186 Mr Glaishsr, On circulating decimals. [Oct. 28, 



a = 6, which is a submultiple of q — 1, =90, and also of 

 </>(?), =72. 



(iii) It is convenient to adopt tlie following definitions. Two 

 digits a, d are complementary if a + a' = 9, and two periods a/3 . . . ^, 

 ci'jS'... f are coinplementary if a + a' = 9, /3 + ^'= 9, ... |^ + f' = 9 ; 

 but two remainders are complementary if their sum is equal to the 

 divisor, viz. if q be the divisor then r and q—r are complementary 

 remainders. It is also convenient to include the dividend among 

 the remainders. 



(iv) If ^ divided by q give a quotient figure a and a remainder 

 r, then q— "p divided by q gives a quotient figure 9 — a and a 

 remainder q — r; and it follows that if _p and q— phe both divided 

 by q, and the same number (ni) of digits in the quotient be 

 obtained in each case, these m quotient digits are complementary, 

 and also the corresponding remainders are complementary. If p 

 be divided by q we shall ultimately arrive at a remainder 

 equal to^, after which the quotient digits recur. Consider however 

 what happens if we arrive at the remainder q — p. We shall then 

 obtain digits in the quotient complementary to those already 

 obtained until we reach the remainder p when the digits begin to 

 recur. Thus if a/3<y...^ be the quotient (of m digits) up to the 

 point at which the complementary remainder occurs then the next 

 TO figures will be a'/S'y...^', where 



ct + a' = 9, /3+^' = d, ... ^ + r=9- 



The following are examples^: 



In the first case the complementary remainder does not occur 

 at all; in the second it occurs after three digits so that the period 

 consists of six digits, and the two halves are complementary. It 

 can be shown that if any one period of a denominator q consists 

 of two complementary portions, all the periods will also consist 

 of two complementary portions. 



1 This mode of arranging divisions, whicli is that employed by Mr Goodwyn in 

 the Appendix to the Tabular series at the end of the Fi7'st centenary, 1818 (see § 5), is 

 very convenient; the two cokimns contain the corresponding quotient digits and 

 remainders, viz. 10 divided by 21 gives quotient and remainder 10, 100 divided by 

 21 gives quotient 4 aiad remainder 16, &c. 



