o48 Mr. W. IT. Walenn on Division-Remainders in Arithmetic. 



theorem is that (10 — 8)t + u has the same remainder to divi- 

 sion by 8 that 10 t + u lias, lOt + u being a given number in 

 terms of its digits, and 8 being any whole number ; this for- 

 mula, which is a function of 8, but not of the quotient of the 

 division, may be expanded to the form 



(lO-8) n -> Chl + (10-8) n - 2 a n _ 1 + ... +(10-8) 2 rt 3 



+ (10-8)a 2 + a 1 . 



This form is well adapted for use in general calculations. 

 Since a 1 is the extreme right-hand digit of any number, it is 

 only necessary that the dividend should be finite ; a Y may 

 either be the unit digit or a digit on the right hand of the de- 

 cimal point. From this it appears that although the divisor 

 must be integral, the dividend may be fractional. That 

 (10 — 8)t + u and 10 t + u have the same remainder to 8 may be 

 seen from the fact, deduced by multiplication, that 



(10-8)t + u = 10t + u-8t. 



In the former paper the name imitation is assigned to the 

 above operation upon a given number, which yields the re- 

 mainder or imitate to a chosen divisor or base. The function 

 called the unitate of x to the base 8 is symbolized by ~U s x. 

 Then 



U^ = (10-o) w -V n + (10-5) n -V-i+ ... +(10-o) 2 « 3 



+ (10-o> 2 + ai, 

 if x = 10 w ~\ + 10 n - 2 a n ^ + . . . + 10 2 a 3 + 10a 2 + a x . 



The function U g ( — x) is easily derived from JJ s x. For —x 

 may have 8 added to it without altering the value of U g ( — x). 

 Hence U 5 (— x)=TJ 8 (8 — x); that is, U 5 ( — x) is the comple- 

 ment of x to 8. For instance, U 9 ( — 3) = U 9 (9 — 3)=6. 

 Thus, for every value of U 5 ( — x) there is a corresponding value 

 of V s x, and for every value of ~U s x there is a, corresponding 

 value of U 5 ( — x). 



The function (JJ-)may be investigated by means of its 



equivalent U g ( x . - J = 1J s (x . y~ ! ). In the case of 8 being a 



prime number, all the values of this function are integral, ex- 



1 2 



cepting when y — 8\ as Uut> = 4, U n ^ =8. When 8 is not 



O O 



a prime number, U 5 - is only integral when V s y is not equal 



