in § 37; A representing the determination of 1⁄20 of 1⁄12 of ¼ of 2880 farthings, and B the conversion of 2880 farthings into £.
(iv.) Properties of Numbers.
(A) Properties not depending on the Scale of Notation.
43. Powers, Roots and Logarithms.—The standard series 1, 2, 3, ... is obtained by successive additions of 1 to the number last found. If instead of commencing with 1 and making successive additions of 1 we commence with any number such as 3 and make successive multiplications by 3, we get a series 3, 9, 27, ... as shown below the line in the margin. The first member of the series is 3; the second is the product of two numbers, each equal to 3; the third is the product of three numbers, each equal to 3; and so on. These are written 3^{1} (or 3), 3^{2}, 3^{3}, 3^{4}, ... where n^{p} denotes the product of p numbers, each equal to n. If we write n^{p} = N, then, if any two of the three numbers n, p, N are known, the third is determinate. If we know n and p, p is called the index, and n, n^{2}, ... n^{p} are called the first power, second power, ... p^{th} power of n, the series itself being called the power-series. The second power and third power are usually called the square and cube respectively. If we know p and N, n is called the p^{th} root of N, so that n is the second (or square) root of n^{2}, the third (or cube) root of n^{3}, the fourth root of n^{4}, ... If we know n and N, then p is the logarithm of N to base n.
0 | 1 = 30 | n^{0} |
1 | 3 = 31 | n^{1} |
2 | 9 = 32 | n^{2} |
3 | 27 = 33 | n^{3} |
4 | 81 = 34 | n^{4} |
. | . . | . |
. | . . | . |
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 n^{p} by n^{q}, we multiply the product of p n’s by the product of q n’s, and the result is therefore n^{p} + ^{q}. Similarly, if we divide n^{p} by n^{q}, where q is less than p, the result is n^{p} − ^{q}. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.
If we divide n^{p} by n^{p}, the quotient is of course 1. This should be written n^{0}. 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 | .. | .. | .. | .. | .. | .. | .. |
2 | 2 | .. | .. | .. | .. | .. | .. |
3 | .. | 3 | .. | .. | .. | .. | .. |
4 | 4 | .. | 4 | .. | .. | .. | .. |
5 | .. | .. | .. | 5 | .. | .. | .. |
6 | 6 | 6 | .. | .. | 6 | .. | .. |
7 | .. | .. | .. | .. | .. | 7 | .. |
8 | 8 | .. | 8 | .. | .. | .. | 8 |
9 | .. | 9 | .. | .. | .. | .. | .. |
10 | 10 | .. | .. | 10 | .. | .. | .. |
11 | .. | .. | .. | .. | .. | .. | .. |
12 | 12 | 12 | 12 | .. | 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.
The following are the most important properties of numbers in reference to factors:—
(i) If a number is a factor of another number, it is a factor of any multiple of that number.
(ii) If a number is a factor of two numbers, it is a factor of their sum or (if they are unequal) of their difference. (The words in brackets are inserted to avoid the difficulty, at this stage, of saying that every number is a factor of 0, though it is of course true that 0·n = 0, whatever n may be.)
(iii) A number can be resolved into prime factors in one way only, no account being taken of their relative order. Thus 12 = 2 × 2 × 3 = 2 × 3 × 2 = 3 × 2 × 2, but this is regarded as one way only. If any prime occurs more than once, it is usual to write the number of times of occurrence as an index; thus 144 = 2 × 2 × 2 × 2 × 3 × 3 = 2^{4}·3^{2}.
The number 1 is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as 1·2^{4}·3^{2}, or as 1^{2}·2^{4}·3^{2}, or as 1^{p}·2^{4}·3^{2}, where p might be anything.
If two numbers have no factor in common (except 1) each is said to be prime to the other.
The multiples of 2 (including 1·2) are called even numbers; other numbers are odd numbers.
46. Greatest Common Divisor.—If we resolve two numbers into their prime factors, we can find their Greatest Common Divisor or Highest Common Factor (written G.C.D. or G.C.F. or H.C.F.), i.e. the greatest number which is a factor of both. Thus 144 = 2^{4}·3^{2}, and 756 = 2^{2}·3^{3}·7, and therefore the G.C.D. of 144 and 756 is 2^{2}·3^{2} = 36. If we require the G.C.D. of two numbers, and cannot resolve them into their prime factors, we use a process described in the text-books. The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa − qb, where p and q are any integers.
The G.C.D. of three or more numbers is found in the same way.
47. Least Common Multiple.—The Least Common Multiple, or L.C.M., of two numbers, is the least number of which they are both factors. Thus, since 144 = 2^{4}·3^{2}, and 756 = 2^{2}·3^{3}·7, the L.C.M. of 144 and 756 is 2^{4}·3^{3}·7. It is clear, from comparison with the last paragraph, that the product of the G.C.D. and the L.C.M. of two numbers is equal to the product of the numbers themselves. This gives a rule for finding the L.C.M. of two numbers. But we cannot apply it to finding the L.C.M. of three or more numbers; if we cannot resolve the numbers into their prime factors, we must find the L.C.M. of the first two, then the L.C.M. of this and the next number, and so on.
(B) Properties depending on the Scale of Notation.
48. Tests of Divisibility.—The following are the principal rules for testing whether particular numbers are factors of a given number. The number is divisible—
(i) by 10 if it ends in 0;
(ii) by 5 if it ends in 0 or 5;
(iii) by 2 if the last digit is even;
(iv) by 4 if the number made up of the last two digits is divisible by 4;
(v) by 8 if the number made up of the last three digits is divisible by 8;
(vi) by 9 if the sum of the digits is divisible by 9;
(vii) by 3 if the sum of the digits is divisible by 3;