VIII. Surds and Logarithms
84. Roots and Surds.—The pth 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 pth roots of the numerator and of the denominator separately. Thus to find the cube root of 1728, we write it in the form 26}·33, and find that its cube root is 22·3 = 12; or, to find the cube root of 1.728, we write it as 1728⁄1000 = 216⁄125 = 23·33/53, 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 pth 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 N0, N1, N2, N3, ..., 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 N1/3 will therefore be such that N1/3 × N1/3 × N1/3 = N1/3 + 1/3 + 1/3 = N; i.e. it will be the cube root of N. By analogy with the notation of fractional numbers, N2/3 will be N1/3 + 1/3 = N1/3 × N1/3; and, generally, Np/q will mean the product of p numbers, the product of q of which is equal to N. Thus N2/6 will not mean the same as N1/3, but will mean the square of N1/6; but this will be equal to N1/3, 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 = 21/3 × 51/2 = 22/6 × 53/6 = 41/6 × 1251/6 = 5001/6 = 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 20, 21, 22, ... and we insert within this series the successive powers of x, where x is such that x100 = 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 = ap, then p is called the logarithm of N to the base a, and is written loga N. As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions. Thus, to find the logarithm of a number to base 2, the number being greater than 1, we first divide repeatedly by 2 until we get a number between 1 and 2; then divide repeatedly by 10√2 until we get a number between 1 and 10√2; then divide repeatedly by 100√2; and so on. If, for instance, we find that the number is approximately equal to 23 × (10√2)5 × (100√2)7 × (1000√2)4, it may be written 23.574, and its logarithm to base 2 is 3.574.
For a further explanation of logarithms, and for an explanation of the treatment of cases in which an antilogarithm is less than 1, see [Logarithm].