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 ℔ 5 oz. of meat at 9½ d. a ℔, 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. § 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 .14285½ and .14286½; but the latter shows that 6⁄7 lies between .85713½ and .85714½, 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 (§ 60 (i)) *(a′ ± α)(b′ ± β) = a′b′ + aβ ± (a′β + b′α)*. If *a′* and *b′* are the stated values, and ± *α* and ± *β* the respective limits of error, we ought strictly to take *a′b′ + αβ* as the product, with a limit of error ± *(a′β + b′α)*. In practice, however, both *αβ* and a certain portion of *a′b′* are small in comparison with *a′β* and *b′α*, and we therefore replace *a′b′ + αβ* 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 × 1.335 = 4.772625 and 3.585 × 1.345 = 4.821825. We might take the product as (3.58 × 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 (§ 82 (i)) if necessary.

VIII. Surds and Logarithms

84. *Roots and Surds*.—The *p*th root of a number (§ 43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the *p*th roots of the numerator and of the denominator separately. Thus to find the cube root of 1728, we write it in the form 2^{6}·3^{3}, and find that its cube root is 2^{2}·3 = 12; or, to find the cube root of 1.728, we write it as 1728⁄1000 = 216⁄125 = 2^{3}·3^{3}/5^{3}, and find that the cube root is 2·3/5 = 1.2. Similarly the cube root of 2197 is 13. But we cannot find any number whose cube is 2000.

It is, however, possible to find a number whose cube shall approximate as closely as we please to 2000. Thus the cubes of 12.5 and of 12.6 are respectively 1953.125 and 2000.376, so that the number whose cube differs as little as possible from 2000 is somewhere between 12.5 and 12.6. Again the cube of 12.59 is 1995.616979, so that the number lies between 12.59 and 12.60. We may therefore consider that there is some number *x* whose cube is 2000, and we can find this number to any degree of accuracy that we please.

A number of this kind is called a *surd*; the surd which is the *p*th root of N is written ^{p}√N, but if the index is 2 it is usually omitted, so that the square root of N is written √N.

85. *Surd as a Power*.—We have seen (§§ 43, 44) that, if we take the successive powers of a number N, commencing with 1, they may be written N^{0}, N^{1}, N^{2}, N^{3}, ..., the series of indices being the standard series; and we have also seen (§ 44) that multiplication of any two of these numbers corresponds to addition of their indices. Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law. The number denoted by N^{1/3} will therefore be such that N^{1/3} × N^{1/3} × N^{1/3} = N^{1/3 + 1/3 + 1/3} = N; *i.e.* it will be the cube root of N. By analogy with the notation of fractional numbers, N^{2/3} will be N^{1/3 + 1/3} = N^{1/3} × N^{1/3}; and, generally, N^{p/q} will mean the product of *p* numbers, the product of *q* of which is equal to N. Thus N^{2/6} will not mean the *same* as N^{1/3}, but will mean the square of N^{1/6}; but this will be *equal* to N^{1/3}, *i.e.* (^{6}√N)^{2} = ^{3}√N.

86. *Multiplication and Division of Surds*.—To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index. Thus ^{3}√2 × √5 = 2^{1/3} × 5^{1/2} = 2^{2/6} × 5^{3/6} = 4^{1/6} × 125^{1/6} = 500^{1/6} = ^{6}√500.

87. *Antilogarithms*.—If we take a fixed number, *e.g.* 2, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of *antilogarithms* of the indices to this base. Thus, if we go to two places of decimals, we have as the integral series the numbers 1, 2, 4, 8, ... which are the values of 2^{0}, 2^{1}, 2^{2}, ... and we insert within this series the successive powers of *x*, where *x* is such that *x*^{100} = 2. We thus get the numbers 2.^{01}, 2.^{02}, 2.^{03}, ..., which are the antilogarithms of .01, .02, .03, ... to base 2; the first antilogarithm being 2.^{00} = 1, which is thus the antilogarithm of 0 to this (or any other) base. The series is formed by successive multiplication, and any antilogarithm to a larger number of decimal places is formed from it in the same way by multiplication. If, for instance, we have found 2.^{31}, then the value of 2.^{316} is found from it by multiplying by the 6th power of the 1000th root of 2.

For practical purposes the number taken as base is 10; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, *i.e.* it means a shifting of the decimal point. In the same way, by dividing by powers of 10 we may get negative indices.

88. *Logarithms*.—If N is the antilogarithm of *p* to the base *a*, *i.e.* if N = a^{p}, then *p* is called the logarithm of N to the base *a*, and is written log_{a} N. As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given