Q. What does tne 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?

§ 43. Each fraction is made up of 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 2 the denominator. In the fraction, 1 is the numerator, and 3 the denominator.

The denominator in 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.

We should then write,

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,

Q. Of how many numbers is each fraction made up? 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?

§ 44. 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. Thus,

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 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 denominator. 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 denominator, the value of the fraction is greater than 1.

Q. When the numerator and denominator are equal, what is the value of the fraction? What is the value of the fraction three halves? 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 twentieths17 thirtieths-41 one thousandths-69 ten thousandths-85 millionths -106 fifths.

OF DENOMINATE NUMBERS.

§ 45. Simple numbers express a collection of units of the same kind, without expressing the particular value of the unit. For example, 40 and 55 are simple numbers, and the unit is 1, but it is not expressed whether the unit is 1 apple, 1 pound, or 1 horse.

A DENOMINATE number expresses the kind of unit which is considered. For example, 6 dollars is a denominate number, the unit 1 dollar being denominated, or named.

When two numbers have the same unit, they are said to be of the same denomination: and when two numbers have different units, they are said to be of different denominations.

For example, 10 dollars and 12 dollars are of the same denomination; but, 8 dollars and 20 cents, express numbers of different denominations, the unit of 8 dollars being 1 dollar, and of 20 cents, 1 cent.

Several numbers of different denominations are often connected together, forming a whole, as 3 dollars 15 cents. Q. What do simple numbers express? What is a denominate number? What is the unit of 6 dollars? When two numbers have the same unit, what do you say of them? When they have different units? Are 6 dollars and 4 dollars of the same denomination? Are

4 dollars and 4 cents? What is the unit of each? Are several numbers of different denominations often connected together? Give an example.

OF FEDERAL MONEY.

§ 46. Federal money is the currency_of_the_United States. Its denominations, or names, are Eagles, Dollars, Dimes, Cents, and Mills.

The coins of the United States are of gold, silver, and copper, and are of the following denominations.

1. Gold-Eagle, half-eagle, quarter-eagle.

2. Silver-Dollar, half-dollar, quarter-dollar, dime, halfdime.

3. Copper-Cent, half-cent.

If a given quantity of gold or silver be divided into 24 equal parts, each part is called a carat. If any number of carats be mixed with so many equal carats of a less valuable metal, that there be 24 carats in the mixture, then the compound is said to be as many carats fine as it contains carats of the more precious metal, and to contain as much alloy as it contains carats of the baser.

For example, if 20 carats of gold be mixed with 4 of silver, the mixture is called gold of 20 carats fine, and 4 parts alloy. The standard of the gold coin in the United States, is 22 carats of gold, 1 of silver, and I of copper.

The standard for silver coins is 1489 parts of pure silver, to 179 of pure copper.

The copper coins are of pure copper.

Q. What is the currency of the United States? What are its denominations? What are the coins of the United States? Which gold? Which silver? Which copper? What do you understand by gold 20 carats fine? What is the standard of the gold coin? What of the silver coin? What of the copper?

TABLE OF FEDERAL MONEY.

10 Mills marked (m) make 1 Cent, marked ct.

1 Dime,
1 Dollar,
1 Eagle,

In this table, 10 units of either denomination make one unit of the next higher denomination, and this is the same way that simple numbers increase from the right to the left. Therefore,

The denominations of federal money here expressed may be added, subtracted, multiplied, and divided, by the same rules that have already been given for simple numbers. From the table it appears,

1st, That cents may be changed into mills by annexing a cipher.

Thus, 8 cents are equal to 80 mills.

2d. That dollars may be changed into cents by annexing two ciphers, and into mills by annexing three.

For example, 12 dollars are equal to 1200 cents, or to 12000 mills. The reason of these rules is evident, since 10 mills make a cent, 100 cents a dollar, and 1000 mills a dollar.

Q. Repeat the table. How many units of either denomination make one of the next higher? How do simple numbers increase from the right to the left? How may Federal Money be added, subtracted, multiplied, and divided? How may cents be changed into mills? How may dollars be changed into cents? How into mills? To how many cents are 12 dollars equal? To how many mills are they equal? How many cents in 4 dollars? How many in 6 dollars? How many mills in 9 dollars? How many mills in 5 dollars? How many cents in 3 dollars? In 8 dollars? In 7 dollars?

NUMERATION TABLE FOR FEDERAL MONEY.

Cents.

Mills.

57, is read, 5 cents and 7 mills, or 57 mills.

1 6 4, 62, 1 2 0, 127, 62 3, 1,

8 9 4 0, 0 4