we are to consider the positive terms as quantities to be added, and the negative terms as quantities to be subtracted; and this is the only signification that need be attached to these signs, at present.

It is obvious that positive and negative terms, as just explained, are, in an important sense, opposites; and it will be shown in a future section (182) that quantities sustaining to each other various other relations of opposition or contrariety are distinguished by the plus and minus signs.

43. In order to establish general rules for algebraic operations, it will be necessary in this place to recognize the following principles, consequent upon the peculiar manner of considering quantities in Algebra:

1. A quantity may be considered as having two kinds of value -an absolute or numerical value determined simply by the number of units it contains, and an algebraic value depending on the sign.

2. Two quantities having the same absolute value, but affected with unlike signs, are not algebraically equal. Thus, 5a is not equal to 5a, for the former expression signifies that a is to be added five times, and the latter signifies that a is to be subtracted five times.

3. Two quantities having the same absolute value, but affected with unlike signs, are together equal to zero. Thus, if a denote the absolute value of any quantity, then

ADDITION.

44. Addition is the process of uniting two or more quantities into one equivalent expression called their sum.

45, The term, addition, has a more general meaning in Algebra than in Arithmetic, because the quantities to be added may be either positive or negative.

46. The Arithmetical Sum of two or more quantities is the sum of their absolute values, and has reference simply to the number of units in the quantities added.

47. The Algebraic Sum of two or more quantities is a quantity which, taken with reference to its sign, is equivalent to the given quantities, each taken with reference to its particular sign.

48. To deduce a rule for addition, which will conform to the nature of positive and negative quantities, let us consider the following examples :

1. Add 4a, 3a, and 5a.

Since in these quantities a is taken positively, 4, 3, and 5 times, or 12 times, the algebraic sum required must be + 12a; or simply, 12a. That is,

Since in these quantities a is taken negatively, 4, 3, and 5 times, or 12 times, the algebraic sum required must be -12a. That is,

Hence,

- 4a - За - ба =

12a.

The algebraic sum of two or more similar terms having like signs. is the sum of their absolute values taken with their common sign. 3. Add a and За.

From Ax. 11 we have

a = 4a+3a.

The sum of 7a and - 3a is therefore the same as the sum of 4a, 3a, and 3a. But since 3a and 3a taken together are equal to nothing (43, 3), the required sum must be the remaining term, 4a. That is,

Ya+(-3a) = 4a+3a3a 4a, Ans.

ra =

The sum of a and 3a is therefore the same as the sum of -4a, 3a, and 3a. But, since 3a and + 3a taken together are equal to nothing (43, 3), the required sum must be the remaining term, -4a. That is,

The algebraic sum of two similar terms having unlike signs, is the difference of their absolute values taken with the sign of the greater term.

It may further be observed,

1st. That three or more similar terms, having different signs, may be added, by first finding the sum of the positive terms, and the sum of the negative terms, separately, and then adding these results.

Thus,

За Ба 4a + 2a + Sa = 13a

9a4a.

2d. Dissimilar terms cannot be united into one term by addition, because the quantities have not a common unit. We can therefore only indicate the addition of dissimilar terms, by connecting them by their respective signs. Thus, the sum of a, b, and -c is a + b

C.

3d. It is immaterial in what order the terms of an algebraic quantity are written, the value being the same so long as the signs of the terms remain unchanged. Thus,

a + b - c = b + a − c = a c + b = - c + a + b. For, each of these expressions denotes the sum of the three terms, a, b, and

C.

49. From these principles and illustrations we deduce the following

RULE.-To add similar terms:

I. When the signs are alike, add the coefficients, and prefix the sum, with the given sign, to the common literal part.

II. When the signs are unlike, find the sum of the positive and of the negative coefficients separately, and prefix the difference of the two sums, with the sign of the greater, to the common literal part.

To add polynomials :

I. Write the quantities to be added, placing the similar terms together in separate columns.

II. Add each column, and connect the several results by their respective signs.

10. Add 12a2x, 5a2x, —4a2x, 6a2x, and —10a2x. Ans. 9a2x. 11. Add 4abd, -2abd2, Tabd, abd2, -5abd, 13abd, and abd. 12. Add 2xy-2a2, 3a2+2xy, a2+xy, 4a2-3xy, and 2xy-2a2.

13. Add 8a2x2 - 3xy, 5ax

Ans.

- abd2.

14. Add a2-2ac+cd+b, 3a2—3ac-3cd-2b, 2a2+ac−5cd+

6b, and a2 4ac2cd3b.

Ans. 7a2. Sac

- 5cd + 2b.

15. Add 2a2x2 — 3mx + 4m2d, 3m2d + 5а2x2 4m2d 3a2x2, and 2mx 3a2x2 3m2d.

16. Add 2bx — 12, 3x2—2bx, 5x2—3√x, 3√√x+12, and x2+3. Ans. 9x2+3.

17. Add 1062-3bx2, 2b2x2-b2, 10-2bx2, b2x2-20, and 3bx2 + b2. Ans. 1062 2bx2 + 3b2x2 - 10. 18. Add 9bc3- 18ac2, 15bc3 + ac, 9ac2 — 24bc3, and 9ac2 - 2. Ans. ac 2.

19. Add 6m2 + 2am + 1, 6am - 2m2 + 4, 2m2 Sam +7, Ans. 9m2 + 11.

20. Add 5x-3x3 + 4x2- 2x + 10, 7x+2x3 + 2x2 + 5x+2, and x3 3x. Ans. 12x+6x2 + 12. 21. Add 3x3y2 — 5x3y3 — 5x2y3 — x2y — xy2 + 5xy, 7x2y3 — 4x3y+2x3y2 + 2xy2 + xy, and x3y2 — xy2 — 2x2y3 + 5xay + 2xy.

Ans. 6x3y2+8xy.

22. Add 5a+3√m2—1+4, 7a−√/m2—1—5, 3a—5√/m2—1 Ans. 17a 8, and 2a + 2√ m2 — 1 + 2.

23. What is the sum of 3a2c - 202a + a1c1, 2a2c‡ + 3c2a¥.

5ac‡, and a2c1—5c2a++8a*c*?

Ans. 6a2c-4c2a+4ate.

24. What is the sum of 9a (a - b) — 4m√m — c, 7m√m-c

-6a (a - b), and 12mm - c - 8a (a - b)?

Ans. 15mm-c5a (a - b).

25. What is the sum of a + b + c + d + m, a+b+c+d—m,

50. The Unit of Addition is the letter or quantity whose coefficients are added, in the operation of finding the sum of two or more quantities. Thus, in the example,

3x+2x+4x= 9x

x is the unit of addition. Also, in the example,

5√ a + c + 4√ a + c −3√ a + c = 6√ a + c

a+c is the unit of addition.

√a