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VI. Note on a Method of Checking Calculations. 

 By W. H. Walenn*. 



THE method of checking calculations by casting out the 

 nines has only been utilized to a limited extent — prin- 

 cipally in schools, to check multiplications and divisions of 

 considerable length. The fact does not seem to have been 

 realized by teachers, although it has long been known to ma- 

 thematicians, that every operation in natural numbers has a 

 corresponding operation in the remainders to any divisor of 

 those natural numbers, which may be used to check a calcu- 

 lation in many cases from beginning to end without appeal to 

 the original numbers, except at the beginning and the end. 



Moreover the application of these remainders is never made, 

 by teachers, to decimal calculations. This is easily under- 

 stood, because the change in the condition of the remainder 

 (when so used) is not realized by them. When employed to 

 check calculations, the remainder is no longer in the condition 



a t 



implied in the equation -=c+ j, but it is simply the number 



of units by which the number is in excess of being exactly 

 divisible by nine. Therefore the checking process is as appli- 

 cable to all terminable decimals as it is to ordinary numbers. 



If, for instance, it be required to ascertain the weight of 

 water in a tank which is 2*375 feet high, 3*25 feet wide, and 

 3*75 feet broad, the formula maybe written io= 62*32 abc, and 

 the state of the question is w = 62*32 . 2*375 . 3*25 . 3*75 in 

 pounds, the result being w= 1803*871875 lbs. The definition 

 of a unitate being the number of units by which the correspond- 

 ing number is in excess of being exactly divisible by a given 

 number (nine in this instance), the operation of checking the 

 calculation by which w was found may be symbolized as fol- 

 lows : — U 9 w* = U 9 (4 . 8 . 1 . 6) = 3. That the above value of w 

 is correct appears because U 9 1803*871875 = 3. It will be 

 found that the formulation of the process to be effected by 

 unitation, as a check upon a main calculation, is of great 

 service. 



Not only is it possible to check calculations involving direct 

 operations, but it is also possible to check inverse operations 

 by unitation. In checking subtraction, it must be remembered 

 that a remainder is not really altered by the addition of the 

 divisor, for instance U 9 3 = U 9 12 ; subtraction may then take 



* Communicated by the Author, having been read before the British 

 Association, Section A, August 23, 1879. 



