FORMATION OF PRODUCTS.

-

81

a, or of + a 4 by

multiplicand has to be taken away as often as is expressed by the negative multiplier; but the taking away of a negative quantity is just the same as the adding of a positive one; and therefore if there is nothing but the product of the two negative factors, that product must be exactly what it expresses greater than 0, that is, it must be the same as if both factors had the sign +, or were positive. Thus, the product of -a by by + a, is equally + a2, or without the sign, a2; is 8, in the same manner as 4 by 2; and the same in all cases. This principle, which must be carefully attended to, is usually stated empirically in the books "like signs give +, unlike signs ;" but this, though true in the case of two factors, or of any even number, as of four, six, eight, and so on, is not true in the case of odd factors, one, three, five, seven, and so on.

2,

We must admit one factor into the series of multiplications; for, as every quantity is positively once itself, and not any other quantity, every quantity must be considered as itself multiplied by + 1, that is, by the integer number 1; and if we were to multiply it by 1 we should change the sign, and along with that the value of the quantity by double of whatever it expressed before being so multiplied. As this co-efficient or factor +1, or 1, is inseparable from the very nature of every quantity, it is never written; we do not write 1a for instance, because when we see a standing alone, we see at once that there is one, and no more. But if we consider it as multiplied by -1, it is quite another matter; for the difference between 1 and 1 is not only 2, but the one is 1 more than 0, and the other is 1 less than 0; and therefore multiplying by - 1 changes the sign, or turns a + quantity to —, and a quantity to +. By separating a into the factors + 1 and a, or 1 and -a, and into +1 and - b, or 1 and b, it would be easy to show the

truth of the rule for the signs in a manner different from the

G

but we trust that which we have said will render it

above; abundantly plain.

When a factor consists of more than one term, the multiplication of it cannot be expressed by annexing the other factor without any sign. Thus, if the one factor were a + b and the other c, then neither a + bc, nor ac + b, would express the product, for the multiplication applies only to b in the one case, and only to a in the other. We must therefore mark the compound factor as a whole, by the vinculum a + b, or rather by the parenthetical characters (a + b); and then (a+b) c expresses the product. If two factors are compound we must inclose each in parentheses, and then it is usual to indicate the multiplication by a dot (.), or rather by the sign x, the last of which is preferable, as the dot is apt to be confounded with the full stop in common language, or the decimal point in arithmetic; thus the product of a + b as one factor by c + d as another factor, is expressed by (a + b) × (c + d).

As these parenthetical characters do not stand for quantities or relations, but merely point out that which is expressed in two or more parts separated by + or -, it may not be amiss to point out, by an example in numbers, the necessity of attending to them. For this purpose let a = : 6, b = 4, c = 8, d = 7, and the above expression will be 6+ 4 × 8+ 7 without the parentheses, and (6 + 4) × (8 + 7) with them. In the first, the multiplication extends no farther than 4 and 8, which produce 32, and there is 6 and 7, or 13 to add, making in all 45, as the whole value. In the second, the multiplication extends to the two sums 10 and 15, and their product, which is 150, is the value, which is very different from the former.

The fourth general relation of the parts of a compound quantity is that in which the one part is a dividend and the other a divisor, the value being a quotient, which we are in the mean

PRINCIPLE OF DIVISION.

83

time merely to express, not actually to find. Indeed, if the expression bears the most simple form, that is, if the divisor and dividend are each expressed by single letter, if these letters

are different, and if nothing farther is stated than that the one of them is to be divided by the other, there are not data sufficient for finding the quotient as a separate quantity. Thus, if the quotient of a divided by b is sought; and if we merely know that a stands for one quantity and b for another, but do not know what kind of quantities they are, whether of the same kind with each other or of different kinds, we cannot tell whether the quotient is or is not a quantity which we can or cannot express in any other way than by indicating it; and, even if we know that both quantities are of the same kind, so that the quotient must be a number, we are not in a condition for stating whether the quotient shall be greater than the number 1, equal to it, or less, unless we know that the quantities are equal or unequal, and in the case of inequality, which is the greater and which the less. Therefore, all that we can do in such cases is to indicate that there is a division to be performed; and this is done generally by writing the dividend above a line, and the divisor below the same. Thus indicates the quotient of a

b

divided by b, though without pointing out what that quotient may be.

The quotients of all quantities may be indicated in the same a + b manner: as, indicates the quotient of the upper quantity с d by the under, whatever may be the forms in which they are expressed. Division may also be indicated by writing the

dividend, then the sign ÷, and lastly the divisor. Thus and ab have the same meaning, and are read " a divided by b."

We shall point out in another section how the division is to be performed so as to obtain the quotient as a separate quantity in all cases where that is possible; but there are some general principles which we can perhaps better explain in this simple view of the matter.

We may, for instance, determine the sign of the quotient, whether we can or cannot express its value by a separate quantity. Here we must bear in mind that the dividend is always equal to the product of the divisor and quotient, so that the finding of a quotient resolves itself into the finding of a quantity the product of which and the quotient shall be equal to the dividend. From this it follows, that if the divisor and dividend have the same sign, the sign of the quotient must be + ; and if they have different signs it must be but that in the case of the same quantities as divisor and dividend, the expression for the quotient will be the same quantities whatever may be the signs.

;

Let us illustrate this by the simplest case that can occur, the

a

division of a quantity by itself, or . The quotient of this, in

a

all cases of the signs, will be expressed by the number 1, because any quantity is, of course, just once itself, and nothing

a

either over or wanting. Now, if it is - or

will be 1, that is, +1; but if it is

will be 1 ; for

a,

a

a

-a

+ a -a

the quotient

or the quotient

a

+a

the divisor in the first case, multiplied

by-1, the quotient, produces +a, the dividend in the first case; anda, the divisor in the second case, multiplied by -1, the quotient, produces

a, the dividend in that case. Hence the quotient of quantities which have the same sign is always a positive quantity, or as much greater than 0 as its

MULTIPLYING DIVISORS AND DIVIDENDS.

85

whole value expresses; but the quotient of two quantities with different signs is always as much less than 0 as its whole value expresses. Hence, in a compound quantity, the

If the divisor were a, and the dividend 3a, it is evident that the quotient would be 3, +3 if the signs were the same, and

- 3 if they were different; and generally, if the divisor were a, and the divident na, that is, any number of times a, whether expressible in terms of the arithmetical scale or not, the quotient would be n, that is, the same number, +n if the signs were the same, and -n if they were different. Now 3 is 1 x 3, and n is 1 x n, whatever number n may stand for; therefore, multiplying the dividend by any quantity has the same effect as multiplying the quotient by the same quantity.

If the dividend were a, and the divisor 3 a, the quotient would be one-third part of 1; and if the dividend were a, and the divisor na, the quotient would be the nth part of 1, or

1

n

; therefore, multiplying the divisor produces the same effect

as dividing the quotient.

Now, if multiplying the dividend multiplies the quotient, and multiplying the divisor divides the quotient, multiplying both by the same quantity, whether that quantity be one single factor or any number of factors, will not alter the value of the

an

a

quotient; or, = whatever n may be, whether large or b n b'

small, simple or compound, provided it is the same in both

cases.

This principle, which is so simple that it is nearly self