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54, 89, 99? What numbers may be expressed by a single figure ? What numbers may be expressed by two figures ?

§ 7. In order to express one hundred, or ten units of the second order, we have to form a new combination. It is done thus,

100 by writing two ciphers on the right of i. This number is read, one hundred. Now this one hundred expresses 10 units of the second order, or one hundred units of the first order. But the one hundred is but an individuai hundred, and in this light may be regarded as a unit of the third order.

We are,

We can now express any number less than one thousand.

For example, in the number three hundred and seventyfive, there are 3 hundreds, 7 tens, and 5 units therefore, to express 3 units of the 3d order, 7 units of the second order, and 5 of the 1st.

Hence, we write and we read from the right, units, tens, hundreds.

In the number eight hundred and ninety-nine, there are 8 units of the 3d order, and 9 of the 2d, and 9 of the 1st.

It is written and read, units, tens, hundreds.

In the number four hundred and six, there are 4 units of the 3d order, 0 of the 2d, and 6 of the 1st. It is written

406 and in a similar manner we may express, by three figures, any number greater than ninety-nine and less than one thousand.

whuns.

tens.
Grunits.

cohuns.
cotens.
counits.

A huns.
Otens.
units.

Q. How do you express one hundred? To how many units of the 2d order is it equal ? To how many of the 1st order? May it be considered a single unit? Of what order is it? How many units of the 3d order in 200? In 300? In 400? In 500? In 600? In 700 ? In 800? In 900 ? Of what is the number 375 composed? The number 406 ? What numbers may be expressed by three figures ?

§ 8. To express ten units of the 3d order, or one thousand, we form a new combination by writing three ciphers on the right of 1; thus,

1000

Now, although this thousand expresses one thousand units of the 1st order, it is, nevertheless, but one single thousand, and may be regarded as a unit of the 4th order.

Proceeding in this way, we may place as many figures in a row as we please. When so placed, we conclude:

1st. That the same figure has different values according to the place which it occupies.

2d. That counting from the right hand towards the left, the first is the place of units ; the second, the place of tens; the third, the place of hundreds; the fourth, the place of thousands ; &c.

3d. That ten units of the first place are equal to one unit of the second place ; that ten units of the second place are equal to one unit of the third place; that ten units of the third place are equal to one unit of the fourth place; and so on, for places farther to the left.

Q. To what are ten units of the 3d order equal ? How do you ex. press them? May thiş be considered a single unit? Of what order ? May any number of figures be written in a row? When so placed has the same figures different values ? On what does the value of the same figure depend? What is the first place on the right called? What is the second called? What is the third called ? What is the fourth called? What are ten units of the first place equal to? What are ten units of the second place equal to? To what are ten units of the third place equal.

§ 9. Expressing or writing numbers by figures, is called NOTATION. Reading the order of their places, correctly, when written, is called NUMERATION.

Q. What is Notation? What is Numeration? Which way do you numerate ? 1. Write three tens.

Ans. 30. 2. Write one hundred and fifty.

Ans. 3. Write twelve tens.

Ans. 120. 4. Write 4 units of the first order, 5 of the 2d, 6 of the 3d, and 8 of the 4th.

Ans. 5. Write 9 units of the 5th order, none of the 4th, 8 of the 3d, 7 of the 2d, and 6 of the 1st. Ans. 90876.

6. Write 1 unit of the 6th order, 5 of the 5th, 4 of the 4th, 9 of the 3d, 7 of the 2d, and none of the 1st.

Ans.

NUMERATION TABLE.*

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for Period

of Quadrillions. Hundreds of Quadrillions. 6th Period Tens of Quadrillions Quadrillions

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for Period

of Thousands. Hundreds of Thousands > 2d Period

Tens of Thousands - Thousands

Hundreds

Tens coer Units

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1

82,

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43,

1

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75

879 6, 0 2 3

301 123,

087 7, 628, 735 210,

4 60 548, 721,

087 6, 2 45, 289, 4 2 1

549, 136, 822 894, 602,

0 43, 288 2 48, 907, 4 5 6 876, 410,

2 8 5 912, 761, 257,

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6 41, 912,

1

84,

826
6,
407,

93 6,
876,

5 41

313
920,
323,
768,

675 The words at the head of the numeration table, units, tens, hundreds, &c., are equally applicable to all numbers, and must be committed to memory, after which, the pupil may read the Table.

* Note.—This Table is formed according to the French method of numeration. The English method gives six places to thousands, &c.

1

327,

57,

289,

2 12, 678, 8 42,

541,

297, 319,

To make the reading of figures easy, they are often separated into periods of three figures each, counting from the right hand.

EXAMPLES IN EXPRESSING NUMBERS BY FIGURES. 1. Write four in figures,

Ans. 4. 2. Write four tens or forty.

Ans. 3. Write four hundred.

Ans. 400. 4. Write four thousand.

Ans. 5. Write forty thousand

Ans. 40,000. 6. Write four hundred thousand. Ans. 7. Write four millions.

Ans. 4,000,000. These examples show us very clearly that the same significant figure will have different values according to the place which it occupies. 8. Write seven.

Write six units of the 2d order. Write nine units of the 3d order. Write six units of the 4th order. Write eight units of the ad order.

Write one unit of the third order. Write nine units of the 6th order. Write two units of the 8th order.

9. Write six hundred and seventy-nine. Ans. 679. 10. Write six thousand and twenty-one. 11. Write two thousand and forty. 12. Write one hundred and five thousand and seven. 13. Write three billions. 14. Write ninety-five quadrillions.

15. Write one hundred and six trillions, four thousand and two.

16. Write fifty-nine trillions, fifty-nine billions, fiftynine millions, fifty-nine thousands, fifty-nine hundreds, and fifty-nine.

17. Write eleven thousand, eleven hundred and eleven. 18. Write nine billions and sixty-five.

19. Write three hundred and four trillions, one million, three hundred and twenty-one thousand, nine hundred and forty-one.

§ 10. There is yet another method of expressing nunbers, called the Roman. In this method the numbers are represented by letters. The letter I stands for one ; V, five; X, ten; L, fifty; C, one hundred; D, five hundred, &c.

I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XX.
XXX.
XL.
L.

ROMAN TABLE.
One

LX.
Two

LXX.
Three

LXXX.
Four

XC.
Five

C.
Six

CC.
Seven

CCC.
Eight

CCCC.
Nine

D.
Ten

DC.
Twenty

DCC.
Thirty

DCCC.
Forty

DCCCC.
Fifty

M.

Sixty
Seventy
Eighty
Ninety
One hundred
Two hundred
Three hundred
Four hundred
Five hundred
Six hundred
Seven hundred
Eight Hundred
Nine hundred
One thousand.

ADDITION OF SIMPLE NUMBERS.

§ 11. John has three apples and Charles has two; how many apples have they between them.

Every boy will answer, five.

Here a single apple is the unit, and the number five contains as many units as the two numbers three and two. The operation by which this result is obtained is called addition. Hence,

Addition is the uniting together of several numbers, in such a way, that all the units which they contain may be expressed by a single number.

Such single number is called the sum or sum total of the other numbers. Thus, 5 is the sum of the apples possessed by John and Charles.

What is the sum of 2 and 4? of 3 and 5 of 6 and 3 ? of 4, 3 and 12 of 2, 3 and 4? of 1, 2, 3 and 4? of 5 and 7? How many units in 4 and 6? How many units in 9 and 4?

Q. What is addition ? What is the single number called which expresses all the units of the numbers added? What is the sum of 2 and 4? What is six called ?

OF THE SIGNS.

$ 12. The sign +, is called plus, which signifies more. When placed between two numbers it denotes that they

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