SECTION IV.

General Rule for Division.

From the foregoing examples and observations, we derive the following general RULE for Division in Algebra, when both the divisor and dividend are compound quantities :

Divide the first term of the dividend by the first term of the divisor, for the first term of the quotient.

Multiply the whole divisor by this term, and subtract the product from the dividend.

Divide the first term of the remainder by the first term of the divisor, for the second term of the quotient. Multiply the whole divisor by this second term, and subtract the product from the remainder.

Continue this series of operations as long as the nature of the question may require.

1. Divide a a + a b + ac + 5a +56+5c by a + b + c.

2. Divide a a - 6a + ab-ac +6c-bcby

a-c.

3. Divide bb b + ccc by b + c.

4. Divide 6 mmmx-3mmxy-2mmx

+mxy+2m x - x y by 2 m — у.

5. Divide x x x + x x + x x y + xy + 3x z + 3 z by x + 1.

..

6. Divide a + b — c — ах - bx + c x by a +

- C.

C

7. Divide a a a - Заах - 3 axx + xxx by a + x.

8. Divide a a a - 3a ay + 3ayy - yyy by

a - y.

9. Divide 4 a aa+3acg+4aac+3ccg -4aag-3cgg bya+c-g.

10. Divide 5 mmn-2myz-5mnn+2nyz +5mnz-2yzzbym - n + z.

11. Divide x x x - 3 x x y + 4 x yy - 4 y YY by x x - 2xy + y y.

xx-2xy+yy) xxx-3xxy+4xyy-4 y y y(x-y+zx-2xy+yy

xxx-2xxy+xyy

As there is a remainder in this example, it is annexed to the quotient in the form of a fraction.

12. Divide a x + bx + ay + by + z by a + b. 13. Divide c d -ch+de-eh+x by d-h. 14. Divide x x xху — хх уу + 5 yyy by x - у. 15. Divide 3 a bbb+2aabbc-abb + b bѣс

+bbx-3aab-2aaac + aa-abc - ах, by 3ab+2aac-a + bc + х.

16. Divide 18 aab+3aасс — 3 ах х + 3abbc-6abb — a bcc + b xx - bb b c + ax + c, by 6 a b + acc - xx + bbc.

17. Divide - 16 aamy - 3aacy + 64 am m + 12 a cm by 16 am + 3ac.

18. Divide 12 a a b - 8a - 144 ab + 96 by 12.

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CHAPTER VI.

FRACTIONS.

SECTION Ι.

Introduction and Definitions.

1. Let it be required to distribute a dollars equally among b poor persons. What will be the share of ANS.

each?

The number of dollars must be divided by the number of persons; but, as the division cannot be actually performed, all that can be done is, to represent it as above. So, too, if we suppose a = 3, and b = 5, we must indicate the answer in a similar manner; thus, .

3

These expressions,,, and others like them, are called Fractions, from a Latin word, which signifies broken, because the value of a fraction is always expressed in parts of a whole one. If, for instance, we cut an apple into 5 equal parts, and give 3 of those parts to a boy, he will have 3 of the apple, which expression is read three fifths. Other fractions are read in a similar manner; as, 1⁄2, one half; 3, two thirds; 1, three fourths; &, five sixths; 1, seven eighths, &c.

The number below the line, which shows into how many parts the unit or quantity is divided, is called the Denominator.

The number above the line, which shows how many of the parts are taken, is called the Numerator.

merators are a, m, 6, 19, a + b, and 3; and the denominators are b, n, 7, 4, x

y, and z.

A Proper fraction is one whose value is less than a unit; that is, whose numerator is less than its denominator; as,

2

3

7

a

hx

5' 12' a+b'x + y'

&c.

An Improper fraction is either equal to or greater 14 49 α ab ab+b than a unit; as, 엔신인

2

α

&c. It

is evident that the division here represented can be performed, either wholly or in part.

b

Expressions consisting of a whole number and a fraction, are called Mixed numbers; as, 24, and 6+; and 2 and b are called integers, or integral quantities. It often happens, both in, Arithmetic and Algebra, that there is a remainder after division, which should be written above the divisor, and annexed to the quotient in the form of a fraction. Hence the origin of mixed numbers. [See Chap. V. Sec. III. and IV.]

Since fractions always imply division, any quotient may be expressed in the form of a fraction, the dividend being the numerator, and the divisor the denominator.

Express the answers of the following examples in the form of fractions:

2. Divide 476 by 19.

476

ANS.

•

19

3. Divide 6 a b c by d.

4. Divide 8 a + b - x byc - d.

5. Divide 17 xy - m + 7 by x + 10.

6. Divide a - m + 7-5 y by 10 b - x + 14.

As the value of any quantity is not altered when it is divided by a unit, we can convert an integer into a fraction by making 1 the denominator. Convert the following quantities into fractions: 7.) 8 =

8 1

10.) 24

1

8.) a = . 11.) a - b.

9.) a + b =

a+b 1

12.) a + b -mn.

The numerator of a fraction being a dividend, and the denominator a divisor, it follows, that when these are equal to each other, as, the value of the fraction is 1; for 2 divided by 2 gives 1. Therefore

1 2 3

4

5

a

, and, are all equal to each

other, the value of each fraction being unity, or 1. Hence, it is evident, that if both the numerator and denominator be either multiplied or divided by the same number or letter, the value of the fraction is not changed. On this principle are founded the rules for bringing fractions to a common denominator, and for reducing them to their least terms.

13. Express 6 in the form of a fraction, having 4

for its denominator.

24

ANS..

4

It is evident that the numerator must be 4 times