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20.

because it would be in the first place. Now the cypher has no value, but it makes the 1 stand in the sccond place, and, therefore, makes it a unit of the second order, which is equal to ten units of the first.

How do you write one ten ? Why do you use the cypher ? Why do you not use more cyphers than one ? Has the cypher any value ? Does the use of it affect the value of other figures ?

Write two tens, or twenty.
Three tens, or thirty.

30.
Four tens, or forty.

40. Five tens, or fifty.

50. Sixty, or six tens.

60. Seventy, or seven tens.

70. Eighty, or eight tens.

80. Ninety, or nine tens.

90. Ten tens, or one hundred. Ten tens, are ten units of the second order, which are equal to one of the third order, or one hundred, written

100 Why do you use two cyphers here? If I were to annex another cypher to 100, (as 1000,) in what place would the 1 stand ? Ans. The fourth place. It would, then, be a unit of what order? Ans. The fourth order.

A UNIT OF THE FOURTH ORDER IS CALLED A THOUSAND.

If a cypher were annexed to 1000, (as 10000) to what order would the 1 belong ?

UNITS OF THE FIFTH ORDER ARE CALLED TENS OF THOUSANDS.

Another cypher would make the 1, a uuit of what order ?

UNITS OF THE SEVENTH ORDER ARE CALLED MILLIONS.
Read the following numbers.

5 Five, or five units.
30 Thirty, or three tens.
35 Thirty-five.

79 63 81 97 83 76 49 87 95
400 Four hundreds.
435 Four hundred and thirty-five.

553 984 659 334 695 843
7000 Seven thousands.
7435 Seven thousand, four hundred, and thirty-five.

2761 1561 4322 5112 7782
80000 Eighty thousand, or eight tens of thousands.
87435 Eighty-seven thousand, four hundred, and thirty-fiye.

55534 60203 88905 76352 99122 500000 Five hundred thousand, or five hundreds of thousands. 587435 Five hundred and eighty-seven thousands, four hun.

dred, and thirty-five.

679458 234567 320320 427427 2000000 Two millions. 2587435 Two million, five hundred and eighty-seven thou.

sand, four hundred and thirty-five.

2305061 8910002 8395421 7878978

2004 17 40002 10203 1000001 2222222 813 617 23904 237801 603545 7897989 6304026 [The teacher should require the pupil to read these numbers, both by the familiar names, and also, by the names of the orders ; as 6 tens of thousands, instead of 60 thousand, &c. So, likewise, in writing the following numbers, the pupil should change the common names into the same terms.]

Write the following numbers in figures. Three

3 Thirty, or three tens

30 Thirty-three, or three tens, and three units. Forty-seven. Eighty-eight. Fifty-six. Thirty-five. Eight hundred

800 Eight hundred and thirty-three. Six hundred and ninety-eight. Five hundred and fifty-five. Eight hundred and fifteen. Two thousand

2000 Two thousand, eight hundred, and thirty-three. Four thousand, nine hundred, and eighty-one. Sixty thousand, or six tens of thousands,

60000 Sixty thousand, eight hundred, and thirty-three. Five hundred thousand, or five hundreds of thousands, 500000

Five hundred and sixty-two thousand, eight hundred, and thirtythree. One million

1000000 One million, five hundred and sixty-two thousand, eight hundred, and thirty-three.

Two hundred and three. Two thousand and three.
Twenty thousand, two hundred, and four.
Three million, eighty-five thousand, and twenty-one.

Six hundred and fifty eight thousand, nine hundred and seventy eight. Three hundred and sixty-five.

Nine Million three hundred and thirty-seven thousand, five hun. dred and fifty-four.

Four hundred and eighty-eight thousand six hundred and seventy five.

Forty-four thousand, eight hundred and ninety-nine.
Five hundred and fifty-three.
Seven hundred and ninety.one.

ġ IV. How many orders of figures have

you learned ? What is the seventh ? The sixth ? Fifth ? Fourth ? Third ? Second ? First?

The other orders, as high as the 24th, may be learned from the following

NUMERATION TABLE.

NAMES.

ORDERS.

7 24 Hund. of Sex.
er 23 T. of Sex.
o 21 Hund. of Quin,

er 19 Quintillions.
set 18 Hund. of Quad.

er 17 T. of Quad.
07 16 Quadrillions.
or 15 Hund. of Trill.

en 22 Sex.
<3 or 14 T. of Trill.
(or 13 Trillions.

en 11 T. of Bill.
Co 12 Hund. of Bill.

er 10 Billions.
et 9 Hund. of Mil,

8 T. of Mil.
7 Millions.
6 Hundred of Th.
5 Tens of Th.
4 Thousands.

3 Hundreds.
Quintillions. 07 20 T. of Quin.

2 Tens.

1 Units.

lor
Sextillions.
Quadrillions.
Thousands or
Trillions.
Millions.
Units.

[graphic]

ORDERS 5 5 5 5 5 5 5 5 5

VIII.

VII.

VI.

IV.

III.

II.

I.

Billions.

PERIODS. In order to read large numbers, it is very convenient to divide them into periods of three figures each, begining at the right. The first is called the periods of units, because it contains units, and tens of units, and hundreds of units. The second, the period of thousands, because it contains thousands, and tens of thousands, and hundreds of thousands, and so on. The names of the periods are given above.

IN READING, YOU SHOULD READ EACH PERIOD AS THOUGA I'Y STOOD ALONE, AND THEN ANNEX THE NAME OF THE PIRIOD.

Thus, to read the period of thousands above, first read it as though it stood alone, and it will be, five hundred and fifty-five. Then annex the name of the period, and it will be, five hundred and fifty-five thousand. Let the pupil point off and read the following numbers.

7 8 6 4 3 7 9 2 5 8 4 2 1 9 5 1 1 1 1 9 9 9 9 3 8 7 6 5 3 1 2 3 4 5 6 7 8 9 0 0 0 0 0 2 4

7 6 0 0 0 0 7 6 0 0 0 0 7 6
9 1 8 7 5 6 7 8 1 2 3 5 4 1 0 0 1

80 90 1 0 3 0 4 0 0 0 2 3 4 5
7 8 9 0 0 0 6 7 8 0 0 5 6 7 0 2

3 4 2 3 1 2 5 6 1 6 5 8 7 1 2 4 8 9 6 7 4 5 0 0 0 2 0 3 0 1 0 2 1 0 78 0 8 It is likewise most convenient to write numbers by periods.

Write three hundred and five thousand, four hundred and fifty-three.

How many periods are here mentioned, and what are they?

Write the period of thousands.. Ans. 305 thousands.
Write the period of units. Ans. 453.
Which ought to stand first?
'Then write them both together. Ans. 305,453.

Write three hundred and five million, four hundred and fifty three. Write first the millions' period.

There are no thousands ; how will you write the thousand period ? Ans. 000 thousands.

Write the units period. Ans. 453.
Arrange the periods in order. Ans. 305,000,453.
Write sixteen thousand and thirty. Ans. 16,030.

Then, to write numbers in figures, BEGIN WITH THE HIGHEST PERIOD AND WRITE EACH PERIOD AS THOUGH IT WAS THE ONLY ONE TO BE WRITTEN. PLACE CYPHERS WHEREVER A PERIOD OR AN ORDER IS OMITTED.

In this manner let the pupil write the following numbers. Write seven millions, six hundred and sixty-two thousand, five hundred and ninety-three.

Four billions, twenty million, two hundred thousand and five.
Sixty-seven trillion-, five hundred and five thousand.
Thirty-five billions and four.

Four hundred and seventy-nine quadrillions, twenty-three million and sixty:

One thousand and three.
One hundred thousand, five hundred and forty.

Seventy-seven million, four hundred and iorty.two thousand, five hundred and nineteen.

Sixty five trillions, seventy four billions, eighty four millions, ninety four thousand, one hundred and five. Two quadrillions and fifty. Nino quintillions and thirty nine. Seven sextillions, eighty five millions, three hundred and four. Five sextillions, three hundred and seventy nine quintillions, four hundred and forty seven quadrillions, eight hundred and twenty three trillions, four hundred and thirty seven billions nine hundred and eighty six millions five hundred and thirty six thousand four hundred and seventy nine. Two sextillions and two.

§ V. Where the figure 1 stands alone, what its value ? Ans. One unit. When the figure 2 stands alone, what is its value ? Ans. Two units. When 3 stands alone what is its value ? When 4 ? 5? 6? 7 ? 8? 9 ?

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THE VALUE OF A FIGURE, STANDING ALONE, IS CALLED ITS SIMPLE VALUE.

If the figure 1 has a cypher at its right hand, what is its value ? Ans. One ten.

If the figure 3 has two cyphers at the right, what is its value ?

What is the value of 7 and four cyphers, or 70000 ? Of 800000 ? Of 2000 ? Of 5000000 ?

Is the value of the figures in these last examples, the same with their simple value, or is it greater, or is it less ?

Why is it greater ? Ans. Because it stands in a higher place. Then,

FIGURES HAVE A VALUE WHICH DEPENDS UPON THEIR PLACE, DIFFERENT FROM THEIR SIMPLE VALUE.

THE VALUE WHICH A FIGURE DERIVES FROM ITS PLACE, IS CALLED ITS LOCAL VALUE.

How many units of the first order are equal to one of the second ?

How many of the second are equal to one of the third ?

How many of the third to one of the fourth ? Of the 4th to one of the 5th ? Of the 6th to one of the 7th? Of the 9th to one of the 10th ? &c. Then

IT TAKES TEN UNITS OF ANY ORDER TO MAKE ONE OF THE NEXT
HIGHER; And,
NUMBERS INCREASS FROM RIGHT TO LEFT IN TENFOLD PROPORTION.
Here is a picture to illustrate this tenfold increase.

The little square on the right, repre-
sents a unit of the first order, and corres-
ponds to the 1 under it. The next dia-
gram represents a unit of the second order,

and corresponds likewise to 1, in the 1

second place. The third diagram represents a unit of the third order, and corresponds to 1, in the third place.

You see, then, how rapidly numbers increase. were to go on, only two orders higher, that is, to the fifth order, the page would not be large enough to contain the diagram for that order.

You have now been learning to write numbers in figures. This is called NOTATION.

If we

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