It is immaterial in what order the quantities are set down, if we take care to prefix to each its proper sign.

Generally speaking, however, it is convenient to arrange algebraical quantities in the order in which the letters occur in the alphabet.

When any terms are similar, they may be incorporated, and the general expression for the sum shortened.

1st When similar quantities have the same sign, their sum is found by taking the sum of the coefficients with that sign, and annexing the common letters.

The reason is evident; 5a to be added (Ex. 3), together with 4a to be added, makes 9a to be added; and 3b to be subtracted, together with 76 to be subtracted, is 10b to be subtracted.

24. If similar quantities have different signs, their sum is found by taking the difference of the coefficients with the sign of the greater, and annexing the common letters as before.

In the first part of the operation (Ex. 5) we have 7 times a to add, and 5 times a to take away; therefore upon the whole we have 2a to add. In the latter part, we have 3 times b to add, and 9 times b to take away; therefore we have upon the whole 6 times b to take away; and thus the sum of all the quantities is 2a 6b.

It must be borne in mind that when any quantity, as a, or x, or x2, has no coefficient expressed, the coefficient 1 is understood.

gd. If several similar quantities are to be added together, some with positive and some with negative signs, take the difference between the sum of the positive and the sum of the negative coefficients, prefix the sign of the greater sum, and annex the common letters.

The method of reasoning in this case is the same as in Ex. 5.

In this example, letters are taken to represent coefficients, and the coefficients of like powers of x are enclosed within brackets; for it is evident that p times a3 together with a times is the same as (p+a) times ; also q times to be subtracted together with b times a to be subtracted is the same as (q+b) times to be subtracted; and r times a to be subtracted together with x, or 1x, to be subtracted, is (r+1) times x to be subtracted.

[Exercises B.]

SUBTRACTION.

86. RULE. Subtraction, or the taking away of one quantity from another, is performed by changing the sign of the quantity to . be subtracted, and then adding it to the other by the rules laid down in Art. 85.

Ex. 1. From 2b take cy, and the difference is properly represented by 2b.x-cy; because the - prefixed to cy shews that

it is to be subtracted from the other; and 2bx-cy is the sum of 2bx and cy, (Art. 85).

Ex. 2. Again, from 2bx take-cy, and the difference is 2bx + cy; because 2bx=2b.x + cy-cy, (Art. 83); take away - cy from these equal quantities, and the differences will be equal (Art. 80); that is, the difference between 2br and cy is 2bx + cy, the quantity which arises from adding +cy to 2bx.

In this example the coefficients of like powers of x are bracketed, for reasons similar to those given in Ex. 11, Art. 85.

[Exercises C.]

ADDITION AND SUBTRACTION BY BRACKETS.

In actual practice it seldom happens that either Addition or Subtraction of Algebraical quantities is presented to us as in the Examples, Art. 85, and 86. All the quantities concerned are more commonly in one line, and are so retained through the whole operation, for the sake of convenience.

This arrangement renders necessary the frequent use of Brackets,
Thus Ex. 3. Art. 85 would stand (5a-3b)+(4a-7b)=9a-10b.
(a+b)-(a−b)=2b,

Ex. 3. Art. 86

...............

In the management of Brackets much care is needed, and the following rules are to be observed:

87. RULE I. If any number of quantities, enclosed within Brackets, be preceded by the sign+, the brackets may be struck out, as of no value or signification.

RULE II. If any number of quantities, enclosed within Brackets, be preceded by the sign, the brackets may be struck out, if the signs of all the quantities within the brackets be changed, namely + into-, and into +.

Rule I. is obviously true; for in this case all that is meant is, that a number of quantities are to be added; and it can clearly make no difference whether they be added collectively or separately. Thus a+b+c) is equivalent to a+b+c; for the former signifies that the sum of b and c is to be added to a, which is evidently the sum of a, b, and c. Also a +(b-c) is equivalent to a+b-c; for the former signifies that a quantity is to be added to a less than b by the quantity c; and the latter, that when b has been added to a, c must be subtracted, which is evidently the same thing.

Rule II. is proved thus:

Let a, b, c represent any Algebraical quantities, simple or compound, of which b+c is to be subtracted from a; this will be expressed by a-(b+c).

Now if from a the portion b be taken, the result is a-b; but there is not enough subtracted from a by the quantity c, since b+c was to be subtracted. Therefore c must also be subtracted, which leaves the result a-b-c; that is,

a-(b+c)=a-b-c.

Again, if b-c is to be taken from a, this will be expressed by a-(b-c).

Now if from a the quantity b be taken, the result is a-b, but there has been too much taken away by the quantity c, since b-c only was to be subtracted; therefore c must be added, and the result becomes a-b+c; that is,

a-(b-c)=a-b+c.

The preceding rules apply also to quantities held together by a vinculum, since a vinculum serves the same purpose as brackets. Art. 67.

N.B. It is immaterial whether a vinculum or brackets be used in any case, that being dependent solely upon the will of the writer: but in some cases it is necessary, for distinction's sake, to use both at the same time, or else two kinds of brackets. Thus, to express a times the difference between b and c-d, we must write either a.(b-c-d), or a.{b-(c-d)}. Consequently it requires to be especially noted, that in all cases when the Student meets with (or or [, he must look, whatever may intervene, for the counterpart) or or] respectively; and all that is included within the complete bracket must be treated, irrespective of other brackets or vincula, as the sign which precedes it directs. So that in striking out brackets by Rules I and II, each pair of symbols, as (), {}, [], must be struck out separately, and not all confusedly and at once.

A few examples will make this clearer.

Ex. 1. Perform the addition expressed by (a+b)+(a−b).

(a+b)+(a−b)=a+b+a-b, by Rule I,

=2a.

Ex. 2. Perform the subtraction expressed by (a+b)-(a−b).

(a+b)-(a−b)=a+b−(a−b), by Rule I,

88.

-[-{-(-a)}]=+{−(-a)}, by Rule II,

=-(-a), by Rule I,
=a, by Rule II.

MULTIPLICATION.

The multiplication of simple algebraical quantities must be represented according to the notation pointed out in Art. 62.

Thus axb, or ab, represents the product of a multiplied by b; abc the product of the three quantities a, b, and c; and so on.

It is also indifferent in what order they are placed, axb and b×a being equal.

For 1×a = axl, or 1 taken a times is the same with a taken once; also b taken a times, or bxa, is b times as great as 1 taken a times; and a taken b times, or axb, is b times as great as a taken once; therefore (Art. 81) bxaaxb. Also abc = cab = bcaacb; for, as in the former case, 1xaxb = axbx1; and cxaxb is c times. as great as 1xaxb; also axbxc is c times as great as axbx1; therefore axbxc=cxaxb (Art. 81); and a similar proof may be applied to the other cases.