190 

 165 



250 

 290 



HO 



275 



25 



Here the remainder 25 ocean again, and therefore the 

 periodical part of the quotient will, after this point, consist of 

 the figures 45 continually repeated. 



EXERCISE 34. 



1. Determine which of the following fractions will produce 

 terminating decimals, and find the equivalent decimals without 

 executing the division : 



A. il. Vft. U. A. i! A!i. *? A'A. VA 1 . 'HP- 



2. Reduce the following fractions to decimals : 



L I. 2. |. S. J. 4. I. 5. I- 7. H* 

 8. 2* x rfj. 9. A 0* 3A- 10. 8ft of l$rf,. 



U -?- 

 271 



83 

 V 



2734 

 375" 



* 



lit 



