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terms as quantities taken additively, and the negative terms as quantities taken subtractively; 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 taken additively five times, and the latter signifies that a is taken subtractively 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

tama=0
+2a-2a=0
+30—3a=0, etc.

ADDITION.

44. Addition, in Algebra, 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 additively, 4, 3, and 5 times, or 12 times, the algebraic sum required must be +12a ; or simply, 12a. That is,

4a+3a+5a=12a 2. Add -40, -3a, and --5a.

Since in these qantities a is taken subtractively, 4, 3, and 5 times, or 12 times, the algebraic sum required must be —12a. That is,

-40-3a-5a=-12a Hence,

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 7a and -3a. From Ax. 11 we have

7a-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

7a+(-3a)=4a+3a43a=4a, Ans.

4. Add –7a and 3a. From Ax. 11,

-7a=-40-3a 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,

-7a+3a=-4a_3a+3a=_4a, Ans. Hence,

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,

3a-5a-4a+2a+8=130–9a=4a 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

a+b-C 3d. It is indifferent in what order the terms of an algebraio quantity are written, the value being the same so long as the signs of the terms remain unchanged. Thus,

atbrc=b+amc=a-c+b=-c+a+b For, each of these expressions denotes the sum of the three terms,

-cis

a, b, and

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.

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(8.)

(9.)
4(0—2a)- m + 4 5(a—)+3Va—.+ 5
3(0—2a)+4m- 8 4(a—«*)—2V ama+ 8
8(0—2a)—3m+12 2(a^2?)—8V a—~—12
12(0–2a)+ m—16 (a—xo)+2V a—x— 1
11(0—2a)+ m~ 8

10(a—x*)---5V ax 10. Add 12aRx, 5aRx, –4aʻx, 6aʻr, and -10aox.

Ans. 9u’x. 11. Add 4abd", ---2abd, Tabd", abd, -5abd", -13abd", and 7abda.

Ans. -aba. 12. Add 2xy2a”, 3a2+2xy, a*+xy, 4a—3xy, and 2.ry-2014

Ans. 4a’+4xy. 13. Add Saʻa?—3xy, 5ax—5xy, 9xy5ax, 2a*x*+ry, and баr3ry. .

Ans. 10aox'—xy+5ax.

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14. Add a?—2ac-6cd+6, 34*—3ac3cd26, 2a*+ac—5cd+ 66, and a’_4ac72cd_36.

Ans. 7a_8ac5cd+20. 15. Add 20'x'-3m.x+4m'd, 3m®+5a2.2-5mm, 6m-4mod --3ac", and 2mx-3._-3mod.

Ans, a?.,.?. 16. Add 26x—12, 3x*—26x, 5.x*—37x, 37 x+12, and aʻ+3.

Ans. 9x”+3. 17. Add 1063_363*, 20*x*—6°, 10—2bx, box?—20, and 36.ro + 39.

Ans. 1002—26x2 +30°.c—10. 18. Add 96c*—18ac-, 15bc%+ac, 9ac-—24bc®, and 9ac-—2.

Ans. ac~2. 19. Add 6m' +2am+1, 6am—2m’+4, 2m?_8am+7, and 3mo -1.

Ans. 9m2 +11. 20. Add 5x4–3x +4x® -2.+10, 7.x* +2.2"+2.0° +5x+2, and 29_3x.

Ans. 12.*+6.co+12. 21. Add 3x*y'_5x'y'--'y-xy' +5xy, 7c'y--4x+y+2x'y" + 2xy +xy, and x*y'—xys- 2.*y* +5x*y+2.y.

Ans. 6x®yo +8xy. 22. Add 5a +31 m-1+4, 7a-V mo--1-5, 30–51 m--1 --8, and 2a+2V m-1+2.

Ans, 17a–V m—1–7. 23. What is the sum of 3ack_2c*a+ac), 2a*c*+30 5ałcą, and a'c*_5coa:+8a2c)? Ans. 6a'c1_4c*a+4ac1.

24. What is the sum of 9a(a6–4mV -C, 6a(a,b), and 12 mV m-c-8a(ab)?

Ans. 15mV m-(450(a-). 25. What is the sum of a+b+c+d+m, a+b+c+d-m, at 6+-d-m, a+b-md-m, and a-bc-d-m?

Ans. 5a+30+c-d-3m. 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+2.c+4.29.C the letter x is the unit of addition. Also, in the example,

51 a+c+41 a+c-31 a+c=6vato the quantity, Va+c, is the unit of addition

7mVm-e

m

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