94 ARITHMETIC. 



is set over a single repetend, and a point over the first and 

 last figures of a compound repetend. 



It may be further observed, that when other decimal fig- 

 ures precede a repetend, in any number, it is called a 



mixed repetend : as '23, or '104123 : otherwise it is a/)//r^, or 



simple, repetend : as '3 and '123. 



Similar repetends begin at the .same place : as '3 and *6, 

 or 1*341 and 2*156. 



Dissimilar repetends begin at different places : as '253 and 

 •4752. 



Conterminous repetends end at the same place : as '125 



and •009, 



Similar and conterminous repetends begin and end at the 



same place : as 2*9104 and '0613. 



ReDUGTION of ClRCVLATING DECIMALS, 



CASE I. 

 To reduce a simple repetend to its equivalent vulgar fraction* 



RULE.* 



I. Make the given decimal the numerator, and let the 

 denominator be a number consisting of as many nines as 

 there are recurring places in the repetend. 



2. If 



* If unity, with cyphers annexed, be divided by 9 ad infinitumy 

 the quotient will be i continually ; i. e. if ^ be reduced to a deci- 

 mal, it will produce the circulate 'i ; and since *! is the decimal 



equivalent to ^, 'z will = J,- '3 r= |, and so on till •9 = ^ = i. 



Therefore, every single repetend is equal to a vulgar fraction, 

 whose numerator is the repeating figure, and denomiiator 9. 



Again, 



