86

ALGEBRAICAL OPERATIONS.

evident, is a very important one in practice; and so is the converse of it, namely, that if both divisor and dividend are divided by the same quantity the quotient is not altered. This last is, in fact, the principle upon which we proceed in the common arithmetical division of one number by another; we consider the divisor as divided by itself, and thus reduced to the number 1; and we divide the dividend also by the divisor, in order to have it expressed in terms of the divisor, considered

as one.

We shall be better able to see the value of these principles afterwards, and shall discover other means of perceiving their truth; so we shall now proceed to show how the elementary operations are performed algebraically.

SECTION VI.

ELEMENTARY OPERATIONS IN ALGEBRA.

As arithmetic is merely the application of the general principles of algebra to those particular cases of quantities which can be expressed by numbers according to the scale and notation of arithmetic, it follows that the elementary operations in the one must be the same as they are in the other, namely, addition, subtraction, multiplication, and division. But if the notation of algebra, as we have attempted to explain it in the preceding section, is properly understood, these operations are far more easily performed by means of the algebraical symbols than the arithmetical ones. If we write down the given quantities with the proper signs, used in the form which indicates the operation, we have an expression for the result of that operation at once; and all that we have to do farther is to find out

whether the expression thus obtained can be reduced to fewer or more simple terms. In this, every case must be considered in itself; and thus every operation in algebra is the discovery of something new, instead of the performing of the same sort of drudgery over and over, as is the case in arithmetic.

The first and most general consideration is, whether the case before us can or cannot be simplified; and as it would be vain to try the cases which cannot be simplified, the knowledge of them is the first point to which we must direct our attention. Now, the principle here is a very simple and self-evident one: if the quantities are all of different kinds, that is, all expressed by different letters, or by different letters combined with different numbers as co-efficients, we cannot shorten or simplify the expression.

Thus, in ADDITION, if the sum is that of a, b, and c, there is no simpler expression for it than a + b + c. Also, if it is 5 a, 3b, and 4 c, there is no simpler expression than 5 a + 3b + 4c. But if it is 5 a, 5 b, and 5 c, we can make it 5 times the sum of a, b, and c, that is (a + b + c) x 5. If the letters are the same, we can bring them into one expression; thus 5 a + 3 a +4a is 12a. Also, if the letters are the same, and some of them and others, we can get one expression for the whole by taking the + into one sum and the into another; subtracting the co-efficients and prefixing the sign of the greater co-efficient to the difference. Thus 5 a 5a = 0; 5 a 4α = a; 5 a 10 a =

2

5a; and so in other cases. Also, if we have a + quantity to add to any expression, and there is a quantity of the same kind, that is, expressed by the same letter or letters in the expression, we get rid of as much of the quantity as is equal to the + one. Thus, if we have to add cb, that is + cb, to a cb, the sum becomes

a; if we have to add 4b to a 6b, the sum becomes a — 2b; and if we have to add 3b to a- 2 b, the sum becomes a + b. When we have mentioned that all the quantities of which the sum is required, express that sum when they are written one after another with their proper signs; that the only cases in which that expression can be shortened are those in which the same quantity occurs more than once; that quantities which do occur more than once in the expression may be reduced to one occurrence by taking their sum; that this sum is the sum of the co-efficients if the signs are the same, but the difference, and having the same sign with the greater, if some have + and others -.

Any instructions more minute than these, and especially any formal or empirical rules for the adding of quantities, are not only superfluous but injurious to those who wish to understand algebra. Algebra is the art of finding out how things are to be done; and thus, if there are rules and formula to be learned, as a child cons by rote a catechism without understanding one word of the reason or truth of the dogma (it is in the manner not the matter that the dogma consists), that which is worked at (we will not say learned) is not algebra, it is the practice of arithmetic in an algebraical dress, more difficult, and therefore less useful than simple arithmetic, just as the common calculations of the schools are more difficult and less useful than the ready-reckoner.

IN SUBTRACTION, we have only to write the quantity to be subtracted after the other quantity, connecting them by the sign —; and the expression thus obtained is the difference, which may or may not be shortened according to circumstances, as is the case with the sum in addition.

We must attend, however, to what is meant by prefixing the

sign -; for, in order that we may fairly express the subtraction, this sign must affect every term which has to be subtracted. An example will perhaps show this more clearly than words.

Let it be required to subtract 5 a + 3b — cd, from 7 a + 3b; and the expression will be

The sign affects all the three terms of the last expression within the parentheses; that is, it makes each of them

1 times itself: - 1 must thus be considered as a factor or multiplier of all the terms; and we already know that the effect of - 1 as a multiplier is to change the signs. Thus, one expression in the example becomes

and, as this expression is perfectly general, for a, b, c, and d may stand for any or for all possible quantities, we have at the same time found this general principle: the subtraction of quantities is expressed by writing them down with their signs changed.

Let us now look back at the expression, and see whether we can shorten it. There are 7 a and - 5 a, which taken together make + 2a, + 5 a and 5 a being = 0. Again, there are +36 and 3b, which together are equal to 0. Therefore 2a + cd is the difference between 7 a +36 and 3 a + 3b-cd. Let us try by addition if the quantity subtracted and the difference make the other quantity, that is, if

5a+3b

cd+2a + cd = 7a + 3b. Looking at the quantity to the left of the sign = we find 3b, and there is 3b in that to the right; thus one term in each agrees. Again, we have 5 a + 2a on the left, and 7 a on the right, all with the sign +, therefore there is in effect 7 a on both sides, and these again agree. Farther, we have cd on the

left, and no cd with any sign on the right, but when we look farther at the left side of the sign we find + c d as well as

-cd; and + cd cd 0. Thus we have in effect 7a + 3b7a+3b, which are not only equal but the same identical quantity.

We may mention that we can never use the sign = unless the quantities to the left of it, taken altogether or as one whole, are exactly equal to those on the right, taken as one whole; and that when we can bring them to an identity of expression without changing their values in respect of each other, we prove this equality.

An expression of this kind is called an EQUATION; and it is the general mode of proceeding in algebra, whether the object be the establishment of a truth, the investigation of a principle, or the finding of an unknown quantity. Indeed, it is the universal formula in the acquiring of all knowledge, of whatever kind it may be; for it is by a perception of the equality either in things themselves or their relations, and by that alone, that we can pass from the known to the unknown. We are on the known bank of the river, and the unknown is the other bank, relation, the foundation and standard of which is equality, is the means by which we are to pass the river. We have the boat in some cases and only "inflated bladders" in others; but in algebra we have the bridge always open, and "no pontage" after we know it.

Even in the most common case of arithmetic, that of the addition of two or more simple numbers, there is an equation involved; and if we wish to understand even that simple case well, it would be better to state this equation at the beginning. The stating of the equation is, as we shall be better able to explain afterwards, nothing more than noting down what we have to do before we begin the doing of it; and everybody