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a (b + c + d) = ab + ac + ad,

(b + c + d) a = ba + ca + da;

and the same reasoning applies to the addition of any number of numbers and their subsequent multiplication.

$5. On Powers.

When a number is multiplied by itself it is said to be squared. The reason of this is that if we arrange a number of lines of equally distant dots in an oblong, the number of lines being equal to the number of dots in each line, the oblong will become a square.

If the square of a number is multiplied by the number itself, the number is said to be cubed; because if we can fill a box with cubes whose height, length, and breadth are all equal to one another, the shape of the box will be itself a cube.

If we multiply together four numbers which are all equal, we get what is called the fourth power of any one of them; thus if we multiply 4 3's we get 81, if we multiply 4 2's we get 16.

If we multiply together any number of equal numbers, we get in the same way a power of one of them which is called its fifth, or sixth, or seventh power, and so on, according to the number of numbers multiplied together.

Here is a table of the powers of 2 and 3:

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The number of equal factors multiplied together is called the index, and it is written as a small figure above the line on the right-hand side of the number whose power is thus expressed. To write in shorthand

the statement that if you multiply seven threes together you get 2187, it is only needful to put down :

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It is to be observed that every number is its own first power; thus 21=2, 3'3, and in general a1=a.

§ 6. Square of a +1.

We may illustrate the properties of square numbers by means of a common arithmetical puzzle, in which one person tells the number another has thought of by means of the result of a round of calculations performed with it.

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This last is always an odd number, and the number thought of is what we may call the less half of it; viz., it is the half of the even number next below it. Thus, the result being given as 7, we know that the number thought of was the half of 6, or 3.

We will now proceed to prove this rule. Suppose that the square of 5 is given us, in the form of twentyfive dots arranged in a square, how are we to form the square of 6 from it? We may add five dots on the right, and then five dots along the bottom, and then one dot extra in the corner. That is, to get the square of 6 from the square of 5, we must add one more than twice 5 to it. Accordingly

36 25+ 10 + 1.

And, conversely, the number 5 is the less half of the difference between its square and the square of 6.

The form of this reasoning shows that it holds good for any number whatever. Having given a square of dots, we can make it into a square having one more dot in each side by adding a column of dots on the right, a row of dots at the bottom, and one more dot in the corner. That is, we must add one more than twice the number of dots in a side of the original square. If, therefore, this number is given to us, we have only to take one from it and divide by 2, to have the number of dots in the side of the original square.

We will now write down this result in shorthand. Let a be the original number; then a +1 is the number next above it; and what we want to say is that the square of a+1, that is (a+1)2, is got from the square of a, which is a2, by adding to it one more than twice a, that is 2a+1. Thus the shorthand expression is

(a + 1)2 = a2 + 2a + 1.

This theorem is a particular case of a more general one, which enables us to find the square of the sum of

any two numbers in terms of the squares of the two numbers and their product. We will first illustrate this by means of the square of 5, which is the sum of 2 and 3.

The square of twenty-five dots is here divided into two squares and two oblongs. The squares are respectively the squares of 3 and 2, and each oblong is the product of 3 and 2. In order to make the square of 3 into the square of 3+2, we must add two columns on the right, two rows at the bottom, and then the square of 2 in the corner. And in fact, 25=9+2×6+4.

§ 7. On Powers of a+b.

To generalise this, suppose that we have a square with a dots in each side, and we want to increase it to a square with a+b dots in each side. We must add b columns on the right, b rows at the bottom, and then the square of b in the corner. But each column and each row contains a dots. Hence what we have to add is twice ab together with b2, or in shorthand :

(a + b)2 = a2+2ab+b2.

The theorem we previously arrived at may be got from this by making b=1.

Now this is quite completely and satisfactorily proved; nevertheless we are going to prove it again in another way. The reason is that we want to extend the proposition still further; we want to find an expression not only for the square of (a+b), but for any other power of it, in terms of the powers and products of powers of a and b. And for this purpose the mode of proof we have hitherto adopted is unsuitable. We might, it is true, find the cube of a+b by adding the proper pieces to the cube of a; but this would be somewhat cumbrous, while for higher powers no such representation can be used. The proof to which we now proceed depends on the distributive law of multiplication. According to this law, in fact, we have

(a + b)2 = (a + b) (a + b) = a (a + b) + b (a + b) = aa + ab + ba + bb

= a2 + 2ab + b2.

It will be instructive to write out this shorthand at length. The square of the sum of two numbers means that sum multiplied by itself. But this product is the first number multiplied by the sum together with the second number multiplied by the sum. Now the first number multiplied by the sum is the same as the first number multiplied by itself together with the first number multiplied by the second number. And the second number multiplied by the sum is the same as the second number multiplied by the first number together with the second number multiplied by itself. Putting all these together, we find that the square of the sum is equal to the sum of the squares of the two numbers together with twice their product.

Two things may be observed on this comparison. First, how very much the shorthand expression gains

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