350 Mr. W. II. Walenn on Division-Remainders in Arithmetic. 



same manner as ordinary division 3 reducing the values from 

 time to time. For instance, to obtain 



TT 269 • 8 , 3 1 „ a _ 

 ^9T«o = i = 1+^=1 + 3x2=*: 

 122 o o 



and 



(»+S) 



4+f=4+6x2=U 9 16 = 7. 



Many theorems that, treated in the ordinary way 3 involve 

 the cumbersome notation of algebra used to deal with scales of 

 notation, can be investigated by the simplest means. For in- 

 stance, the proof that the addition of the digits of a number 

 casts out the nines of that number is manifest from the sub- 

 stitution of 9 for 8 in the general formula of imitation given 

 above: for in that case each term has 1 for its coefficient. In 

 the function U n .i' the coefficients are, beginning with the unit, 

 + 1 and — 1 alternately. 



These relations of U 9 # (generally written U# as being the 

 most common of all the systems of unitation) and of U ni /\ give 

 the following singular property, an extension of a well-known 

 fact. If dots or marks be made at equal intervals over the 

 digits of a given number, or part of a given number, and 

 (having regard to the relative position of the marked digits 

 in the number) if the marked digits be transposed among 

 themselves, the remainder obtained by the subtraction of the 

 transposed number from the original number — or vice versa, 

 according to which number is the greatest — will be divisible 

 by a number composed entirely of ones, as 1111, also by some 

 other number composed ejrtirely of nines, as 9999, the number 

 of ones or of nines in the divisor being a + 1. if there be a in- 

 tervals. For instance, 



5 8 2 1 2 3 1 - 1 2 3 5*8 2 1 = 4 5 8 5 4 1 0, 



and 4585410 is exactly divisible by 111 and by 999. 



An easy and practical method of constructing decimal equi- 

 valents for reciprocals is as follows : — Take a reciprocal with 

 any number for its denominator that has 9 for its unit figure, 

 and employ as a multiplier a number greater by 1 than that 



represented by the remaining figure or figures. In -j 5 is 



the multiplier. Beginning with 1, this multiplier is used to 

 obtain the next figure towards the left from the previous one 

 as from a multiplicand, adding-in the tens digit or other digits 



