xxi.] THEORY OF APPROXIMATION. 48 1 



Arithmetic of Approximate Quantities. 



Considering that almost all the quantities which we 

 treat in physical and social science are approximate only, 

 it seems desirable that attention should be paid in the 

 teaching of arithmetic to the correct interpretation and 

 treatment of approximate numerical statements. We seem 

 to need notation for expressing the approximateness or 

 exactness of decimal numbers. The fraction '025 may 

 mean either precisely one 4Oth part, or it may mean 

 anything between -0245 and '0255. I propose that when 

 a decimal fraction is completely and exactly given, a 

 small cipher or circle should be added to indicate that 

 there is nothing more to come, as in -0250. When the 

 first figure of the decimals rejected is 5 or more, the first 

 figure retained should be raised by a unit, according to a 

 rule approved by De Morgan, and now generally recog- 

 nised. To indicate that the fraction thus retained is more 

 than the truth, a point has been placed over the last figure 

 in some tables of logarithms ; but a similar point is used 

 eo denote the period of a repeating decimal, and I should 

 therefore propose to employ a colon after the figure ; thus 

 025: would mean that the true quantity lies between 

 0245 and -025 inclusive of the lower but not the higher 

 limit. When the fraction is less than the truth, two dots 

 might be placed horizontally as in '025. . which would 

 mean anything between -025 and "0255 not inclusive. 



When approximate numbers are added, subtracted, mul- 

 tiplied, or divided, it becomes a matter of some complexity 

 to determine the degree of accuracy of the result. There 

 are few persons who could assert off-hand that the sum 

 of the approximate numbers 3470, 52-693, 8o - i, is 167*5 

 within less than '07. Mr. Sandeman has traced out the 

 rules of approximate arithmetic in a very thorough manner, 

 and his directions are worthy of careful attention. 1 The 

 third part of Sonuenschein and Nesbitt's excellent book 

 on arithmetic z describes fully all kinds of approximate 

 calculations, and shows both how to avoid needless labour 



1 Sandeman, Felicotetics, p. 214. 



2 The Science and Art of Arithmetic for the Use of Schoolt 

 (Whitakcr and Co.) 



1 I 



