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FISH'S ARITHMETIC.

NUMBER TWO.

1. Arithmetic is the science of numbers and the art of computation.

As a science, arithmetic treats of the nature and properties of numbers. As an art, it teaches how to apply a knowledge of numbers to practical and business purposes.

2. A unit is one thing, or a group of things regarded as one. Thus, one day, one man, one ten, one dozen, are units.

3. A number is a unit, or a collection of units.

Thus, one, six; five books, eight days, are numbers.
A number answers to the question, "How many?"

4. An integral number or integer is a number representing whole things.

Thus, 5, 7; 12 books, 9 men, 20 days, are integers or whole things.

5. The unit of a number is one of the collection forming that number.

Thus, 1 is the unit of 4; 1 dollar is the unit of 5 dollars.

6. Like numbers are numbers having the same kind of unit, or that express the same kind of quantity.

Thus, 3 and 5,2 days and 6 days, 5 cents and 10 cents are like numba,

NOTATION AND NUMERATION.

7. Notation is a method of writing numbers. Numbers are expressed by words or characters. The characters may be either figures or letters.

8. Numeration is the method of reading numbers expressed by characters.

9. Arabic notation employs ten different characters called figures, to express numbers, viz.:

0 1 2 3 012

4 5 6 7 8 9

Naught, one, two, three, four, five,

six,

seven, eight, nine.

The first character, naught, is also called cipher, or zero, and when standing alone, has no value.

The other nine are called significant figures, because each has a value of its own. They are also called digits.

These ten characters, when combined according to certain principles, can be made to express any number.

10. In representing numbers, objects are supposed to be arranged in groups of ten, each group being ten times as great as the next lower group, and having a different name. Hence, we have single things, or units; next, groups containing ten units, or one ten; next, groups containing ten tens, or one hundred; and again, groups containing ten hundreds, or one thousand, etc.

11. This method of grouping is called the decimal system, from the Latin decem, which signifies ten.

The successive order of units in a number form a scale; and when the increase or decrease is uniformly ten, it is called a decimal scale.

12. The first nine numbers are each expressed by a single figure, and are called units of the first order.

13. Since there is no single character to express the number ten, by grouping ten ones or units of the first order into a larger collection, there is formed a unit of the second order, called ten, 'written 10.

In the same manner are expressed,

2 tens, or Twenty, by 20.
3 tens, or Thirty,
tens, or Forty,
5 tens, or Fifty, by 50.

1 Unit.

1 Ten.

by 30.
by 40.

6 tens, or Sixty,
7 tens, or Seventy, by 70.
8 tens, or Eighty, by 80.
9 tens, or Ninety, by 90.

by 60.

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The numbers from one hundred to nine hundred ninetynine inclusive are expressed by writing hundreds in the third place from the right, tens in the second place, and units in the first place.

The greatest number that can be expressed by three figures is 999.

15. By grouping ten units of the third order or ten hundreds into a larger collection, there is formed a unit of the fourth order, called one thousand, written 1000.

In like manner are expressed,

2 thousand, by 2000.
3 thousand, by 3000.
4 thousand, by 4000.

5 thousand, by 5000.

6 thousand, by 6000.

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The numbers from one thousand to nine thousand nine hundred ninety-nine are expressed by writing thousands in the fourth place from the right, hundreds in the third place, tens in the second place, and units in the first place.

The greatest number that can be expressed by four figures is 9999.

16. In the same manner, other new orders are formed to represent larger numbers, by grouping ten units of the fourth order to form the fifth order, or tens of thousands; and ten units of the fifth order to form the sixth order, or hundreds of thousands, etc.

Thus, 432076 represents 6 units of the first order, 7 units of the second order, 0 units of the third order, 2 units of the fourth order, 3 units of the fifth order, and 4 units of the sixth order, and is read, Four hundred thirty-two thousand seventy-six.

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17. Orders of units are denoted by the position of the figures used in expressing a number.

Thus, 532 represents 2 units of the first order, 3 units of the second order, or 3 tens, and 5 units of the third order or 5 hundreds, and is read five hundred thirty-two.

PRINCIPLES.-I. Ten units of any order in a number make one unit of the next higher order.

II. When a figure is moved one place to the left, it represents units of ten times the value it did before.

III. When a figure is moved one place to the right, ît represents units of one tenth the value it did before.

When any order of units in a number is vacant, the place should be filled with a cipher.

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18. Copy and read the following numbers, naming in each the number of units of each order:

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1. Seven units of the fourth order, eight of the second

order, and three of the first.

2. Three units of the fifth order, two of the third, and one

of the first.

3. Two units of the sixth order, nine of the fifth, four of the third, one of the second, and seven of the first.

4. Forty-seven thousand six hundred forty.
5. One hundred forty-six thousand five hundred.
6. Ninety-five thousand three hundred ten.
7. Four hundred seventy thousand two hundred.
8. Eight hundred five thousand one hundred three.

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