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Write the following:

1. Fourteen and five tenths. 2. Eighty-four and twenty-five hundredths.

3. Two hundred and three, and sixty-seven hundredths.

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4. Seven hundred and ninety- 8. Eight hundred and sixtysix, and eight hundred seventy-four dollars, thirty-seven cents five thousandths. and five mills.

NOTE.-It is recommended that only advanced or finishing classes study the English Method of Numeration.

ENGLISH METHOD OF NUMERATION.

46. The method of numeration by dividing numbers into periods of three figures each, is called the French Method.

47. The English Method uses periods of six figures each, calling the first period units, the second millions, the third billions, the fourth trillions, etc.

48. The places in each period are units, tens, hundreds, thousands, tens of thousands, hundreds of thousands. The method is represented in the following table:

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NOTE.-The French Method is used throughout the United States, France, etc., and being much more convenient, is very likely to supersede the other method in England.

EXAMPLES FOR PRACTICE.

1. Write by the English method, one million; one billion; one trillion; one quadrillion.

Hundreds of Thousands.
Tens of Thousands.
Thousands.

- Hundreds.
→ Tens.
Units.

6,

ROMAN NOTATION.

49. The Roman Method of Notation employs seven let ters of the Roman alphabet. Thus, I represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand.

50. To express other numbers these characters are combined according to the following principles:

1. Every time a letter is repeated its value is repeated. 2. When a letter is placed before one of greater value, the DIFFERENCE of their value is the number represented.

3. When a letter is placed after one of a greater value, the SUM of their values is the number represented

4. A dash placed over an expression increases its value a thousand fold. Thus, VII denotes seven thousand.

51. These principles are exhibited in the following table, which the pupil will examine carefully :

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XIX

XX

Nineteen. MCLX One thousand one hundred
Twenty.
MDCCCLXXVI, 1876 [and sixty

52. The Roman Method is named from the Romans, who invented and used it. It is now only employed to denote the chapters and sections of books, pages of preface and introduction, and in other places for prominence and distinction.

WRITTEN EXERCISES.

Express the following numbers by the Roman method: 1. Twenty-seven. 2. Seventy-seven. 3. Two hundred and one. 4. Six hundred and fifty-six. 5. One thousand seven hundred and seventy-six. 6. Four thousand seven hundred and fifty-seven. 7. 25007. 8. 206484.

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53. Lumbermen in marking lumber employ a modifica tion of the Roman Method of Notation. The first four characters are like the Roman. The others are as follows:

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NOTE. For a full discussion of Arithmetical Language, see the author's

Philosophy of Arithmetic.

INTRODUCTION TO ADDITION.

MENTAL EXERCISES.

1. If there are 7 boys in one class and 6 in another class, how mary boys in both classes?

SOLUTION.—If there are 7 boys in one class and 6 in another class, in both classes there are 7 boys plus 6 boys, which are 13 boys.

2. In a garden there are 9 apple-trees and 10 pear-trees; how many trees in the garden?

3. Anna bought a bonnet for 15 dollars and a cloak for 24 dollars, how much did both cost her?

4. Mary gave 19 apples to one of her schoolmates and 15 to an other; how many did she give to both?

5. If a boy gave 75 cents for a pair of skates and 25 cents for a ball, what did they both cost him?

6. A lady gave 10 cents for needles, 14 cents for thread, and 15 cents for muslin; what did she give for all?

7. A farmer sold some wheat for 15 dollars, some corn for 25 dollars, and some oats for 30 dollars; what did he receive?

8. How many are 6 and 21? 9 and 22? 8 and 31? 10 and 17? 4 and 18? 7 and 27? 9 and 287? 12 and 11? 13 and 11?

9. How many are 6 and 17? 8 and 16? 9 and 18? 8 and 19? 7 and 12? 10 and 21? 12 and 20? 14 and 18? 11 and 21? 10. How many are 3 and 14? 4 and 17? 5 and 15? 7 and 17? 11 and 21? 10 and 20? from 20 to 40;

6 and 16? 8 and 18? 9 and 19?

from 40 to 60; from

11. Count by 2's from 2 to 20: 1 to 21; from 21 to 41; from 41 to 61. 12. Count by 3's from 3 to 21; from 21 to 42; from 1 to 22; from 22 to 43; from 2 to 23; from 23 to 44.

13. Count by 4's from 4 to 40; from 40 to 80; from 1 to 25; from 25 to 41; from 2 to 30; from 30 to 50; from 3 to 35; from 35 to 55. 14. Count by 5's from 5 to 60; from 1 to 61; from 2 to 62; from 3 to 63; from 4 to 64.

15. Count by 6's, 7's, 8's, 9's, 10's, 11's, and 12's, as in the previous problems.

The uniting of two or more numbers into one sum is called Addi tion. The sign of addition is +, and is read plus.

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SECTION II.

FUNDAMENTAL OPERATIONS.

ADDITION.

54. Addition is the process of finding the sum of two or more numbers.

55. The Sum of several numbers is a number which contains as many units as the numbers added.

56. The Sign of Addition is +, and is read plus. It denotes that the numbers between which it is placed are to be added.

57. The Sign of Equality is, and is read equals. Thus, 10 4+6 is read 10 equals 4 plus 6.

NOTES.-1. The Sign of Addition consists of two short lines bisecting each other, the one in, and the other perpendicular to, the line of writing. 2. The symbol + was introduced by Stifelius, a German mathematician, in a work published in 1544.

PRINCIPLES.

1. The numbers added must be similar.

2. The sum is a number similar to the numbers added.

CASE I.

58. To add when the sum of no column exceeds nine units of that column.

1. What is the sum of 34, 23, and 32?

OPERATION.

34

23

32

SOLUTION. We write the numbers so that figures of the same order stand in the same column, draw a line beneath, and begin at the right to add. 2 units and 3 units are 5 units, and 4 units are 9 units, which we write under the column of units; 3 tens and 2 tens are 5 tens, and 3 tens are 8 tens, which we write under the column of tens. Hence the sum of 34, 23, and 32 is 8 tens and 9 units, or eighty-nine. Hence the following

89

Rule.-I. Write the numbers to be added so that terms of the same order stand in the same column.

II. Begin at the right, add each column separately, and write each sum under the column which produced it

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