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4. From their names we see that small numbers are reckoned by ones, larger numbers by tens, and still larger numbers by hundreds, as far as ten hundred. To ten hundred we give the simple name thousand.

Above a thousand numbers are reckoned by thousands, by tens of thousands, and by hundreds of thousands, up to ten hundred thousands, or a thousand thousands, which we call a million.

Above a million numbers are reckoned by millions, by tens of millions, and by hundreds of millions, up to a thousand millions, which we call a billion.

Above a billion numbers are reckoned by billions, by tens of billions, and by hundreds of billions, up to a thousand billions, which we call a trillion. And so we go on with higher numbers as far as we choose.

5. One, Ten, a Hundred, a Thousand, Ten-thousand, a Hundred-thousand, a Million, etc., are called units, because they are used in reckoning or measuring other numbers. *

6. To distinguish these units, we call one a unit of the first order, ten a unit of the second order, a hundred a unit of the third order, a thousand a unit of the fourtb order, ten thousand a unit of the fifth order, and so on. When we speak of a unit without mentioning the order, we usually mean a unit of the first order, or one. 7. These units form a scale; and because ten units of

order make a unit of the next higher order, the scale is called a scale of tens, or a decimal scale.

any

* A unit is a fixed quantity of any kind used to measure other quantities of the same kind. Thus, a foot, a yard, a meter, are units, being fixed lengths used to measure other lengths; a pound, an ounce, a dollar, a cent, an hour, a second, are units, used to reckon or measure weight, value, or time. The word unit is also much used as a name for one, and nits for ones.

8. A system of numbers whose successive units form a scale of tens is a decimal system of numbers. The system of numbers in common use is a decimal system.

9. Table of Units of the Different Orders.

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Ten ones (or units)

make a Ten, Ten tens

make a Hundred, Ten hundreds .

make a Thousand, Ten thousands

make a Ten-thousand, Ten ten-thousands .

make a Hundred-thousand, Ten hundred-thousands

make a Million, and so on.

10. It will be convenient to remember that
A thousand ones (or units). are a Thousand,
A thousand thousands

are a Million, A thousand millions

are a Billion, A thousand billions

are a Trillion

and so on.

11. Exercises.

5. Count by hundreds to a thousand; to two thousand ; to two thousand five hundred.

6. Count by thousands to ten thousand ; by tens of thousands to a hundred thousand; by hundreds of thousands to a million.

7. Count by millions to ten million ; by tens of millions to a hundred million ; by hundreds of millions to a billion.

8. How many units of each order are there in twenty-five ? seventeen? 'eleven? ninety? ninety-nine? four hundred ? five hundred forty-four ? one thousand eight hundred ?

NOTE. This kind of exercise may be extended at the discretion of the teacher.

Writing Numbers.

one,

seven,

12. Besides being expressed in words, numbers are expressed by writing the signs 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are called figures. These signs are also called Arabic numerals, because they were first made known to Europeans by the Arabs.*

13. The first of these signs, 0, is called zero, or cipher and is used to stand for no number; the others are used to stand for the first nine numbers, and take their names, thus: 1, 2,

3, 4, 5, 6, 7, 8, 9. two, three, four, five, sisc,

eight, nine 14. Numbers higher than nine have no single signs foi themselves, but are expressed by writing side by side two or more of the figures above.

15. Tens are expressed by writing a figure to tell how many tens, and then writing a zero at the right of it. The tens' figure is then said to stand in the second place, the first or units' place being filled by a zero. Thus, we write Ten (one ten), 10. Forty (four tens), 40. Seventy (seven tens), 70. Twenty (two tens), 20. Fifty (five tens), 50. Eighty (eight tens), 80. Thirty (three tens), 30. Sixty (six tens), 60. Ninety nine tens), 90.

16. Numbers that are made up of tens and ones are expressed by writing a figure in the second place for the tens, and a figure in the first place for the ones. Thus,

Eleven (ten and one), 11. Twenty-one (two tens and one), 21. Twelve (ten and two), 12. Twenty-two (two tens and two), 22.

Let the teacher dictate numbers between ten and a hundred for the pupil to write.

* For an account of the Roman numerals, which were displaced by the Arabic, see Appendix, p. 299.

17. Hundreds are expressed by writing a figure in the third place, the second and first places being filled by zeros. Thus, One hundred, 100. Four hundred, 400. Seven hundred, 700. Two bundred, 200. Five hundred, 500. Eight hundred, 800. Three hundred, 300. Six hundred, 600. Nine hundred, 900.

18. Numbers made up of hundreds, tens, and ones are expressed by writing a figure in the third place for the hundreds, a figure in the second place for the tens, and a figure in the first place for the ones. Thus,

Four hundred eighty-three (4 hundreds, 8 tens, 3 ones), 483. Nine hundred sixty (9 hundreds, 6 tens, no ones), 960. Nine hundred six (9 hundreds, no tens, 6 ones), 906. Let the teacher dictate numbers between a hundred and a thousand for the pupil to write.

19. Thousands, tens of thousands, and hundreds of thousands are expressed by writing figures in the fourth, fifth, and sixth places respectively. The figures in these three places taken together form the Thousands' group; while the figures in the hundreds', tens', and units' places taken together form the Units' group. These groups are usually separated by a comma. Thus, One thousand, 1,000. Three thousand,

3,000. Ten thousand,

.

.

10,000. Twenty thousand, 20,000. A hundred thousand, 100,000. Five hundred thousand, 500,000. Five hundred twenty-three thousand, .

523,000. Six hundred eight thousand, seven hundred twenty-eight, . 608,728.

20. Exercises. Write in figures :

9. Four thousand. 10. Four thousand four hundred. 11. Four thousand forty. 12. Four thousand four.

13. Eight thousand, four hundred twenty-two. 14. Three hundred fifty-six thousand, eight hundred ninety. 15. Sixty thousand, sixty-five. 16. Eighteen hundred seventy-eight.

Let the teacher dictate other numbers, to a million, for the pupil to write.

21. The examples above given illustrate the principle on which all numbers are written, and which is this: Units of any order are expressed by writing a figure in the place corresponding to that order.

If the units of any orders are wanting in the number, the corresponding places are filled by zeros.

22. The general method of writing numbers on the principle above stated is shown by the following

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NOTE. For the names of higher numbers, see Appendix, page 300.

23. This table shows that the figures used to express a number fall into groups of three figures each. The first group expresses simple units, tens, and hundreds; the second, units, tens, and hundreds of thousands; the third, units, tens, and hundreds of millions; and so on.

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