DIVISIONS OF ARITHMETIC..

106. The science of arithmetic, which treats of numbers, may be divided into three parts:

1st. That which treats of the properties of entire units, called the Arithmetic of Whole Numbers;

2d. That which treats of the parts of unity, called the Arithmetic of Fractions; and

3d. The application of the science of numbers to practical and useful purposes.

A portion of the first part has already been treated under the heads of Numeration, Addition, Subtraction, Multiplication, and Division.

The second part comes next in order, and naturally divides itself into two branches: viz.,

Vulgar or Common Fractions, in which the denominators are any number whatever; and Decimal Fractions, in which the unit is divided according to the scale of tens, hundreds, thousands, &c.

The third part embraces the applications of the principles of entire and fractional numbers to the ordinary transactions and business of life.

The uses and applications of figures are so numerous and so important, that the business of a single day cannot be conducted without them; and hence, no element of education is of greater value than a knowledge of the science of numbers.

106. Of what does the science of arithmetic treat? Into how many parts may it be divided? Of what does the first part treat? Of what does the second part treat? What is the third part? Which part has been treated? Under how many heads? Into how many heads is the second part divided? What are they called? What distinguishes them? What does the third part embrace? knowledge of the science of numbers important?

Is a

OF VULGAR OR COMMON FRACTIONS.

107. The unit 1 represents an entire thing; as 1 apple, 1 chair, 1 pound of tea.

If we suppose one thing, as one apple, or one pound of tea, to be divided into two equal parts, each part is called one half.

If the unit be divided into 3 equal parts, each part is called one third.

If the unit be divided into 4 equal parts, each part is called one fourth.

If the unit be divided into 12 equal parts, each part is called one twelfth ; and when it is divided into any number of equal parts, we have a similar expression for each of the parts.

The equal parts of a thing are expressed thus:

is read one half.

is read one seventh.

The expressions 1, 1, 1, &c., are called fractions.

108. Each fraction is expressed by two numbers; the number which is written above the line is called the numerator; and the one below it is called the denominator, because it gives a denomination or name to the fraction. For example, in the fraction, 1 is the numerator, and

107. What does the unit 1 represent? If we divide it into two equal parts, what is each part called? If it be divided into three equal parts, what is each part? Into 4, 5, 6, &c., parts? What are such expressions called?

108. By how many numbers is each fraction expressed? What is the one above the line called? The one below the line? What does the denominator show? What does the numerator show? In the three-eighths, which is the numerator? Which the denominator? Into how many parts is the unit divided? How many parts are expressed? In the fraction nine-twentieths, into how many parts is the unit divided? How many parts are expressed?

2 the denominator.

In the fraction, 1 is the numera

tor, and 3 the denominator.

The denominator of every fraction shows into how many equal parts the unit, or single thing, is divided. For example, in the fraction, the unit is divided into 2 equal parts; in the fraction it is divided into 3 equal parts; in the fraction, it is divided into 4 equal parts, &c. In each of the fractions one of the equal parts is expressed. But suppose it were required to express 2 of the equal parts, as 2 halves, 2 thirds, 2 fourths, &c.

If it were required to express three of the equal parts, we should place 3 in the numerator; and generally, The numerator shows how many of the equal parts are expressed in the fraction.

For example, three eighths are written,

109. When the numerator and denominator are equal, the numerator expresses all the equal parts into which the unit has been divided: therefore, the value of the fraction is equal to 1.

But if we suppose a second unit, of the same kind, to be divided into the same number of equal parts, those parts may also be expressed in the same fraction with the parts of the first unit.

109. When the numerator and denominator are equal, what is the value of the fraction? What is the value of the fraction threehalves? Of seven-fourths? Of sixteen-fifths? Of eighteen-sixths? Of twenty-five sevenths? Repeat the six principles. Write the fraction nineteen-fortieths :-also, 60 fourteenths-18 fiftieths-16 twentieths-17 thirtieths-41 one thousandths-69 ten thousandths -85 millionths?-106 fifths.

Thus,

is read three halves.

18

2,5

seven fourths.

sixteen fifths.

eighteen sixths.

twenty-five sevenths.

The denominator of the first fraction shows that a unit has been divided into 2 equal parts, and the numerator expresses that three such parts are taken. Now, two of the parts make up one unit, and the remaining part comes from the 2d unit: hence, the value of the fraction is 1; that is, one and one half.

The denominator of the second fraction shows that a unit has been divided into four equal parts, and the numerator expresses that 7 such parts are taken. Four of the 7 parts come from one unit, and the remaining 3 from a second unit: the value of the fraction is therefore equal to 13; that is, to one and three fourths. In the third fraction, the unit has been divided into 5 equal parts, and 16 such parts are taken. Now, since each unit has been divided into 5 equal parts, 15 of the 16 parts make 3 units, and the remaining part is 1 part of a fourth unit. Therefore, the value of the fraction is 3; that is, three and one fifth. The value of the fourth fraction is three, and of the fifth, three and four-sevenths. From what has been said, we conclude:

1st. That a fraction is the expression of one or more parts of unity.

2d. That the denominator of a fraction shows into how many equal parts the unit or single thing has been divided, and the numerator expresses how many such parts are taken in the fraction.

3d. That the value of every fraction is equal to the quotient arising from dividing the numerator by the denom

inator.

4th. When the numerator is less than the denominator, the value of the fraction is less than 1.

5th. When the numerator is equal to the denominator, the value of the fraction is equal to 1.

6th. When the numerator is greater than the denomina. tor, the value of the fraction is greater than 1.

MENTAL EXERCISES IN COMMON FRACTIONS.

1. If a unit be divided into two equal parts, what is each part called? How do you express one of the parts? 2. If a unit be divided into three equal parts, what is each part called? How do you express one of the parts? How do you express two of them? How do you express three of them?

3. If a unit be divided into four equal parts, how do you express one of the parts? Two of the parts? Three of the parts? Four of the parts?

4. How many halves are there in one thing? How many fourths or quarters are there? How much greater is a half than a quarter?

5. If a unit be divided into five equal parts, what is each part called? How do you express three of the parts? Four of them? Five of them?

6. If a unit be divided into six equal parts, what is each part called? How do you express one-sixth? How do you express two of the parts? Three of them? How do you express six of them?

7. How many thirds are there in a unit? How many sixths are there? How much greater then is one-third

than one-sixth?

8. If a unit be divided into seven equal parts, what is each part called? How do you express one part? Two parts? Four parts? Six parts? Seven parts?

9. If a unit be divided into eight equal parts, what is each part called? How do you express four of the parts? Five of them? Six of them? Seven of them? Eight of them?

10. How many fourths or quarters are there in a unit ? How many eighths are there? How much greater, then, is a quarter than an eighth? How many eighths are equal to two quarters? How many to three quarters?

11. If a unit be divided into nine equal parts, what is