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DRILL TABLES, 29, 44, 71, 89, 146, 159, 181.

MISCELLANEOUS EXAMPLES AND REVIEWS, 41, 67, 87, 90, 103, 135,

142, 156, 194, 233, 289, 365, 394.

NEW

FRANKLIN ARITHMETIC.

SECOND BOOK.

SECTION I.

NUMBERS AND THEIR EXPRESSION.

ARTICLE 1. A collection of similar objects suggests the ideas of one and more than one. One of any kind is a unit. A unit or a collection of units is a number.

2. Arithmetic is the science of numbers and the art of using them in computation.

INTEGRAL NUMBERS.

3. An integral number is a number whose unit is a whole or undivided thing, as seven apples, the unit of which is one apple; ten days, the unit of which is one day; fifteen dozen, the unit of which is one dozen.

4. Small numbers are reckoned by ones and by tens, larger numbers by tens of tens, or hundreds, and still larger numbers by tens of hundreds, or thousands. Above thousands, numbers are reckoned by ten-thousands, hundredthousands, millions, ten-millions, hundred-millions, and so on.

5. One, ten, a hundred, a thousand, etc., are all called units, because each is used as a single thing in reckoning other numbers.

6. To distinguish these units, one is called 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 fourth order, ten thousand a unit of the fifth order, and so on. When a unit is spoken of without the order being named, a unit of the first order, or one, is meant.

7. These units form a scale; and because ten units of any order make a unit of the next higher order, the scale is called a scale of tens, or a decimal scale.

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.

NOTATION AND NUMERATION.

9. Notation is the art of writing numbers, and numeration is the art of reading them.

10. There are two systems of notation in common use, the Roman and the Arabic. The Roman notation employs the characters I. V. X. L. C. D. and M. and combinations of these to express numbers. It is not used in arithmetical computation.

11. The Arabic notation employs ten characters called figures.

5 6 7 8 9

O 1 2 3 4
Zero One Two Three Four Five

Six Seven Eight Nine

12. The first of these figures, 0, is called zero or cipher, and stands for no number. The others stand respectively for the numbers whose names are written beneath.

13. Numbers larger than nine require two or more figures for their notation. These are written side by side, and so the place occupied by each figure becomes important.

14. Tens are expressed by writing a figure to denote how many tens, and then placing a zero at the right of it. The figure so written is said to occupy the second place, or the tens' place, while the first or units' place is filled by the zero. Thus,

Ten (one ten), 10. Forty (four tens), 40. Seventy (seven tens), 70.

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

Eleven (one ten and one), 11. Twenty-four (two tens and four), 24. 16. Hundreds are expressed by writing a figure in the third place, the second and first places being filled with Thus,

zeros.

One hundred, 100.

Five hundred, 500.

17. Numbers made up of hundreds, tens, and units 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 units. Thus,

430.

Three hundred twenty-seven (3 hundreds, 2 tens, 7 units), 327.
Four hundred thirty (4 hundreds, 3 tens, 0 units),
Five hundred six (5 hundreds, 0 tens, 6 units),

506.

18. Thousands are expressed by writing a figure in the fourth place, tens of thousands by writing a figure in the fifth place, and hundreds of thousands by writing a figure in the sixth place. The figures in these three places taken together form a group called the thousands' group; while the figures in the hundreds', tens', and units' places form the units' group. These groups are usually separated by a comma. Thus,

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Nine hundred seventy-eight thousand eight hundred six, 978,806.

19. Millions, tens of millions, and hundreds of millions are expressed by writing figures in the seventh, eighth, and ninth places. These figures taken together form the millions' group. Thus,

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Five hundred fifty-five millions, 555,000,000.

20. The above illustrations show the principles upon which all numbers are written, which are

(1) Units of any order are expressed by writing a figure in the place corresponding to that order, and filling vacant places with zeros.

(2) The value expressed by a figure is increased tenfold by each removal one place to the left, and decreased tenfold by each removal one place to the right.

21. The general method of writing integral numbers is shown by the following

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480, 297, 034, 508, 672 FIGURES.

[blocks in formation]

NOTE. The groups above billions are seldom used.

Their names

are trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, tredecillions, quatuordecillions, quindecillions, sexdecillions, septendecillions, octodecillions, novemdecillions, vigintillions, etc.

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