PRIME NUMBERS.

132. No general method of determining prime numbers, beyond a certain limit, has yet been discovered, although much time has been spent in the investigation.

133. We give the following practical method, which consists in writing a series of numbers, and sifting out those which are composite.

METHOD. Since the even numbers after 2 are composite, we write the series of odd numbers; thus,

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41,

43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81,

Now, commencing at 3, since every third term is divisible by 3, every third number is composite, which we indicate by putting the figure 3 over it.

Commencing at 5, every fifth number is divisible by 5, and is therefore composite, hence we place a figure 5 over every fifth number. Proceed in the same manner with 7, and the numbers unmarked will be the prime numbers up to 100.

This method was discovered by Eratosthenes, a Greek mathematician, He inscribed the series of odd numbers on parchment, and then cutting out the composite numbers, his parchment with its holes resembled a sieve; hence the method has been called Eratosthenes Sieve.

NOTE. This page will be of interest to the student to read, but is not tc be recited.

INTRODUCTION TO FRACTIONS.

MENTAL EXERCISES.

1. If an apple is divided into two equal parts, what is one of these parts called?

2. What are two of these parts called? How many halves in anything?

3. What is of 6? of 4? of 12? of 16? of 10? of 18? of 20? of 24? of 28? of 36?

4. If I divide an apple into 3 equal parts, what is one of these parts called?

5. What are 2 and 3 of these parts called? How many thirds in anything?

6. The number of equal parts into which a unit may be divided is represented by a figure below the line; thus z represents halves; thirds; fourths, etc.

7. The number of fractional parts taken may be represented by a figure above the line; thus, represents 2 thirds; †, 3 fourths; §, 5 sixths, etc.

8. What is of 6? of 9? of 12? of 18? of 15? of 21? of 27? What are of 12? of 15? of 21? of 18? of 24? of 33?

9. If I divide an apple into 4 equal parts, what is one of these parts called? If I divide in 5, or 6, etc., equal parts ?

10. How many fourths make a whole? How many fifths? Sixths? Sevenths? Eighths? Ninths? Tenths?

11. What is of 12? of 20? of 40?

of 35?

of 24? of 16? of 30? of 281

12. What is of 20? of 15? of 12? of 24? § of 27? † of 30! of 22

of 64?

13. If a yard of muslin cost 24 cents, what will of a yard cost? What will of a yard cost?

14. Henry's age is 36 years, and his wife's age is as much; what is his wife's age?

15. If 5 melons cost 60 cents, what will 7 melons cost at the same rate?

16. What will 4 yards of satin cost at the rate of $6 for of a yard?

17. What must I pay for three-fourths of a ton of hay if five-sixths of a ton cost $20?

18. What will of a ton of coal cost at the rate of $4.50 for of a ton?

SECTION IV,

COMMON FRACTIONS.

134. A Fraction is a number of the equal parts of a unit.

135. Fractions are divided into two classes; common fractions and decimal fractions.

136. A Common Fraction is one in which the unit is divided into any number of equal parts.

137. A Decimal Fraction is one in which the unit is divided into tenths, hundredths, etc.

138. A Common Fraction is expressed by two numbers, one written above the other, with a short line between them; thus, expresses 3 fourths.

139. The Denominator of a fraction denotes the number of equal parts into which the unit is divided.

140. The Numerator of a fraction denotes the number of equal parts which are taken

141. The Numerator and Denominator are called the Terms of the fraction. The numerator is written above the line, and the denominator below it.

142. Common Fractions consist of three principal classes; namely, Simple, Compound, and Complex.

143. A Simple Fraction is a fraction having a single integral numerator and denominator; as,, etc.

144. A Proper Fraction is a simple fraction whose value is less than a unit; as 3, 4.

145. An Improper Fraction is a simple fraction whose value is equal to or greater than a unit; as §, 7, 12, etc. 146. A Compound Fraction is a fraction of a fraction; ar of, of § of 7, etc.

147. A Complex Fraction is one whose numerator, or

denominator, or both, are fractional; as

5 of

of g

148. A Mixed Number consists of an integer and a fraction; as, 4, 5, etc.

149. The Reciprocal of a number is a unit divided by that number; thus, the reciprocal of 3 is

150. The Number of Cases of common fractions is eight. They are as follows:

1. Reduction,

2. Addition,

3. Subtraction,

4. Multiplication,

5. Division,

6. Relation of Fractions,

7. Greatest Common Divisor, 8. Least Common Multiple.

NOTES.-1. Each fractional part is used as a single thing and is therefore a unit; hence, we have Units and fractional units.

2. The primary conception of a fraction is that it is a number of equal parts of a unit. It may, however, be regarded as a number of parts of one thing, or as one part of a number of things. Thus, may be regarded as of one or of three.

NUMERATION AND NOTATION.

151. Numeration of Fractions is the art of reading a fraction when expressed by figures.

Rule. Read the number of fractional units expressed by the numerator, and give them the name indicated by the denominator.

Name the kind and read the following:

152. Notation of Fractions is the art of expressing fractions by means of figures.

Rule. Write the number of fractional units, draw a line beneath, under which write the number which indicates the kind of fractional units.

Write the following fractions:—

1. Two-thirds.

2. Five-sixths. 3. Six-eighths. 4. Nine-tenths.

5. Seven-elevenths.

6. Eight-tenths.

7. Eleven-fifteenths.

8. Twelve-twentieths.

ANALYSIS OF FRACTIONS.

153. To Analyze a fraction is to explain what is ex· pressed by the fractional notation.

1. Analyze the fraction .

SOLUTION.-In the fraction, the denominator 5 indicates that the unit is divided into 5 equal parts, and the numerator 4 denotes that 4 of these parts are taken.

Analyze the following fractions:

154. There are Two Methods of treating common fractions, which may be distinguished as the Inductive and Deductive Methods.

155. By the Inductive Method we solve all the different cases by analysis, and derive the rules or methods of operation from these analyses by inference or induction.

156. By the Deductive Method we first establish a few general principles, and then derive all the rules or methods of operation from these general principles.

NOTE.-The Inductive Method will be used with the mental exercises; with the written exercises the method which is thought to be the simplest is used.

PRINCIPLES OF FRACTIONS.

1. Multiplying the numerator of a fraction by any number multiplies the value of the fraction by that number.

If we multiply the numerator of a fraction by any number, as 5, the resulting fraction will express 5 times as many fractional units, each of the same size as before, hence the value of the fraction is 5 times as great.

2. Dividing the numerator of a fraction by any number divides the value of the fraction by that number.

If we divide the numerator of a fraction by any number, as 4, the re sulting fraction will express as many fractional units, each of the same size as before, hence the value of the fraction is divided by 4.

3. Multiplying the denominator of a fraction by any num her divides the value of the fraction by that number.