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THE

COMPLETE ARITHMETIC.

1. A Unit is a single thing, or one; as one hour, one dollar, one.

2. A Number is a unit, or a collection of units; as one month, four hours, seven.

3. Arithmetic treats of numbers and their use.

NOTATION AND NUMERATION.

4. Notation is writing numbers in figures.

5. Numeration is reading numbers written in figures. 6. Figures are characters used to express numbers. Ten different figures are used; they are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Zero, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.

7. Zero, or cipher, used alone expresses no units, or nothing.

The other nine figures express the number of units shown by their names.

8. To express numbers larger than nine, two or more figures are written side by side.

9. A figure written alone has only a simple name and value; but when used with other figures it has also a place-name and value.

10. The Place of a figure is its position with reference to another figure in a number.

11. When two figures are written side by side, the figure at the right occupies the first place, has the place-name ones, and expresses units of the first order. The figure at the left occupies the second place, has the place-name tens, and expresses units of the second order, each of which is ten times larger than a unit of the first order. Thus,

80 is read 8 tens, 0 ones, or briefly eighty.

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By the use of the other possible combinations of two figures any number between nine and one hundred may be written.

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14. Sixty-four. 17. Four tens, eight ones. 20. Twenty-nine. 21. Sixty-seven.

15. Seventy-nine. 18. Ninety-two.

12. A figure at the left of the tens' figure occupies the third place, has the place-name hundreds, and expresses a third order of units, each ten times as large as one of those in the second place. Thus,

659 is read 6 hundreds, 5 tens, 9 ones, or briefly, six hundred fifty-nine. 508 is read 5 hundreds, 0 tens, 8 ones, or briefly, five hundred eight.

By using the different combinations of three figures any number between ninety-nine and one thousand may be expressed.

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34. Eight hundred forty-four. 38. Five hundred eighty-eight. 35. Two hundreds, seven ones. 39. Seven hundred.

36. Seven hundred fifty-three. 40. Six hundred sixty-six. 37. Nine hundred ninety. 41. Nine hundred four.

13. A figure at the left of the hundreds' figure occupies the fourth place, has the place-name thousands, and expresses the fourth order of units, each ten times as large as one in the third place. Thus,

8976 is read 8 thousands, 9 hundreds, 7 tens, 6 ones, or briefly, eight thousand nine hundred seventy-six.

In this way the various combinations of four figures may be used to express the numbers between nine hundred ninety-nine and ten thousand.

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51. Six thousands, no hundreds, five tens, seven ones. 52. Eight thousands, two hundreds, no tens, no ones. 53. How may the preceding number be read more briefly? 54. Three thousand eight hundred forty-six.

55. Nine thousand two hundred seven.

56. Five thousand seven hundred forty.

57. Two thousand eight.

58. Four thousands, four tens, four ones.

59. Eight ones, three tens, nine hundreds, seven thousands.

14. On the same principle larger numbers are written by using five, six, seven, eight or more figures, each additional figure filling a new place, named successively towards the left, ten-thousands, hundred-thousands, millions, ten-millions, and so on, and forming a new order of units, each of which is ten times as large as the next smaller.

15. The place-names and values of figures are shown in the following

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16. For convenience in reading and writing numbers, each three places in a number, beginning with the ones, form a group. Each group is named from its right-hand order of units.

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The comma is used to mark off the groups.

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