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T'wenty-one, twenty-two, twenty-three, twenty-four, twenty five, twenty-six, twenty-seven, twenty-eight, twenty-nine.

20. Continuing in the same manner, we would have three. tens, four-tens, five-tens, etc. By this principle were derived the following ordinary names:

Thirty, forty, fifty, sixty, seventy, eighty, ninety.

21. A group of ten tens is called a hundred; a group of ter. hundreds, a thousand; the next group receiving a new name consists of a thousand thousands, called a million; the next group of a thousand millions, called a billion, etc.

22. After a thousand, the two intermediate groups between those having a distinct name, are numbered by tens and hundreds, as ten thousand and hundred thousand.

Norts.-1. The above shows the principle by which the numbers were named. The names, however, were not derived from the particular expressions given, but originated in the Saxon language.

2. Eleven is from the Saxon endlefen, or Gothic ainlif (ain, one, and lif, ten); twelve is from the Saxon twelif, or Gothic tvalif (tva, two, and lif, ten). Some have supposed that eleven meant one left after ten, and twelve, two left after ten.

3. Twenty is from the Saxon twentig (twegen, two, and tig, a ten); thirty is from the Saxon thritig (thri, three, and tig, a ten), etc.

4. Hundred is a primitive word ; thousand is from the Saxon thusend, or Gothic thusundi, (thus, ten, and hund, hundred); million, billion, etc., are from the Latin.

NOTATION. 23. Notation is the method of writing numbers. Numbers may be written in three ways:

1st. By words, or common language. 2d. By figures, called the Arabic Method. 3d. By letters, called the Roman Method. NOTE.—The method by words is that of ordinary written language and peods nc explanation.

ARABIC NOTATION. 24. The Arabic System of Notation is the method of expressing numbers by characters called figures.

25. In this system numbers are expressed according to the following principle:

Principle.- We employ characters to represent the first nine numbers, and then use these characters to number the groups, the group numbered being indicated by the position of the character.

AND VALUES. one, two, three, four, five, six, seven, eight, pine, cipher or zero.
12
FIGURES.
NAMES

26. Figures.--Figures are characters used in expressing numbers. There are ten figures used, as follows: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

naught,

27. By the combination of these figures all numbers may be expressed ; hence they are appropriately called the alpha bet of arithmetic.

28. Combination. These figures are combined accord. ing to the following principle:

1. A figure standing alone, or in the first place at the right of other figures, expresses UNITS or ONES.

2. A figure standing in the second place, counting from the right, expresses TENS; in the third place, HUNDREDS ; in the fourth place, THOUSANDS, etc.; thus, 10 is 1 ten, or ten.

100 is 1 hundred. 20 2 tens, or twenty.

200 “ 2 hundred. 30 3 tens, or thirty.

300 3 hundred. 4 tens, or forty.

400 “ 4 hundred. 50 5 tens, or fifty.

1000 1 thousand. 60 " 6 tens, or sixty.

2000 “ 2 thousand. 90 " 9 tens, or ninety. 4000 4 thousand.

29. The name of each of the first twenty-one places is represented by the following

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30. Periods. For convenience in writing and reading aumbers, the figures are arranged in periods of three places each, as shown in the table. The first three places constitute the first, or units period; the second three places constitute the second, or thousands period, etc.

1. Required the names of the following places: First; third ; second ; sixth ; fourth ; eighth; tenth ; ninth ; twelfth; fifth ; seventh ; eleventh ; thirteenth ; seventeenth ; fourteenth ; sixteenth ; eighteenth ; fifteenth ; nineteenth ; twenty-first; twentieth.

2. Required the places of the foilowing:

Tens; hundreds ; thousands ; millions ; ten-thousands ; hundred thousands ; ten-millions ; billions ; hundred-millions ; hundred-bil. lions ; units ; ten-billions ; trillions ; quadrillions ; hundred-quintillions ; ten-trillions; ten-quintillions ; hundred-quadrillions ; quintillions; hundred-trillions ; ten-quadrillions. 3. Required the names of the following periods: 1. First.

3. Second.

5. Fourth. 2. Third.

4. Fifth.

6. Seventh. 4. Required the period and place of the following:

Thousands ; millions ; ten-thousands ; hundred-millions ; billions ; hundred-trillions ; trillions ; ten-trillions ; quadrillions ; ten-quadrillions; hundred-trillions ; quintillions ; hundred-quintillions ; hun. dred-thousands; ten-millions.

31. The combination of figures to express a number forms a numerical expression. Thus, 25 is a numerical expression which denotes the same as the common word twenty-five.

32. The different figures of a numerical expression are called terms. Terms are also used to indicate the numbers represented by the figures.

NOTE.—The use of the word term, to indicate both the figures and the numbers represented by them, enables us to avoid the error of using the word figure for the word number.

EXERCISES IN NUMERATION. 33. The pupils are now prepared to learn to read numbers when expressed by figures. From the preceding explana tions we have the following rule for numeration :

Rule.-I. Begin at the right hand, and separate the nice merical expression into periods of three figures each.

II. Then begin at the left hand and read each period in succession, giving the name of each period except the last.

Note.-The name of the last period is usually omitted, because it la understood.

1. What number is expressed by 6325478? SOLUTION.-Separating the numerical expression OPERATION. into periods of three figures each, beginning at the 6,325,478 right hand, we have 6,325,478. The third period is 6 millions, the second period is 325 thousands, and the first is 478 units, hence the number is 6 millions, 325 thousands, 478. Read the following numerical expressions : 3876 10. 468217 18.

80305072 3. 2185 11. 654879 | 1y.

65073058 4. 3072 12.

803006 20.

484378513 5. 5678 13. 1234567 21.

123456789 6. 12630 14. 3078560 22.

854327031 7. 70851 15. 8507032 23.

80735468579 8. 32468 | 16. 54372568 24. 20650708462067 9. 507035 17. 87072135 25. 798653013285678521

NOTE.-After pupils are familiar with reading by dividing into periods, the division may be omitted or performed mentally.

EXERCISES IN NUTATION.

.

proper order.

34. Having learned to read numerical expressions, we are now prepared to write them. From the principles which have been explained, we derive the following rule:

Rule.-I. Begin at the left and write the hundreds, tens, and units of each period in their

II. When there are vacant places, fill them with ciphers

1. Express in figures the number four thousand three hundred and four.

SOLUTION.—We write the 4 thousands in the 4th place, 3 hundreds in the 3d place, a cipher in the 2d 4,304 place, there being no tens, and '4 units in the 1st place, and we have 4305.

Express the following numbers in figures: 2. One hundred and six.

8. Three hundred and ifty 3. One hundred and ten. 4. Two hundred and forty.

9. Four hundred and twenty 5. Two hundred and sixty-five. eigiit. 6. Two hundred and nine.

10. Seved hundred and eighty 7. Three hundred and twelve fou“.

OPERAMON.

sevun.

seven

11. Nine hundred and thirty 27. Eight million two thousand

and sixty-seven. 12. Eight hundred and ninety 28. Five million two hundred Qine.

and ninety-six thousand. 13. Four thousand and seven. 29. Seventy million, one thous.

14. Five thousand two hundred and and forty-five. and thirty-six.

30. One million, two hundred 15. Six thousand and eighty- and thirty thousand, four hundred five.

and fifty-six. 16. Twenty-three thousand six 31. Four million, three hundred hundred and forty-seven.

and seven thousand, four hundred 17. One hundred and forty-five and nine. thousand seven hundred and six.

32. Fifteen million, four hun. 18. Three hundred and eight dred and eight thousand, and thousand three hundred and eight. eighty-four. 19. Six hundred and four thous

33. Twenty-eight million, five and three hundred and sixty-eight. hundred and ninety-four thou.

20. Eight hundred and seventy- sand, seven hundred and nine. four thousand one hundred and 34. Forty-seven million, thirty. twenty.

eight thousand, two hundred and 21. Seven hundred and seven eight. thousand

hundred and 35. Two hundred million, fortyseven.

nine hundred and twenty-eight. 22. One million.

36. Six billion, seven hundred 23. Two million and six. and five million, thirty-five thou24. Three million and twelve.

sand and six. 25. Forty-five million and 37. Forty-nine trillion, fiftytwenty-four.

eight thousand, seven hundred 26. Six million worty-seven and ninety-eight. thousand.

seven

35. Orders.--Since we may have 2 tens, 3 tens, etc., 2 hundreds, 3 hundreds, etc., the same as 2 apples, 3 apples, 2 books, 3 books, etc., these different groups may be regarded as units of different orders; thus,

are called

UNITS
TENS
HUNDREDS
THOUSANDS
TEN-THOUSANDS

etc.,

Units of the 1st order.
Units of the 2d order.
Units of the 3d order.
Units of the 4th order.
Units of the 5th order.

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etc.

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