Chapter VII: Approximation

81. _Approximate Character of Numbers._--The numbers (integral or decimal) by which we represent the results of arithmetical operations are often only approximately correct. All numbers, for instance, which represent physical measurements, are limited in their accuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations. Even where numbers are supposed to be exact, calculations based on them can often only be approximate. We might, for instance, calculate the exact cost of 3 lb. 5 oz. of meat at 9-1/2 d. a lb., but there are no coins in which we could pay this exact amount.

When the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the _error_.

82. _Degree of Accuracy._--There are three principal ways of expressing the degree of accuracy of any number, i.e. the extent to which it is equal to the number it is intended to represent.

(i) A number can be _correct to_ so many _places of decimals_. This means (cf. S 71) that the number differs from the true value by less than one-half of the unit represented by 1 in the last place of decimals. For instance, .143 represents 1/7 correct to 3 places of decimals, since it differs from it by less than .0005. The final figure, in a case like this, is said to be _corrected_.

This method is not good for comparative purposes. Thus .143 and 14.286 represent respectively 1/7 and 100/7 to the same number of places of decimals, but the latter is obviously more exact than the former.

(ii) A number can be correct to so many _significant figures_. The significant figures of a number are those which commence with the first figure other than zero in the number; thus the significant figures of 13.027 and of .00013027 are the same.

This is the usual method; but the relative accuracy of two numbers expressed to the same number of significant figures depends to a certain extent on the magnitude of the first figure. Thus .14286 and .85714 represent 1/7 and 6/7 correct to 5 significant figures; but the latter is relatively more accurate than the former. For the former shows only that 1/7 lies between .142855 and .142865, or, as it is better expressed, between .14285-1/2 and .14286-1/2; but the latter shows that 6/7 lies between .85713-1/2 and .85714-1/2, and therefore that 1/7 lies between .14285-7/12 and .14285-9/12.

In either of the above cases, and generally in any case where a number is known to be within a certain limit on each side of the stated value, the _limit of error_ is expressed by the sign [+-]. Thus the former of the above two statements would give 1/7 = .14286 [+-] .000005. It should be observed that the numerical value of the error is to be subtracted from or added to the stated value according as the error is positive or negative.

(iii) The limit of error can be expressed as a fraction of the number as stated. Thus 1/7 = .143 [+-] .0005 can be written 1/7 = 143(1 [+-] 1/286).

83. _Accuracy after Arithmetical Operations._--If the numbers which are the subject of operations are not all exact, the accuracy of the result requires special investigation in each case.

Additions and subtractions are simple. If, for instance, the values of a and b, correct to two places of decimals, are 3.58 and 1.34, then 2.24, as the value of a-b, is not necessarily correct to two places. The limit of error of each being [+-].005, the limit of error of their sum or difference is [+-].01.

For multiplication we make use of the formula (S 60 (i)) (a' [+-] [alpha])(b' [+-] [beta]) = a'b' + [alpha][beta] [+-] (a'[beta] + b'[alpha]). If a' and b' are the stated values, and [+-][alpha] and [+-][beta] the respective limits of error, we ought strictly to take a'b' + [alpha][beta] as the product, with a limit of error [+-](a'[beta] + b'[alpha]). In practice, however, both [alpha][beta] and a certain portion of a'b' are small in comparison with a'[beta] and b'[alpha], and we therefore replace a'b' + [alpha][beta] by an approximate value, and increase the limit of error so as to cover the further error thus introduced. In the case of the two numbers given in the last paragraph, the product lies between 3.575 X 1.335 = 4.772625 and 3.585 X 1.345 = 4.821825. We might take the product as (3.58 X 1.34) + (.005)^2 = 4.797225, the limits of error being [+-].005(3.58 + 1.34) = [+-].0246; but it is more convenient to write it in such a form as 4.797 [+-] .025 or 4.80 [+-] .03.

If the number of decimal places to which a result is to be accurate is determined beforehand, it is usually not necessary in the actual working to go to more than two or three places beyond this. At the close of the work the extra figures are dropped, the last figure which remains being corrected (S 82 (i)) if necessary.