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Three places of figures, beginning on the right hand, are called a period, and each successive three places, another period; the first period, on the right hand, being called the period of units; the second, the period of thousands, and so on, as in the Table. There is an obvious reason for this division into periods; for at the beginning of each period, there is a new denomination of units, of which the tens and hundreds are numerated, as in the first period. The names of the periods are derived from the Latin numerals, and they may be continued without end. They are as folother ratio. The method of counting by tens, which is now in general use, originated from the practice of counting on the fingers, which are ten in number.

This method of enumerating figures to the left hand of the place of hundreds of millions, differs from that which has hitherto been in general use in this country. This method of dividing lines of figures into periods, and of naming those periods, is used by the French and Italians; and it has lately been adopted by a number of respectable British and American Authors on Arithmetic. The method is strongly recommended by its simplicity and elegance, and it is probable that it will soon be much more universally used.

The usual method is to divide the numbers into periods of six figures each; which periods have the same names as those in Table 2d, except thousands, for which there is not a distinct period. The common method of enumerating figures is exemplified in the following

TABLE.

&c.

Hund. of thou. of bill.
Tens of thou. of bill.
Thousands of billions.
Hundreds of billions.
Tens of billions.
∞ Billions.

Hund. of thou. of mill.
Tens of thou. of mill.
Thousands of millions.
Hundreds of millions.
Tens of millions.
Millions.

Hundreds of thousands.
Tens of thousands.
Thousands.
Hundreds.
∞ Tens.

Units.

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

Period of
Billions.

Period of
Millions.

Period of

Units.

It will be seen that the two methods of enumerating lines of figures, agree as far as hundreds of millions; and, as it is very rarely necessary to name larger numbers, it is evident that the proposed change from the old to the new method, cannot be attended with much inconvenience.

lows, for twenty-two periods, viz. Units, Thousands, Millions, Billions, Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions.

For the convenience of reading large numbers, the periods of figures which express them are often separated by commas, as in Table 2d.

To enumerate any parcel of figures.

RULE.

1. In order to ascertain the local values of the given figures, divide them into periods of three figures each, beginning at the right hand, and proceeding towards the left, till not more than three figures remain. Then the first period towards the right hand will contain units or ones, the second thousands, the third millions, &c. as in Table 2d.

2. Then, in reading the number, begin at the left hand, and to the value expressed by the figures of each period, join the name of the period, excepting the name of the period of units, which need not be mentioned.

Ex. The number in Table 2d, viz. 651,237,428,714,970,305,082, is read thus, six hundred fifty-one quintillions, two hundred thirty-seven quadrillions, four hundred twentyeight trillions, seven hundred fourteen billions, nine hundred seventy millions, three hundred five thousand, and eighty-two.

Note. By practice, the pupil will soon be able to enumerate numbers, or lines of figures, which are not very large, without dividing them into periods; for the local values of the figures may be readily ascertained by merely enumerating them from right to left, thus, tens, hundreds, thousands, &c. as in the Numeration Tables,

10=Ten.

11=Eleven.

12=Twelve.

13 Thirteen.

14-Fourteen.

15=Fifteen.

More Examples in Numeration.

| 20=Twenty.
21=Twenty-one.
30 Thirty.

32 Thirty-two.
40-Forty.

43 Forty-three.

16 Sixteen. 50=Fifty.
17=Seventeen. 54=Fifty-four.

18-Eighteen. 60-Sixty.
19-Nineteen. 65=Sixty-five.

70=Seventy.

76-Seventy-six.
80=Eighty.

87 Eighty-seven.
90-Ninety.
98-Ninety-eight.

100 One hundred.

200-Two hundred. 1000-One thousand.

2000 Two thousand.

four hundred and seven.

148, read, one hundred and forty-eight.

407
950

4,549 90,170 800,000

7,154,918

987,654,321

5,000,217,080

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nine hundred and fifty.

four thousand, five hundred and 49. ninety thousand, one hundred and 70. eight hundred thousand.

7 millions, 154 thousand, 918.

987 millions, 654 thousand, 321.
5 billions, 217 thousand, and 80.

Read, or write in proper words, the following numbers.

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To set down in figures any number proposed in words, observe the following

RULE.

Set down the figures instead of the words, or names belonging to them, taking care to supply with ciphers all the places below the highest whose names are not mentioned in words.

Exercises in Notation.

Write down, in proper figures, the following numbers.

Four hundred and forty-seven.

Four thousand four hundred and eight.

Ninety-seven thousand six hundred and

fourteen.

Sixty thousand and twenty one.

447.

4,408.

60,021.

Four hundred thousand.
Ninety-eight millions.

Nine hundred and six millions, five hundred
thirty thousand, two hundred and one.
Nine hundred fifty billions, twenty-four
millions, six hundred ten thousand and
ninety.

98,000,000.

950,024,610,090.

ROMAN METHOD OF NOTATION.

The Romans and several other ancient nations, made use of certain letters of the alphabet to express numbers. The Romans used only seven numeral letters, being the seven following capitals, viz. I for one, V for five, X for ten, L for fifty, C for an hundred, D for five hundred, and M for a thousand. The other numbers they expressed by various repetitions and combinations of these letters, after the following

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Note. As often as any letter is repeated, so many times is its value repeated: thus, II=2, III=3, XX=20, &c.

A less literal number placed before a greater, diminishes the value of the greater: thus, V denotes 5, but IV only 4; and X=10, but IX=9, &c.

A less literal number placed after a greater, increases its value: thus, VI=6, and XI=11, &c.

A bar or line over any literal number, increases it a thousand fold: thus, V-5000, and XX=20000, &c.

20=XX

5000=V

30=XXX

&c.

THEORETICAL QUESTIONS.

1. What is Arithmetic? 2. Into how many parts is it divided? 2. What does Theoretical Arithmetic explain? 4.

What does Practical Arithmetic teach? 5. What is a nugiber? 6. What is unity, or a unit? 7. What is an integer, or a whole number? 8. What are fractions? 9. What is meant by quantity? 10. What is a simple quantity? 11. What is a compound quantity? 12. What is Notation? 13. What is Numeration? 14. What is meant by the simple value of any figure? 15. What by its local value? 16. What does the right hand figure of any whole number denote? and what the next figure to the left? and so on? 17. Does the cipher denote any number? 18. What effect does it have when annexed to other figures? 19. What is the rule for enumerating any parcel of figures? 20. What is the rule for expressing by figures any number proposed in words?

FUNDAMENTAL RULES OF ARITHMETIC.

There are four rules which are called the Fundamental Rules, because all operations in Arithmetic are performed by the use of them. They are Addition, Subtraction, Multiplication, and Division.

ADDITION,

Is putting together two or more numbers, so as to find their amount, or total value, which is called their sum. It is called Simple Addition, or, Addition of Whole Numbers, when the numbers to be put together are all simple, or whole numbers, of the same denomination; and Compound Addition, when the numbers are compound, or of different denominations.

SIMPLE ADDITION,

Is putting together two or more whole numbers, or quantities, of the same denomination; as 3 dollars and 2 dollars, added together, make a sum of 5 dollars: Or, it is simply adding together two or more whole numbers, without regard to their signification; as 5 added to 4, makes 9.

To perform the operation of addition, it is necessary that the learner should be able to assign the sum of any two of the small numbers less than 10; and for this purpose he should be exercised in the following table.

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