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ARITHMETIC.

ARITHMETIC is the art of computing by numbers. It has five principal rules for its operations; viz. numeration, addition, subtraction, multiplication, and division.

NUMERATION

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Numeration teaches to write or express numbers by figures, and to read numbers thus written or expressed.

In treating of numbers, the following terms are employed: viz. unit, ten, hundred, thousand, and million; as also billion, trillion, and some others. But the latter are seldom used.

A unit is a single one.
A ten is ten units.
A hundred is ten tens.
A thousand is ten hundreds.
A million is ten hundred thousands.

Note. -As it takes ten hundred thousands to make a million, when we express a number, greater than a thousand, and less than a million, we use tens of thousands, or hundreds of thousands, or both, as the case requires. Likewise, to express a number, greater than a million, we employ tens of millions, or hundreds of millions, &c.

The following are the figures used in numeration, with their names above them. One two three four five six seven eight nine

1 2 3 4 5 6 7 8 9

Each of these figures represents the number which its name denotes; but it is understood to be that number of units, or that number of tens, or that number of hundreds, &c. according to its relative place: which is exemplified in the following tables.

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TABLE FIRST.

TABLE SECOND.

Hundred million

Ten million
Hundred thousand

Ten thousand
-Million
-Thousand
Hundred

Ten
Unit

Hundreds of millions
Tens of millions
Hundreds of thousands

Tens of thousands
COMillions
Ő UThousands
Hundreds

Tens
Units

These tables show that in using figures to express numbers they are placed in a horizontal row—the first figure at the right hand representing one or more units, the next tens, the next hundreds, &c. Thus a l is one unit, or one ten, or one hundred, &e. according to the place in which it stands; and in like manner, a 2 is two units, or two tens, or two hundreds, &c. The same rule determines the value of each of the other figures.

In reading numbers, the units and tens are taken together. 1 ten and 1 unit are read eleven; 1 ten and 2 units, twelve; 1 ten and 3 units, thirteen, &c.: 2 tens and 1 unit are read, twenty-one; 3 tens and 1 unit, thirty-one, &c. Thus the number expressed by the row of figures in table first is read-one hundred and eleven millions, one hundred and eleven thousands, one hundred and eleven. That expressed by the figures in table second is read-two hundred and twenty-two millions, two hundred and twenty-two thousands, two hundred and twenty-two..

The succeedingtables will further illustrate the subject.

TABLE THIRD.

Millions
Tens
Units
Hundreds
Hundreds of millions
Tens of millions
Thousands
Tens of thousands
Hundreds of thousands on

1
1 2
1 2 3
1, 2 3 4

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

One
Twelve
One hundred and twenty-three

1 thousand 234
12 thousand 345
123 thousand 456

1 million 234 thousands 567 12 millions 345 thousands 678 123 millions 456 thousands 789

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In writing numbers which have no units, or no tens, or no hurdreds, &c. the order observed in the foregoing tables must be maintained by filling the vacant places with a character called a nought or cypher, (0) which, of itself, represents no number. See

TABLE FOURTH.

Millions
Tens
Tens of millions
Thousands
Units
Hundreds
Hundreds of millions
Tens of thousands
Hundreds of thousands

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10 Ten
1 0 0 One hundred
1,000

1 thousand
-1 0,000 10 thousand
100,000 100 thousand
1,000,000

1 million 10,000,000 10 millions 10 0,00 0,00 0 100 millions 20.0,00 0,00 2 200 millions and 2 30 0,30 0,0 3 0 300 millions 3 thou. and 30 4 0 4,0 4 0,4 0 0 404 millions 40 thou. 4 hun. 5 5 0,5 0 0,0 0 0 550 millions 500 thousand

EXAMPLES.
Read the following numbers or write them in words.

Note.- Making a point or dot after every third figure, counting from the units place, greatly facilitates the reading of large numbers. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 30, 31, 32, 40, 43, 44, 50, 55, 56, 60, 67, 68, 70, 71, 79, 80, 182, 83, 90, 92, 100, 101, 111, 112, 113, 114, 120 128, 130, 132, 200, 203, 210, 300, 320, 332, 400, 500, 600, 700, 800, 900, 1000, 2001, 3010, 4020, 5200, 10250, 23450, 356789, 6789402, 76450791, 20156789, 1304136784.

Write the following numbers in figures. Ten. Twelve. Fifteen. Seventeen Twenty-six. Thirtynine. Fifty-two. Seventy-four. Eighty-one. Ninetysix. One hundred and fifteen. Two hundred. Thrče hundred and twenty.

Nine hundred and nine. One thousand two hundred. Seven thousand seven hundred

and thirty. One hundred and forty thousand. Seven hundred thousand five hundred and sixty-three. Seventeen millions. Eighty-four millions two thousand and forty-nine. Two hundred millions and fifteen.

SIMPLE ADDITION.
Addition teaches to collect several numbers into one:

The number formed by adding several numbers is called the amount or sum of those numbers.

RULE. Place the numbers one under another, with units under units, tens under tens, &c. then, beginning with the units, add up all the columns successively, and under each column set down its amount. But if either of the amounts (except the last) be more than 9, set down its right hand figure only, and add the number expressed by its left hand figure or figures into the next column. The whole amount of the last column must be set down.

PROOF.
Perform the addition downwards.

EXAMPLES
4 1 3 3
4 8 3 2

5 1 3 0
4 2 1 1
8 5 2 3

4 3 2 0 3 0 2 2

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

7 2 4 4 2 0 7 2 4 0

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

8.0 4 67 0 9.4 2 8 2 0 170

100 2:04

2.0

37 4 3 1 5 6 0 2 9 0

200 3 2 0

7 0 2 5 OO

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Add the following numbers, viz. 14, 18, 99,-45,"28, 27% 19,-38, 16; 39, 48, 29, 260, 148.

Add, six hundred and forty, seventy-nine, eighty, one hundred, two hundred and ten, four hundred and fifty.

Add, nineteen thousands, fifty thousands, one million one hundred and one, one hundred and twenty-five.

APPLICATION. 1. If John give Charles twenty nuts, and James give him fifty-six, and Joseph give him ninety-five, how many will he have?

Answer 171. 2. A person went to collect money, and received of one man ninety dollars; of another, one hundred and forty dollars; of another, one hundred and one dollars; and of another, twenty-nine dollars. How much did he collect in all?

Ans. 360 dollars. 3. Deposited in bank, fisty dollars in gold; three hundred dollars in silver, and five thousand dollars in notes. What is the whole amount deposited? Ans. 5350dols.

4. The distance from Philadelphia to Bristol is 20 miles; from Bristol to Trenton, 10 miles; from Trenton to Princeton, 12 miles; from Princeton to Brunswick, 18 miles; from Brunswick to New York, 30 miles. How many miles from Philadelphia to New York? Ans. 90.

5. A merchant bought of one person 50 barrels of flour for 300 dollars; of ano:her person, 75 barrels for 525 dollars; and of another person, 125 barrels for 1000

The nu

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