xliv HKFIMTIONS ANI> KX 1'LANATIONS. 



Thus, calling the last place retained the rth place, the limits of the rejection 

 error entering into that place are +0.5 and 0.5, and as all amounts between 

 these limits are equally likely to occur, the average rejection error in the long 

 run will be 0.25 in the rth place. 



Law and Amount of Accumulated Rejection Error. Let the last place 

 of significant figures retained in a number, e.g. the fourth, fifth, etc., be called 

 the rth place. Then the error entering from rejection of figures beyond the rth 

 will be at most 5 in the (r -f i) place, and any error between these limits, 

 -f 5 and 5 in the (r-f- i) place, will be equally likely to occur in any given 

 case, and therefore will be of equal frequency in the long run. The average 

 rejection error in any considerable number of rejections will therefore be 2.5 

 in the (r + i) place. If then in direct processes of multiplication, division, evolu- 

 tion, or involution (separate or combined) each factor, product, and quotient in 

 the operation be carried out to the same number r of places, what will be the accu- 

 mulated fractional rejection error if n such rejections are made during the entire 

 operation ? This error we shall call for brevity the computation error, or 

 simply the rejection error. This question might easily be answered in a form 

 giving the average accumulated error, but we are at present concerned chiefly 

 with the maximum possible error, since our object is to frame rules which will 

 reduce the worst possible computation error to negligible dimensions. The 

 maximum computation error would arise when every single rejection error was 

 5 in the (r + i ) place, and all had the same sign ; and this would be the greatest 

 fraction of the final result when that result and all the factors (and therefore all 

 the intermediate products, quotients, etc.) began with i and had o in the other 

 places, i.e. were each io r ~ l with the decimal point wherever it might happen to 

 stand. If there were n of these factors, products, quotients, etc., at which 

 rejections were made, the maximum possible computation error in the result 

 would, therefore, be n x 5 in the (r -f i) place, and the fractional error would 

 be 5 n/i<y. Since this maximum error would be exceedingly rare unless n were 

 very small, and any approaching one-half of it would be very rare, we may 

 properly assume that our rules will be sufficiently stringent if we allow this 

 maximum error to have the same magnitude as the desired accuracy in the 

 result as expressed on the basis of the precision measure, or deviation measure, 

 explained at page xlii. To determine most simply what number r of places this 

 limitation would call for in the processes under consideration, let us take specific 

 Suppose the work is desired to possess an accuracy of i per cent. Then 

 5 n/io r must be equal to or less than i per cent; 1.6.5 <i/ioo. Hence, we 

 have to solve 



5ft = J_ 



IO r IOO 



By inspection, if r = 3, 5 n = 10, .*. n = 2, 

 if r = 4, 5 n = 100, .'. n = 20. 



But obviously n will almost never be as small as 2, and rarely as large as 20, 

 lying with greatest frequency between 5 and 10 and averaging below 7. This 

 will give a maximum error of 0.0035 or per cent with n = 7, r = 4, which would 

 be insignificant, and rising to i per cent only when n = 20. Hence, in work of 

 multiplication, division, etc., where i per cent accuracy, or a little better, is 

 desired (remembering that by this we mean only a deviation measure of i per 

 cent) r = 4 will be au entirely safe but not an excessive value ; that is, the reten- 



