COMMON FRACTIONS.

123. AN INTEGRAL, or whole number, is the unit 1, or a collection of such units.

NOTE.-All integral numbers are formed by the continual addition of 1: as, 1+1 = 2, 2 + 1 = 3, &c.

124. A UNIT is a single thing; as, an apple, a chair, a hat, &c.; and is denoted by 1.

If a unit be divided into two equal parts, each part is called, one-half.

If a unit be divided into three equal parts, each part is called, one-third.

If a unit be divided into four equal parts, each part is called, one-fourth.

If a unit be divided into twelve equal parts, each part is called, one-twelfth; and if it be divided into any number of equal parts, we have a like expression for each part.

The, is an entire half; the , an entire third; the , an entire fourth; and the same for each of the other equal parts; hence, each equal part is an entire thing, and is called a frac tional unit.

123. What is an integral, or whole number? numbers formed?

How are integral

124. What is a unit? By what is it denoted? What is each part called when the unit 1 is divided into two equal parts? When it is divided into three? Into four? Into five? Into twelve?

125. THE UNIT OF A FRACTION is the single thing that is divided into equal parts

126. THE FRACTIONAL UNIT is one of the equal parts of the unit that is divided.

127. A FRACTION is a fractional unit, or a collection of such units.

NOTE.-In every fraction, let the pupil distinguish carefully between the unit of the fraction and the fractional unit. The first is the whole thing from which the fraction is derived; the second, one of the equal parts into which that thing is divided.

128. Every whole number, except 1, has a fractional unit corresponding to it: thus the numbers,

2, 3, 4, 5, 6, 7, 8, 9, 10, &c.,

have, corresponding to them, the fractional units,

1, 1, 1, 1, 1, 1, 1, 1, 1, &c.

67

129. Expressing Fractions.

Each fractional unit may, like the unit 1, become the base of a collection: thus, suppose it were required to express 2 of each of the fractional units, we should then write

which is read 2 halves = × 2.

22 23 24 25

If it were required to express 3 of each of the fractional units, we should write

Hence, if we suppose a second unit to be divided into the same number of equal parts, such parts may be expressed in the same collection with the parts of the first: thus,

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

Thus,

and is read, 3 ones.

But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 8 ones to 8; hence, the value of a number is not changed by placing 1 under it for a denominator: Hence, we see,

1. That Fractions are expressed by two numbers, one written above the other, with a line between them.

2. That every fraction may be divided into two factors, one of which is the fractional unit, and the other the number denoting how many times the fractional unit is taken.

130. THE DENOMINATOR is the number written below the line. It shows into how many equal parts the unit of the fraction is divided.

125. What is the unit of a fraction ?-126. What is the fractional unit?-127. What is a fraction ?-128. What fractional unit corresponds to the whole number 5? What to 6? What to 14?

129. May a fractional unit become the base of a collection? How are fractions expressed? Into how many factors may every fraction be divided? What are they?

130. What is the denominator of a fraction?

131. THE NUMERATOR is the number written above the line. It shows how many fractional units are taken.

132. THE TERMS of a fraction are the numerator and denominator taken together: hence, every fraction has two terms.

133. THE VALUE of a fraction is the quotient of the numer ator divided by the denominator.

134. THE ANALYSIS of a fraction is the naming of its unitits fractional unit-and the number of fractional units taken.

135. Analysis of Fractions.

How is the fraction

to be interpreted?

1. The unit of the fraction is 1.

2. The unit of the fraction is divided into 8 equal parts; hence, the fractional unit is one-eighth.

3. Seven fractional units are taken. In the fraction, the base of the collection of fractional units is, but it is not the primary base. For, is one-eighth of the unit 1; hence, the primary base of every fraction is the unit 1.

The expression may also be interpreted as the quotient of 7 divided by 8. In the latter case, the thing divided is the number 7; in the former, it was the number 1. The value in both cases is the same; for, seven times one-eighth of 1, is equal to one of the 8 equals of 7. Hence, a fractional expression has the same form as an unexecuted division.

136. Principles and Properties of Fractions.

1. A fraction is a fractional unit, or a collection of such units. 2. The denominator shows into how many equal parts the unit of the fraction is divided.

131. What is the numerator of a fraction?-132. What are the terms of a fraction? How many terms has every fraction ?—133. What is the value of a fraction?-134. What is the analysis of a fraction?

135. Analyze the fraction. What is the base of the collection? What is the primary base? What else does express?

136. Explain the principles and properties of Fractions.

3. The numerator shows how many fractional units are taken. 4. The value of every fraction is equal to the numerator divided by the denominator.

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

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

7. When the numerator is greater than the denominator, the value of the fraction is greater than 1.

137. Writing and Reading Fractions.

1. Read and analyze the following fractions:

흡, 구, 흙, 15, 21, 16,

18

2. Write 15 of the 19 equal parts of 1. Also, 37 of the 49 equal parts of 1. Write 24-thirtieths.

3. If the unit of the fraction is 1, and the fractional unit one-fortieth, express 27 fractional units. Also, 95. Also, 106. Also, 87. Also, 41.

4. If the unit of the fraction is 1, and the fractional unit one-68th, express 45 fractional units. Also, 56. Also, 85. Also, 95. Also, 37.

5. If the unit of the fraction is 1, and the fractional unit one90th, express 9 fractional units. Also, 87. Also, 75. Also, 65.

138. Six Kinds of Fractions.

1. A PROPER FRACTION is one whose numerator is less than the denominator.

The following are proper fractions:

137. Give an example in writing, reading, and analyzing fractions 138. How many kinds of fractions are there? Name and describe each.