§ 3. A Product is Independent of the order of Multiplying.

Now let us suppose that we take six groups of things which all contain the same number, say 5, and that we want to count the aggregate group which is made by putting all these together. We may count the six groups of five things one after another, which amounts to the same thing as adding 5 five times over to 5. Or if we like we may simply mix up the whole of the six groups, and count them without reference to their previous grouping. But it is convenient in this case to consider the six groups of five things as arranged in a particular way.

Let us suppose that all these things are dots which are made upon paper, that every group of five things is five dots arranged in a horizontal line, and that the six groups are placed vertically under one another as in the figure.

We then have the whole of the dots of these six groups arranged in the form of an oblong which contains six rows of five dots each. Under each of the five dots belonging to the top group there are five other dots belonging to the remaining groups; that is to say, we have not only six rows containing five dots each, but five columns containing six dots each. Thus the whole set

of dots can be arranged in five groups of six each, just as well as in six groups of five each. The whole number of things contained in six groups of five each, is called six times five. We learn in this way therefore that six times five is the same number as five times six.

As before, the remark that we have here made about two particular numbers may be extended to the case of any two numbers whatever. If we take any number of groups of dots, containing all of them the same number of dots, and arrange these as horizontal lines one under the other, then the dots will be arranged not only in lines but in columns; and the number of dots in every column will obviously be the same as the number of groups, while the number of columns will be equal to the number of dots in each group. Consequently the number of things in a groups of b things each is equal to the number of things in b groups of a things each, no matter what the numbers a and b are.

The number of things in a groups of b things each is called a times b; and we learn in this way that a times b is equal to b times a. The number a times b is denoted by writing the two letters a and b together, a coming first; so that we may express our result in the symbolic form ab=ba.

Suppose now that we put together seven such compound groups arranged in the form of an oblong like that we constructed just now. They cannot now be represented on one sheet of paper, but we may suppose that instead of dots we have little cubes which can be put into an oblong box. On the floor of the box we shall have six rows of five cubes each, or five columns of six cubes each; and there will be seven such layers, one on the top of another. Upon every cube therefore which is in the bottom of the box there will be a pile of six

cubes, and we shall have altogether five times six such piles. That is to say, we have five times six groups of seven cubes each, as well as seven groups of five times six cubes each. The whole number of cubes is independent of the order in which they are counted, and consequently we may say that seven times five times six is the same thing as five times six times seven.

But it is here very important to notice that when we say seven times five times six, what we mean is that seven layers have been formed, each of which contains five times six things; but when we say five times six times seven, we mean that five times six columns have been formed, each of which contains seven things. Here it is clear that in the one case we have first multiplied the last two numbers, and then multiplied the result by the first mentioned (seven times five times six = seven times thirty), while in the other case it is the first two numbers mentioned that are multiplied together, and then the third multiplied by the result (five times six times seven thirty times seven). Now it is quite

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evident that when the box is full of these cubes it may be set upon any side or upon any end; and in all cases there will be a number of layers of cubes, either 5 or 6 or 7. And whatever is the number of layers of cubes, that will also be the number of cubes in each pile. Whether therefore we take seven layers containing five times six cubes each, or six layers containing seven times five cubes each, or five layers containing six times seven cubes each, it comes to exactly the same thing.

We may denote five times six by the symbol 5 x 6, and then we may write five times six times seven, 5 × 6 × 7. But now this form does not tell us whether we are to multiply together 6 and 7 first, and then take 5

times the result, or whether we are to multiply 5 and 6 first, and take that number of sevens. The distinction between these two operations may be pointed out by means of parentheses or brackets; thus, 5 × (6 × 7) means that the 6 and 7 must be first multiplied together and 5 times the result taken, while (5 × 6) × 7 means that we are to multiply 5 and 6 and then take the resulting number of sevens.

We may now state two facts that we have learned about multiplication.

First, that the brackets make no difference in the result, although they do make a difference in the process by which the result is attained; that is to say, 5 × (6 × 7) = (5 × 6) × 7.

Secondly, that the product of these three numbers is independent of the order in which they are multiplied together.

The first of these statements is called the associative law of multiplication, and the second the commutative law.

Now these remarks that we have made about the result of multiplying together the particular three numbers, 5, 6, and 7, are equally applicable to any three numbers whatever.

We may always suppose a box to be made whose height, length, and breadth will hold any three numbers of cubes. In that case the whole number of cubes will clearly be independent of the position of the box; but however the box is set down it will contain a certain number of layers, each layer containing a certain number of rows, and each row containing a certain number of cubes. The whole number of cubes in the box will then be the product of these three numbers; and it will be got at by taking any two of the three

numbers, multiplying them together, and then multiplying the result by the third number.

This property of any three numbers whatever may now be stated symbolically.

In the first place it is true that a(bc) = (ab)c; that is, it comes to the same thing whether we multiply the product of the second and third numbers by the first, or the third number by the product of the first and second.

In the next place it is true that abc=acb=bca, &c., and we may say that the product of any three numbers is independent of the order and of the mode of grouping in which the multiplications are performed.

We have thus made some similar statements about two numbers and three numbers respectively. This naturally suggests to us that we should inquire if corresponding statements can be made about four or five numbers, and so on.

We have arrived at these two statements by considering the whole group of things to be counted as arranged in a layer and in a box respectively. Can we go any further, and so arrange a number of boxes as to exhibit in this way the product of four numbers? It is pretty clear that we cannot.

Let us therefore now see if we can find any other sort of reason for believing that what we have seen to be true in the case of three numbers-viz., that the result of multiplying them together is independent of the order of multiplying-is also true of four or more numbers.

In the first place we will show that it is possible to interchange the order of a pair of these numbers which are next to one another in the process of multiplying, without altering the product.