The third and fourth laws embody an entirely analogous prin ciple in connection with multiplication ; viz., that when integers are multiplied, the final product obtained is the same, no matter in what arrangement the operations of multiplication are per formed. However the reason for this principle is not quite so immediate. It may be formulated as follows : First consider two distinct classes A and B with a and b elements respectively. A class may be formed whose elements are all possible pairs of elements, one from A and the other from B. For each element of A there will be b elements of B which may be associated with it. Consequently by the definition of multiplication there will be a x b pairs in all, or equally bXa of course. Thus we obtain the symbolism we write s=a+b. The operation of determining s when a and b are given is called addition. Thus the operation of addition corresponds to the fundamental logical process of com bination of two classes.
Similarly there may be a classes, each of which contains the same number b of elements, while no two of these classes have an element in common. If these classes be combined to form a new class composed of all these elements, its integer p is deter minable from a and b, as follows in a similar manner. The integer p is called the "product of a and b": p=aXb, and the corre sponding operation is called multiplication. Thus the operation of multiplication corresponds to the combination of a number of distinct classes, themselves in one-to-one correspondence with each other.
The two operations of addition and multiplication are usually considered to be the two fundamental operations. The inverse operations of subtraction and division may be defined by means of the respective equations Consequent the symbol a—b will represent an integer if and only if b is smaller than a (i.e., some class B can be put in one to-one correspondence with a part of some class A). Likewise a/b will represent an integer if and only if a contains b as a fac tor. The successive integers may be defined in terms of addition as follows : • , • • ., Here I is to be regarded as the integer for any unit class. The definition of multiplication makes it clear that multiplication of any integer by I does not affect the integer.
of the integer, as based upon the one-to-one correspondence of classes, shows immediately that certain simple laws must hold. For instance since the class C, obtained by the combination of two classes A and B, is the same whichever class is mentioned first, we infer that the sum of two integers is independent of the order in which they are taken. This law has been termed the commuta tive law of addition (see COMMUTATIVE Laws), and is the first of the five fundamental laws written below in algebraic form: third commutative law of multiplication. The same justification may be presented in a simple graphical form. A rectangular array of a X b dots may be regarded either as a rows of b dots each or as b rows of a dots each.
Secondly, consider three distinct classes A, B and C with a, b, and c elements. A class may be formed whose elements are all possible triples of elements, one from A, one from B, and one from C. Evidently the class of triples so obtained may also be formed by taking the class of pairs from A and B mentioned above, and associating any such pair with any element of C to form a triple. Likewise the same class of triples may be con structed by taking the class of pairs from B and C and associating any such pair with any element of A. In this way the fourth associative law of multiplication is justified. This justification may also be given graphical form by use of an array of aXbXc dots in space arranged in the form of a rectangular parallelepiped.
The fifth distributive law of multiplication (see DISTRIBUTWE LAW) may be seen to hold as follows. The class of pairs of elements taken one from the class A and the other from the com bination of B and C is evidently merely the combination of the class of pairs taken from A and from B, with the class of pairs taken from A and C. This, too, admits of graphical 4ustification, since an array of a X (b+c) dots is evidently made up of an array of aXb dots and an array of a X c dots.
These five general laws contain all of the principles necessary for the manipulation of the integers, provided that we take for granted the special law, a X i =a. In particular we can deduce the usual addition and multiplication tables for the integers by the aid of these laws ; as an illustration we find