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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 following:

Tens; hundreds; thousands; millions; ten-thousands; hundred thousands; ten-millions; billions; hundred-millions; hundred-billions; 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.
2. Third.

3. Second.

4. Fifth.

5. Fourth.

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; hundred 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 numerical 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 is understood.

OPERATION.

6,325,478

1. What number is expressed by 6325478? SOLUTION.-Separating the numerical expression into periods of three figures each, beginning at the 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:

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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 NOTATION.

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 proper order.

II. When there are vacant places, fill them with ciphers. 1. Express in figures the number four thousand three hundred and four.

4,304

SOLUTION.-We write the 4 thousands in the 4th OPERATION. place, 3 hundreds in the 3d place, a cipher in the 2d place, there being no tens, and 4 units in the 1st place, and we have 4304.

Express the following numbers in figures:

2. One hundred and six.

3. One hundred and ten.

4. two hundred and forty.

5. Two hundred and sixty-five. 6. Two hundred and nine.

7. Three hundred and twelve.

8. Three hundred and fifty.

seven.

9. Four hundred and twenty eight.

10. Seven hundred and eighty. four.

11 Nine hundred and thirty

even

12. Eight hundred and ninetynine.

13. Four thousand and seven.

14. Five thousand two hundred and thirty-six.

27. Eight million two thousand and sixty-seven.

28. Five million two hundred and ninety-six thousand.

29. Seventy million, one thousand and forty-five.

30. One million, two hundred

15. Six thousand and eighty- and thirty thousand, four hundred five. and fifty-six.

16. Twenty-three thousand six hundred and forty-seven.

17. One hundred and forty-five thousand seven hundred and six. 18. Three hundred and eight thousand three hundred and eight. 19. Six hundred and four thousand three hundred and sixty-eight. 20. Eight hundred and seventyfour thousand one hundred and twenty.

21. Seven hundred and seven thousand seven hundred and

seven.

22. One million.

23. Two million and six.

24. Three million and twelve.

25. Forty-five

million and

twenty-four.

26. Six million

thousand.

31. Four million, three hundred and seven thousand, four hundred and nine.

32. Fifteen million, four hundred and eight thousand, and eighty-four.

33. Twenty-eight million, five hundred and ninety-four thou sand, seven hundred and nine.

34. Forty-seven million, thirtyeight thousand, two hundred and eight.

35. Two hundred million, forty. nine hundred and twenty-eight.

36. Six billion, seven hundred and five million, thirty-five thousand and six.

37. Forty-nine trillion, fiftyeight thousand, seven hundred forty-seven and ninety-eight.

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,

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36. From this it is seen that ten units of a lower order make one unit of the next higher order; the system of notation is therefore called the Decimal System, from the Latin, decem, ten.

WRITTEN EXERCISES.

1. Write and read two units of the second order, and four of the first.

2. Write and read four units of the third order, and six of the second.

3. Write and read nine units of the third order, and three of the first.

4. Write and read three units of the fifth order, six of the third, and seven of the first.

5. Write and read five units of the sixth order, four of the third, and eight of the second.

6. How many units in one ten? In 2 tens? In 3 tens? In 4 hundreds?

thousands? In 5 thousands? In 60 tens? In 3 millions?

9. What is the result of changing a figure one place to the left Two places? Three places?

10. What is the result of chang ing a figure one place to the right? Two places? Three places?

11. What is the result of placing one naught to the right of one or more figures? Two naughts? Three naughts?

12. What place is occupied by tens figure? Hundreds figure? Ten-thousands? Millions? Bil

lions?

13. What is the denomination of a figure in third place? In fifth place? In fourth place? In sev

7. How many tens in 1 hundred? In two hundreds? 3 hun-enth place? In ninth place? In dreds? 2 thousands? sixth place? In eighth place?

8. How many hundreds in 3

37. The table which has been given enables us to read a numerical expression consisting of twenty-one figures; the periods which follow them in order are as follows:

Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions Duodecillions, Tertio-decillions, Quarto-decillions, Quinto-decillions, Sexto-decillions, Septo-decillions, Octo-decillions, Nono-decillions, Vig'lions. With these, and those already given, we can write and read a numerical expression consisting of sixty-six places.

NOTES.-1. The first of the nine Arabic characters are called digits, from the Latin word digitus, a finger, owing to the fact that the ancients reckoned by counting the fingers. They are also called significant figures, because they always indicate a definite number of units. The character

0, called zero, cipher, or naught, always indicates an absence of units.

2. The Arabic Notation is named from the Arabs, who introduced it into Europe by their conquest of Spain during the 11th century. The Arabs obtained it from the Hindoos, by whom it was probably invented more than 2000 years ago.

3. There are three theories for the origin of the Arabic characters, for which see Brooks's Philosophy of Arithmeti

THE DECIMAL SCALE.

88. Since figures express different orders of units in dif ferent places, they are said to have two kinds of values; a simple value and a local value.

39. The Simple Value of a figure is its value when it stands alone or in units place. The Local Value of a figure s its value when in some other place than units.

40. The value of a figure is increased tenfold for every place it is moved from right to left; and is decreased tenfold for every place it is moved from left to right. This law is called the scale of the system.

41. Since the value of terms decreases from left to right at a tenfold rate, if we fix the place of units by a point (.), we may extend the decimal scale to the right of units.

42. The first place on the right of the point will be onetenth of units or tenths, the second place one-tenth of tenths, or hundredths, the third place, thousandths, etc.

43. Such terms are called decimals, and the point is called the decimal point. The expression 48.37 is read 48 units, 3 tenths, and 7 hundredths, or 48 and 37 hundredths.

44. The Money of the United States is expressed by the decimal system. The dollar is the unit, and is indicated by the symbol $. The first place at the right of the decimal point is called dimes; the second place, cents, etc.

45. Dimes and Cents are usually read as a number of Thus, $4.65 is read 4 dollars and 65 cents; and $72.48 is read 72 dollars and 48 cents.

cents.

The Decimal system of numeration had its origin in the practice common to all nations, of counting by groups of tens.

EXAMPLES FOR PRACTICE.

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