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2. a+b+c-(a—b)—(b—c). 3. 4a2-b-(2a-3b+1)+3a. 4. a+2b-(3m—2a+3x2).

Ans. b+2c.

Ans. 4a2+a+26−1.

5. 3a3-2a2+a+1-(2a3—a2-a+5)-(a3—a2-5a-4). 6. a+b-(2a-3b)–(5a+7b)—(−13a+2b).

7. 8a2xy—5bx2y+7cxy2—3y3 — (a2xy+3bx2y—4cxy2+9y3). 8. 7ax2-10a2x2+11a2x-5a-(9ax2+12a2x2-6a2x-9a1).

51. Hence we see that when an expression is inclosed in a parenthesis, the essential sign of a term depends not merely upon the sign which immediately precedes it, but also upon the sign preceding the parenthesis.

Thus m+(+n) is equivalent to m+n,

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The sign immediately preceding n is called the sign of the quantity; the sign preceding the parenthesis may be called the sign of the operation; while the sign resulting from the operation is called the essential sign of the term. We perceive that when the sign of the quantity is the same as the sign of operation, the essential sign of the term is positive; but when the sign of the quantity is different from the sign of operation, the essential sign of the term is negative.

52. Use of Negative Quantities.-The introduction of nega tive quantities into Algebra enables us not only to compare the magnitude, but also to indicate the relation or quality of the objects about which we are reasoning. This peculiarity will be understood from a few examples:

1st. Gain and Loss in Trade.-Suppose a merchant to gain in one year a certain sum, and in the following year to lose a certain sum; we are required to determine what change has taken place in his capital. This may be indicated algebraically by regarding the gains as positive quantities, and the losses as negative quantities. Thus, suppose a merchant, with a capital of 10,000 dollars, loses 3000 dollars, afterward gains 1000

dollars, and then loses again 4000 dollars, the whole may be expressed algebraically thus,

10,000—3000+1000-4000,

which reduces to +4000. The sign of the result indicates that he has now 4000 dollars remaining in his possession. Suppose he further gains 500 dollars, and then loses 7000 dollars. The whole may now be expressed thus,

10,000-3000+1000-4000+500—7000,

which reduces to -2500. The sign of the result indicates that his losses exceed the sum of all his gains and the property originally in his possession; that is, he owes 2500 dollars more than he can pay.

53. 2d. Motion in Contrary Directions.-Suppose a ship to sail alternately northward and southward, and we are required to determine the last position of the ship. This may be indicated algebraically, if we agree to consider motion in one direction as a positive quantity, and motion in the opposite direction as a negative quantity.

Suppose a ship, setting out from the equator, sails northward 50 miles, then southward 30 miles, then northward 10 miles, then southward again 20 miles, and we wish to determine the last position of the ship. If we call the northerly motion +, the whole may be expressed algebraically thus,

50-30+10-20,

which reduces to +10. The positive sign of the result indicates that the ship was north of the equator by 10 miles.

Suppose the same ship sails again 40 miles north, then 70 miles south, the whole may be expressed thus,

50-30+10-20+40-70,

which reduces to -20. The negative sign of the result indicates that the ship was now south of the equator by 20 miles. We have here regarded the northerly motion as +, and the

southerly motion as ; but we might with equal propriety have regarded the northerly motion as -, and the southerly motion as +. It is, however, indispensable that we adhere to the same system throughout, and retain the proper sign of the result, since this sign shows whether the ship was at any time. north or south of the equator.

In the same manner, if we regard westerly motion as +, we must regard easterly motion as -, and vice versa; and, generally, when quantities which are estimated in different directions enter into the same algebraic expression, those which are measured in one direction being treated as +, those which are measured in the opposite direction must be regarded as —.

54. The same principle is applicable to a great variety of examples in Geography, Astronomy, etc. Thus, north latitude is generally indicated by the sign+, and south latitude by the sign West longitude is indicated by the sign +, and east longitude by the sign -.

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Degrees of a thermometer above zero are indicated by the sign+, while degrees below zero are indicated by the sign -. A variation of the magnetic needle to the west is indicated by the sign+, while a variation to the east is indicated by the sign

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The date of an event since the birth of Christ is indicated by the sign+; the date of an event before the birth of Christ, by the sign; and the same distinction is observed in a great variety of cases which occur in the application of the mathematics to practical problems. In all such cases the positive and negative signs enable us not merely to compare the magnitude, but also to indicate the relation of the quantities considered.

CHAPTER IV.

MULTIPLICATION.

55. Multiplication is the operation of repeating one quantity as many times as there are units in another.

The quantity to be multiplied is called the multiplicand; and that by which it is to be multiplied is called the multiplier.

56. When several quantities are to be multiplied together, the result will be the same in whatever order the multiplication is performed.

In order to demonstrate this principle, let unity be repeated five times upon a horizontal line, and let there be formed four such parallel lines, thus,

Then it is plain that the number of units in the table is equal to the five units of the horizontal line repeated as many times as there are units in a vertical column; that is, to the product of 5 by 4. But this sum is also equal to the four units of a vertical line repeated as many times as there are units in a horizontal line; that is, to the product of 4 by 5. Therefore the product of 5 by 4 is equal to the product of 4 by 5. For the same reason, 2×3×4 is equal to 2x4x3, or 4x3x2, or 3×4×2, the product in each case being 24. So, also, if a, b, and c represent any three numbers, we shall have abc equal to bca or cab.

CASE I.

When both the factors are monomials.

57. Suppose it is required to multiply 5a by 46. The product may be indicated thus, 5a x 4b.

But since the order of the factors may be changed without affecting the value of the product, the factors of the same kind may be written together thus,

4× 5ab;

or, simplifying the expression, we have

20ab.

Hence we see that the coefficient of the product is equal to the product of the coefficients of the multiplicand and multiplier.

58. The Law of Exponents.-We have seen, in Art. 16, that when the same letter appears several times as a factor in a product, this is briefly expressed by means of an exponent. Thus, aaa is written a3, the number 3 showing that a enters three times as a factor. Hence, if the same letters are found in two monomials which are to be multiplied together, the expression for the product may be abbreviated by adding the exponents of the same letter. Thus, if we are to multiply a3 by a2, we find a3 equivalent to aaa, and a2 to aa. Therefore the product will be aaaaa, which may be written a3, a result which is obtained by adding together 3 and 2, the exponents of the common letter a. Hence we see that the exponent of any letter in the product is equal to the sum of the exponents of this letter in the multiplicand and multiplier.

59. Hence, for the multiplication of monomials, we have the following

RULE.

Multiply together the coefficients of the two terms for the coefficient of the product.

Write after this all the letters in the two monomials, giving to each letter an exponent equal to the sum of its exponents in the two factors.

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