01 = 30n0
13 = 31n1
29 = 32n2
327 = 33n3
481 = 34n4
..   ..
..   ..

The calculation of powers (i.e. of N when n and p are given) is involution; the calculation of roots (i.e. of n when p and N are given) is evolution; the calculation of logarithms (i.e. of p when n and N are given) has no special name.

Involution is a direct process, consisting of successive multiplications; the other two are inverse processes. The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.

The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.

44. Law of Indices.—If we multiply np by nq, we multiply the product of p n’s by the product of q n’s, and the result is therefore np + q. Similarly, if we divide np by nq, where q is less than p, the result is np − q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

If we divide np by np, the quotient is of course 1. This should be written n0. Thus we may make the power-series commence with 1, if we make the index-series commence with 0. The added terms are shown above the line in the diagram in § 43.

45. Factors, Primes and Prime Factors.—If we take the successive multiples of 2, 3, ... as in § 36, and place each multiple opposite the same number in the original series, we get an arrangement as in the adjoining diagram. If any number N occurs in the vertical series commencing with a number n (other than 1) then n is said to be a factor of N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, 6 and 12 are factors of 12.

1..............
22............
3..3..........
44..4........
5......5......
666....6....
7..........7..
88..8......8
9..9..........
1010....10......
11..............
12121212..12....
........
........
........

A number (other than 1) which has no factor except itself is called a prime number, or, more briefly, a prime. Thus 2, 3, 5, 7 and 11 are primes, for each of these occurs twice only in the table. A number (other than 1) which is not a prime number is called a composite number.

If a number is a factor of another number, it is a factor of any multiple of that number. Hence, if a number has factors, one at least of these must be a prime. Thus 12 has 6 for a factor; but 6 is not a prime, one of its factors being 2; and therefore 2 must also be a factor of 12. Dividing 12 by 2, we get a submultiple 6, which again has a prime 2 as a factor. Thus any number which is not itself a prime is the product of several factors, each of which is a prime, e.g. 12 is the product of 2, 2 and 3. These are called prime factors.