39. Quotients.—The converse of subdivision is the formation of units into groups, each constituting a larger unit; the number of the groups so formed out of a definite number of the original units is called a quotient. The determination of a quotient is equivalent to completion of the diagram by entry of the number when the unit and the product are given. There is no satisfactory name for the operation, as distinguished from partition; it is sometimes called measuring, but this implies an equality in the original units, which is not an essential feature of the operation.
40. Division.—From the commutative law for multiplication, which shows that 3 × 4d. = 4 × 3d. = 12d., it follows that the number of pence in one-fourth of 12d. is equal to the quotient when 12 pence are formed into units of 4d.; each of these numbers being said to be obtained by dividing 12 by 4. The term division is therefore used in text-books to describe the two processes described in §§ 38 and 39; the product mentioned in § 34 is the dividend, the number or the unit, whichever is given, is called the divisor, and the unit or number which is to be found is called the quotient. The symbol ÷ is used to denote both kinds of division; thus A ÷ n denotes the unit, n of which make up A, and A ÷ B denotes the number of times that B has to be taken to make up A. In the present article this confusion is avoided by writing the former as 1⁄n of A.
Methods of division are considered later (§§ 106-108).
41. Diagrams of Division.—Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplication-diagram. Thus the uncompleted diagram for partition is F or G, while for measuring it is usually H; the vacant compartment being for the unit in F or G, and for the number in H. In some cases it may be convenient in measuring to show both the units, as in K.
42. Successive Division may be performed as the converse of successive multiplication. The diagrams A and B below are the converse (with a slight alteration) of the corresponding diagrams 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 31 (or 3), 32, 33, 34, ... where np denotes the product of p numbers, each equal to n. If we write np = 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, n2, ... np are called the first power, second power, ... pth 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 pth root of N, so that n is the second (or square) root of n2, the third (or cube) root of n3, the fourth root of n4, ... If we know n and N, then p is the logarithm of N to base n.