A Treatise on Algebra

CHAPTER III.

ADDITION. SUBTRACTION. BRACKETS.

ADDITION.

46. WE have already seen that any term is added by writing it down, with its sign unchanged, after the expression to which it is to be added; and we have also seen that to add any expression as a whole gives the same result as to add all its terms in succession. We therefore have the following rule:-to add two or more algebraical expressions, write down all the terms in succession with their signs unchanged.

Thus the sum of a -2b+3c and -4d-5e+6f is

a-2b+3c-4d5e+6 f.

47. If some of the terms which are to be added are 'like' terms, the result can, and must, be simplified before the process is considered to be complete.

Now two 'like' terms which have the same sign are added by taking the arithmetical sum of their numerical coefficients with the common sign, and affixing the common letters.

For example, to add 2a and 5a in succession gives the same result, whatever a may be, as to add 7a; that is, +2a+5a=+7a. Also, to subtract 2a and 5a in succession gives the same result as to subtract 7a; that is, 2a-5a7a.

Also two 'like' terms whose signs are different are added by taking the arithmetical difference of their numerical coefficients with the sign of the greater, and affixing the common letters.

also

For example,

+5a-3a+2a+3a − 3a: = +2a,

+3a-5a+3a-3a - 2a = -2a.

Thus, when there are several 'like' terms some of which are positive and some negative, they can all be reduced to one term.

The sum is 3a2 - 5ab+7b2 - 4a2 - 2ab+3b2+2a2 + 5ab - 8b2.

The terms 3a2, - 4a2, and +2a2 can be combined mentally; and we have a2. Similarly we have - 2ab and +2b2.

Thus the required sum is a2 - 2ab+2b2.

SUBTRACTION.

48. We have already seen that any term may be subtracted by writing it down, with its sign changed, after the expression from which it is to be subtracted; and we have also seen that to subtract any expression as a whole gives the same result as to subtract its terms in succession. We therefore have the following rule: To subtract any algebraical expression, write down its terms in succession with all the signs changed.

Thus, if a 26+ 3c be subtracted from 2a-3b-4c, the result will be 2a-3b-4c-a+2b3ca-b-7c.

49. The expression which is to be subtracted is sometimes placed under that from which it is to be taken, 'like' terms being for convenience placed under one another;

and the signs of the lower line are changed mentally before combining the 'like' terms.

Thus the previous example would be written down as under:

As another example, if we have to subtract 3ab-5ac+c2 from a2-5ab2ac - 262, the process is written

50. To indicate that an expression is to be added as a whole, it is put in a bracket with the + sign prefixed. But, as we have seen in Art. 46, to add any algebraical expression we have only to write down the terms in succession with their signs unchanged.

Hence, when a bracket is preceded by a + sign, the bracket may be omitted.

Thus +(2a5b+7c) = + 2a − 5b + 7c.

Hence also, any number of terms of an expression may be enclosed in brackets with the sign + placed before each bracket. Thus

3a-2b+4c-d+e-f=3a-2b+ (4c-de-f)= 3a +(-2b+4c) — d + (e−ƒ).

When the sign of the first term in a bracket is + it is generally omitted for shortness, as in the preceding example.

51. To indicate that an expression is to be subtracted as a whole, it is put in a bracket with the sign prefixed. But, as we have seen in Art. 48, to subtract any algebraical expression we have only to write down the terms in succession with all their signs changed.

Hence, when a bracket is preceded by a sign, the bracket may be omitted, provided that the signs of all the terms within the bracket are changed. Thus

a − (2b − c + d) = a -2b+c - d.

Hence also, any number of terms of an expression may be enclosed in a bracket with the sign - prefixed, provided that the signs of all the terms which are placed in the bracket are changed. Thus

a-2b+3c-d=a-(2b - 3c + d) = a -2b-(-3c+d).

52. Sometimes brackets are put within brackets: in this case the different brackets must be of different shapes to prevent confusion.

Thus a-[2b- {3c − (2d − e)}]; which means that we are to subtract from 26 the whole quantity within the bracket marked {}, and then subtract the result from a; and, to find the quantity within the bracket marked {}, we must subtract e from 2d, and then subtract the result from 3c.

When there are several pairs of brackets they may be removed one at a time by the rules of Arts. 50 and 51.