Positive and Negative Quantities.

(24.) A positive quantity is one which enters additively, and a negative quantity is one which enters subtractively, into a calculation,

A positive quantity is denoted by the sign+, and a negative quan tity by the sign, prefixed to it.

In the polynomial a+b-c, the quantity b is positive, while c is negative.

A quantity with neither + nor be positive.

prefixed to it, is understood to

Thus in the preceding polynomial, the first term a is understood to be +a; for it may be regarded as 0+a.

(25.) A negative quantity has an effect contrary to that of ar equal positive quantity, in the expression, or calculation, into which it

enters.

This is evident with regard to the addition and subtraction of the same quantity; the effect of subtracting it is contrary to that of adding it.

The effect of multiplying by a negative quantity is contrary to that of multiplying by an equal positive quantity.

For example; 3a X2, 3a multiplied by positive 2, denotes that 3a is to be repeated additively, and is equivalent to 3a+3a.

But 3aX-2, 3a multiplied by negative 2, denotes that 3a is to be repeated subtractively, and is equivalent to -3a-3a.

The same thing is true with respect to dividing by a negative, and an equal positive quantity.

The positive and negative signs are sometimes used to distinguish quantities as estimated in contrary directions from a given point or line.

Thus if Motion in one direction be denoted by the sign+, then motion in the contrary direction will be denoted by the sign If North Latitude be considered as positive, South Latitude will be negative.

CHAPTER II.

ADDITION.-SUBTRACTION.-MULTIPLICATION.-DIVISION.

ADDITION.

(26) Algebraic ADDITION consists in finding the simplest expres sion for the value or two or more quantities connected together by the sign+ or -, and this equivalent expression is called the sum of the quantities.

The simple expression for the value of 5a+3a, 5 times a plus 3 times a, is Sa; then 8a is the sum of 5a and 3a.

The simplest expression for the value of 5a-3a, 5 times a minus 3 times a, is 2a; then 2a is the sum of 5a and -3a.

In the second example, observe that adding -3a is equivalent to subtracting 3a; the Adding of a negative quantity being the same as the Subtracting of an equal positive quantity.

Addition of Monomials.

(27.) Similar terms with like signs, are added together by taking the sum of their coefficients, annexing the common literal factor, and prefixing the common sign.

Thus 3ax+2ax is 5ax; just as 3 cents + 2 cents is 5 cents. And -3a-2a is -5a; for 3a to be subtracted and 2a to be subtracted make 5a to be subtracted.

What is the Sum of 4a and 3a? and -7x? Of 5a2, 2a2, and 4a2? Of

Of ax a Бах? Of 2 -26, -36, and —56?

(28.) Two equal similar terms with contrary signs, when added together, mutually cancel each other; that is, their sum is 0.

Thus 3a-3a is 0; that is, 0 is the simplest expression for the value of 3a-3a, is therefore the sum of the two terms.

ar

So 5a+x-5a 15 equal to x; for 5a and --5a cancel each other, r is therefore the sum of the three given terms

What is the Sum of 2ax and pression for the value of 2ax-2ax? -76? Of a, 3x, and -a? Of 3y2, and -5? Of -abc, +a2x, and abc?

-2ax, that is, the simplest ex-
What is the Sum of 76 and
-3y2, and 562 ?
Of ax2, 5,

(29.) Two unequal similar terms with contrary signs, are added together by taking the difference of their coefficients, annexing the common literal factor, and prefixing the sign of the greater term.

And 3a--7a is -4a. For -7a is equal to -3a-4a (27); then 3a-7a is equal to 3a-3a-4a; 3a and -3a cancel each other (28), and leave -4a.

What is the Sum of 5ab and -3ab? Of 9ax2 and -4ax2? Of -a2x and 3a2x?

What is the simplest the Sum of 26 and -56?

Of 3a2c and -a2c? Of 5ac and -5ac?

expression for the value of 26-56, that is, How do you reason in finding that sum?

What is the Sum of 3a2 and -8a2? and how do you reason in finding it? What is the Sum of ax2 and -2ax2 ? and how do you reason in finding it? What is the Sum of -11a2 and 5a2? and how do you reason in finding it?

(30.) The Sum of two or more dissimilar terms can only be indicated by arranging them in a Polynomial, so that each term shall have its given sign prefixed to it.

Thus the Sum of ax and bc2 can only be indicated as ax+bc2; and the Sum of ax and -bc2 as ax-bc2.

What is the Sum of 3a and 56? Of 5a and -b? Of 3a2 and 2x2 ? Of ax and by? Of -36 and -4y?

What is the Sum of 3b, 56, and 6c? 4c and -3y? Of -ab, 3xy, and 2c? 5a, -5, and 56 ?

Of a2, bx, and 3a2? Of
Of 3a2, 46, and 2c? Of

The preceding principles. and the following Rule, provide for all the cases in Algebraic Addition.

RULE I.

(31) For the Addition of Algebraic Quantities.

1. Find the positive and the negative Sum of similar terms, sepa rately, (27), and then add together the similar sums, (28) and (29).

2. Connect the results thus found, and the dissimilar terms, in a Polynomial, prefixing to each term its proper sign. (30).

EXAMPLE.

To add together 4a2+2bc-xy, 2a2-3bc+5y, bc-a2+3xy, and cy-2a2-5bc.

4a2+2bc-xy 2a2-3bc+5y -a2+ bc+3xy

-2a2-5bc+cy

3a2-5bc+2xy+cy+5y

The sum of the positive terms 2a2 and 4a2 is 6a2, and the sum of the similar negative terms -2a2 and -a2 is -3a2, (27). Adding together these two similar sums, 6a2 and −3a2, the result is 3a2, (29).

The sum of the positive terms bc and 2bc is 3bc, and the sum of the similar negative terms -5bc and -3bc is -8bc. Adding together these two similar sums, 3bc and -8bc, the result is -5bc.

The sum of the similar terms 3xy and -xy is 2xy.

The results thus found, namely, 3a2, -5bc, and 2xy, and the dissimilar terms cy and 5y, are connected in a Polynomial.

EXERCISES.

1. Add together ab+3c2-2x, 3ab-c2+5x, 5c2-2ab+y, and 4ab-c2+x. Ans. 6ab+6c2+4x+y. 2. Add together 4a2-6b+3, 3a2+66-7, 5b+a2-bc, and a2bc+10.

3. Add together 262 +be+x, b2+bc+3y.

4 Add together 5a-663+3, 362+2c+4.

5. Add together a2b-5c+xy, a2+3c-13.

Ans. 9a2+5b+6—2bc. 362-2bc+y, bc-362+5x, and Ans. 362+bc+6x+4y. 263-4a-7, 7a-b3+c, and a+ Ans. 9a—5b3+3c+3b2. 4a2b+8c-xy, a2b+c+3xy, and Ans. 6a2b+7c+3xy+a2-13.

7. Add together 2x+ab2 −3y2,

6. Add together a3+2a2-3bc, 2a3-a2+bc, —a3—a2+b2, and 3a3+3a2+362. Ans. 5a3+3a2-2bc+4b2. 2ab2 −3x+y2, —ab2+5x—y2, Ans. 9x+3ab2-3y2-y. b2-3a+3, 2a+362-cx2, and Ans. a +362 — 6.

and 5x+ab2—y.

8. Add together 4a-362+cx2, 262-2a-9.

9. Add together 76—2a2—xy, 5a2−6b+3xy, b−3a2+4, and a2-b-3xy. Ans. b+a2-xy+4. 10. Add together 362-2a3+13, 3a3-262-5, 4ab+7a3-3, and 262-a3+ab. Ans. 362+7a3+5+5ab.

11. Add together ax2-2y+b, 2y+2ax2-3b, 4ax2—y—b, and 26-3ax2+b. Ans. 4ax2-y. 12. Add together 2c2+a2+3bc, 5c2-3a2-2bc, c2+2a2 —bc, and b+bc-3.

13. Add together ab+a2c-5, and ab+a2c+5.

14. Add together 36-2a2x+b, and 3626-3c.

15. Add together 2a2+3-ac, 3ac+4-a2.

together b2c+2-y2,

16. Add and b2 —y2+5.

Ans. Sc2+be+b−3. 3ab-3a2c+7, 2a2c-2ab-3, Ans. 3ab+a2c+4. b2+3a2x-3b, b2-a2x+c, Ans. 6b2-b-2c. 3a2-7+ac, 3ac-5a2+9, and Ans -a2+9+6ac.

y2-3b2c-10, 262c-3+2y2, Ans. 6+y2+b2.

17. Add together a3+bc-c, 2a3-be+c, 3a3+3bc-4c, and a3+bc+c. Ans. 7a3+4bc+c. 18. Add together 63-3a2+2c2, 262 +5a2+c2, b2-c+2, and 562+3a2-2c+3y. Ans. 63+5a2+11c2+852-2c+2+3y.

(32.) When the same Polynomial is enclosed in two or more parentheses, (11), this polynomial enters into a calculation in the same manner as the common factor of similar monomials.

Thus the Sum of 3(a+b) and 5(a+b), is evidently 8(a+b).

19. Add together 2(a-3b)+c, 3(a-3)-3c, -4(a-3b)+c, and 5(a-36)+3c. Ans 6(a-3b)+2c.

20. Add together 3a2+2(a-c2), a2 −3(a−c2), 4a2+5(a-c2) and -7a2+(a-c2). Ans. a2+5(a-c2). 21. Add together 5(a+b−c)+3b2, 3(a+b−c)—2b2, 2(a+b−c) and 362+24, +62. Ans. 10(a+b-c)+5b2+2y..

and

22. Add together 2ab2+3a(b+c2), ab2-2a(b+c2), a(b+c2), ab2-3a(b+c2). Ans. 2ab2-a(b+c2).

23. Add together 5+(c-d+m)+2b, 1+3(c-d+m)—b+2, and 2(c-d+m)+1b. Ans. 8+31(c-d+m)+11⁄2b.