9^ ARITHMETIC. 



2. If the whole denominator vanish in diviciing by 2, 5 

 or 10, the decimal will be finite, and will consist of so ma- 

 ny places as you perform divisions. 



3. If it do not so vanish, divide pppp, ^cc by the result, 

 till nothing remain, and the number of 9s used will shew 

 the number of places in the repetend ;* wliich will begin 

 after so many places of figures, as there were los, 2s or 5s, 

 used in dividing. 



EXAMPLES. 



T. Required to find whether the decimal equal to -"--""- 

 be finite or infinite ; and, if infinite, how many places the 

 repetend will consist of. 



a a 2 

 ■rrr^~ 2itV [^ I 4 | 2 ] i ; therefore the decimal 



is finite, and consists of 4 places. 



2. Let 



than icooo, &c. by i, therefore 9999, &c. divided by any num. 

 ber whatever will leave o. for a remainder, wlien the repeating fig- 

 ures are at their period. Now whatever niunber of repeating figures 

 we have, when the dividend is i, there will be exactly the same 

 number, when the dividend is any other number whatever. For 

 the product of any circulating number, by any other given num- 

 ber, will consist of the same number of repeating figures as before. 

 ^Thus, let '507650765076, &c. be a circulate, whose repeating 

 part is 5076. Now every repetend (5076) being equally multi- 

 plied, must produce the same product. For though these products 

 will consist of more places, yet the overplus in each, being ahke, 

 will be carried to the next, by which means, each product will be 

 equally increased, and consequently every four places will con- 

 tinue alike. And the same will hold for any other number what- 

 ever. 



Now hence it appears^ that the dividend m^ay be altered at 

 pleasure, and the number of places in the repetend will still be the 



same : thus -r^=:'9o, and -rV> or ttX3='27, where the number 

 of places in each is alike, and the same will be true in all cases. 



