788 



8CIENCE. 



[Vol.. II., No. 46. 



A NEW RULE FOR DIVISION IN 

 ARITHMETIC. 



The ordinary process of long division is 

 rather difficult, owing to the necessity of 

 guessing at the successive figures which form 

 the divisor. In case the repeating decimal 

 expressing the exact quotient is required, the 

 following method will be found convenient. 



Rule for division. 



First, Treat the divisor as follows : — 



If its last figure is a 0, strike this off, and 

 treat what is left as the divisor. 



If its last figure is a 5, multiply the whole 

 by 2, and treat the product as the divisor. 



If its last figure is an even number, multiplj' 

 the whole hy 5, and treat the product as a di- 

 visor. 



Repeat this treatment until these precepts 

 cease to be applicable. Call the result the 

 prepared divisor. 



Second, From the prepared divisor cut off 

 the last figure ; and, if this be a 9, change it to 

 a 1 , or, if it be a 1 , change it to a 9 : otherwise 

 keep it unchanged. Call this figure the extra- 

 neous midtiplier. 



Multiply the extraneous multiplier into the 

 divisor thus truncated, and increase the pro- 

 duct by 1, unless the extraneous multiplier be 

 7, when increase the pi'oduct b}' 5. Call the 

 result the current midtiplier. 



Third, Multiply together the extraneous 

 multiplier and all the multipliers used in the 

 process of obtaining the pi-epared divisor. 

 Use the product to multiplj' the dividend, call- 

 ing the i-esult t\\e prepared dividend. 



Fourth, From the prepared dividend cut 

 oflF the last figure, multiply this by the current 

 multiplier, and add the product to the trun- 

 cated dividend. Call the sum the modified 

 dividend, and treat this in the same waj^ 

 Continue this process until a modified dividend 

 is reached which equals the oiiginal prepared 

 dividend or some previous modified dividend ; 

 so that, were the process continued, the same 

 figures would recur. 



Fifth, Consider the series of last figures 

 which have been successively cut off from the 

 prepared dividend and from the modified divi- 

 dends as constituting a number, the figure first 

 cut off being. in the units' place, the next in 

 the tens' place, and so on. Call this the first 

 infinite number, because its left-hand portion 

 consists of a series of figures repeating itself 

 indefinitel3- toward the left. Imagine another 

 infinite number, identical with the first in the 

 repeating part of the latter, but differing from 

 this in that the same series is repeated unin- 



terruptedly and indefinitely toward the right, 

 into the decimal places. 



Subtract the first infinite number from the 

 second, and shift the decimal point as many 

 places to the left as there were zeros dropped 

 in the process of obtaining the prepared divisor. 



The result is the quotient sought. 



Examples. 



1. The following is taken at random. Divide 

 1883 by 365. 



First, The divisor, since it ends in 5, must 

 be multiplied by 2, giving 730. Dropping the 

 0, we have 73 for the prepared divisor. 



Second, The last figure of the prepared 

 divisor being 3, this is the extraneous multi- 

 plier. Multiplying the truncated divisor, 7, 

 bj' the extraneous multiiDlier, 3, and adding 1, 

 we have 22 for the current multiplier. 



Third, The dividend, 1883, has now to be 

 multiplied by the product of 3, the exti-aneous 

 multiplier, and 2, the multii>lier used in pre- 

 paring the divisor. The product, 11298, is 

 the prepared dividend. 



Fourth, From the prepared dividend, 11298, 

 we cut ofl:' the last figure, 8. and multiply this 

 by the current multiplier, 22. The ])roduct, 

 176, is added to the truncated dividend, 1129, 

 and gives 1305 for the first modified divisor. 

 The whole operation is shown thus : — 

 1 S S 3 



112 9l8 



13 0|5 

 110 



19|S 

 17 6 

 19|5 

 110 



12[9 



2|10 



2 2 



AVe stop at this point because 24 was a 

 previous modified dividend, written under the 

 form 240 above. Onr two infinite numbers 

 (which need not in practice be written down) 

 are, with their difference, — 



iO,9.-.S,904,058 



10,958, 904, 109. 5890410958904 

 51.5890410958904' 

 Hence the quotient sought is 5.158904109. 



