When a less number is to be divided by a greater, the division is performed by writing the divisor under the dividend, and drawing a line between them. Thus we have, at one view, in a fractional expression, the divisor, dividend and quotient. As a quotient, it expresses the ratio of dividend to divisor. The divisor, or figure below the line, is called the denominator, because it gives name to the parts, or shows into how many parts the unit is divided. If the denominator be 2, the unit is divided into two parts; if 3, three parts; if 4, four parts, &c. The figure above the line is called the numerator, because it numbers the parts, or shows how many parts are contained in the fraction. If the numerator be 2, the fraction contains two parts; if 3, three parts; if 4, four parts, &c.

A fraction is also the result of division, when the dividend is greater than the divisor, but will not contain it without a remainder.

NOTE. The idea connected with the numerator and denominator of a fraction, may be familiarly illustrated thus: Suppose I have a box of 12 oranges, labelled on the outside, thus:

12 Oranges.

The number above the line shows how many things are contained in the box, and the word below the line shows what kind of things they are. If we write, instead of the word Oranges,' the figure 1, it would then show that the hox contained 12 things, or units. Again, should we write the numbers, 2, 3 or 4, instead of the word Oranges,' they would show, not what kind of things were in the box, but into what parts the things were divided. Thus:

Were the question, How many miles in 123 furlongs, (p. 46,) 8 would still be the divisor. As 8 furlongs are equal to 1 mile, (8÷÷8=1) 1 furlong is equal one eighth of a mile, (184) and 2 furlongs to, and 3 furlongs to. Increasing the numerator is only repeating the units to be divided; and as 123 units are to be divided by 8, we may write the whole in the form of a fraction, thus: 123. The question now is, How many miles in 123 eighths of a

QUESTIONS. 1. What are Fractions? 2. What is their origin? 3. If a thing be divided into two equal parts, what are these parts called? 4. Into four equal parts? 5. To what does the dividing figure give name? 6. How are frac tions expressed? 7. What is the figure below the line called? 8. What is the figure above the line called? 9. and what does it show? 10. What does the denominator show in the fraction? 11. The numerator?

mile? The same is required as at the first, viz. the quotient of 123 divided by 8.

Operation. 8)123(15

120

3

We find by trial, that 15 is not the exact quotient of 123 divided by 8, and 16 would be too large; therefore, the true quotient is between those numbers, and must be expressed by a fraction. Having divided 120, the greatest number of units contained in the dividend of which 8 is a factor, we have 3 units of the dividend left, as a remainder, each of which must be divided by 8. 18, and 3÷8-3, which evidently is the fraction required to express the exact quotient; for 120+3 equals the dividend, and 120+3÷8=15+3=153. All questions in Division might be written in the form of a fraction, and to all fractional expressions the language employed in Simple Division might be applied, viz. What is the quotient of divided by ? Hence it ap

pears,

1. That the value of a fraction is the quotient of the numerator divided by the denominator.

2. If the numerator be less than the denominator, the value of the fraction is less than a unit, or 1.

3. If the numerator be equal to the denominator, value of the fraction is equal to a unit, or 1.

the

4. If the numerator be greater than the denominator, the value of the fraction is greater than a unit, or 1.

Fractions are of two kinds-Vulgar, or common, and Decimal. They differ in the form of expression and mode of operation.

In Decimal Fractions, the unit, or integer, is divided into 10, 100, 1000, &c., equal parts; or the denominator is always 1, with as many ciphers annexed as the numerator has places.

In Vulgar Fractions the integer may be divided into any number of parts; and the denominator being always expressed, may be any thing but 1, with or without a cipher or ciphers annexed.

Vulgar fractions are either proper, improper, compound or mixed.

QUESTIONS. 12. Fractions are of how many kinds, and what are they? 13. In what do they differ? 14. How is the unit divided in Decimal Fractions? 15. What is always the denominator? 16. In Vulgar Fractions, how is the Integer divided? 17. What may the denominator be? 18. How are Vulgar Fractions divided?

1. A Proper Fraction is one whose numerator is less than the denominator, as 1, 3, 4, &c.

9

2. An Improper Fraction is one whose numerator is equal to, or greater than the denominator, as §, g, †, &c. 3. A Compound Fraction is a fraction of a fraction, as of of 4, &c.

4. A Mixed Number is a whole number and fraction written together, as 23, 14, 15.5, &c.

5. A Mixed Fraction is one which has a fraction either in its numerator or denominator, or in both of them, thus: 61, 8, 31.

8 91 5

6. A Common Divisor, or Common Measure of two or more numbers, is a number which will divide each of them without a remainder. 4 is the common measure of 12.

7. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide those numbers without a remainder. Thus, 12 is the greatest common measure of 12.

8. Two or more fractions are said to have a common denominator, when the denominator of each is the same.

9. A Common Multiple of two or more numbers, is a number which may be divided by each of those numbers without a remainder.

10. The Least Common Multiple of two or more numbers, is the least number which may be divided by those numbers, without a remainder. Thus 8 is the least common multiple of 8, 4, and 2.

11. A Prime Number is that which can be measured only by itself or a unit.

12. An Aliquot Part of any number, is such a part of it as, being taken a certain number of times, will exactly make that number.

13. A Perfect Number is equal to the sum of all its aliquot parts.

The smallest perfect number is 6, whose aliquot parts are 3, 2, 1; and 3+2+1-6. The next perfect number

QUESTIONS. 19. What is a proper fraction? 20. An improper fraction? 21. A compound fraction? 22. A mixed number? 23. What is a mixed fraction? 24. What is a common divisor, or common measure of two numbers? 25. The greatest common divisor? 26. When are fractions said to have a common denominator? 27. What is meant by a common multiple of two or more numbers? 28. The least common multiple? 29. What is a prime number? 30. What is a perfect number? 31. What is meant by the terms of a fraction? 32. When is a fraction said to be in its lowest terms?

F

is 28, the next 496, and the next 8128. Only ten perfect numbers are yet known.

14. The numerator and denominator of a fraction, taken together, are called the terms of the fraction.

15. A fraction is said to be in its lowest terms, when no number greater than 1, or unity, will divide the terms of the fraction without a remainder.

EXERCISES.

1. If I divide an apple into 8 parts, by what fraction will one of those parts be expressed? 2 of those parts? 3, 4, 5, 6, 7, 8? Answer, §, 7, 8, 1, §, §, 7, §. 2. If 1 be divided by 4, what will be the quotient? if by 6, what? if by 7? if by 8? if by 9? if by 10? if by 11? if by 12? Answer,,, t. I, I, To, T's T2. 3. If 2 be divided by 4, what will be the quotient? if by 6? by 7? by 8? by 13? Answer,,,, 2, f. 4. If a bushel be divided equally among 4 persons, what part of a bushel does each receive?

5. If 2 bushels of apples be divided equally among 4 persons, what will each receive?

6. If a bushel of corn be divided into four parts, what are those parts called? if into 5 into 6? into 7? 8? 9? 10?

7. If I give away 6 quarts of nuts, what part of a peck is it? if 7? if 8 ? if 9?

8. How many of the four last questions are proper fractions? Are any improper, and which are they?

9. How is the quotient of 4 divided by 3 expressed? What is the expression called?

10. If I divide an apple into halves, and give away of , what part of the apple do I give away? What is the expression, of called?

In the foregoing question, the unit is divided into two equal parts, and each part is of the unit. A division is again made of one of these parts into two other equal parts, and each part is of, or of the unit first divided. The expression,, as it respects the unit of which it is a part, is a fraction; (see definition, p. 58,) but as it respects itself, or a subsequent division, it is to be regarded as itself a unit, and may be divided into halves, or any number of parts. A quarter, or of a thing, is a whole quarter; and is made up of as many parts as the thing of which it is a

part. It is, therefore, in relation to a division already made, that an expression is to be regarded as a fraction. As it respects itself, or a subsequent division, it is to be considered a unit. Example. A yard may be divided into 3 equal parts, or feet. A foot, when spoken of in relation to the yard, is ; but of a yard is one foot, and may be divided into 12 equal parts, or inches, and each inch is of a foot, or of of a yard. The inch may be divided into 3 equal parts, or barley corns, and each barley corn is itself a unit of less value, and it is also a fraction of a unit of a higher value; that is, 1 barley corn is of of a yard. That the terms, unit, and fraction, are merely relative, may be seen by the following formula.

yd.

ft.

ft. in.

in.

of

in. bar.

yd. ft.
1+3=1, and 1÷121, and 1÷÷3=3=1.

To reduce a Whole Number to an equivalent fraction, having a given denominator.

1. Reduce 8 to a fraction whose denominator shall be 4. As in 1 unit there are 4 fourths, so in 8 units there must be 8X4 32 fourths, expressed thus: 32; therefore the RULE.

Multiply the whole number by the given denominator, and set the product over the said denominator.

7.

9.

2. Reduce 16 to a fraction whose denominator shall be Answer, 112.

3. Reduce 40 to a fraction whose denominator shall be

Answer, 360.

4. Reduce 129 to a fraction whose denominator shall be 21.

Answer, 2702.

5. Reduce 339 to a fraction whose denominator shall be 39. Answer, 13221.

A whole number may be expressed fractionally, by writing 1 under it for a denominator.

Thus 2 may be written; and read 2 ones;

As the expression, , is equal to 2, and to 3, the value of a number is not affected by writing 1 under it, as a denominator.

QUESTION. HOW is a whole number reduced to an equivalent fraction, having a given denominator?