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13. The other numbers, up to one thousand, may be expressed by putting a significant figure in the place of one or cach of the ciphers in the above numbers; thus,

Two hundred and three, expressed in figures, is
Six hundred and eighty,

Nine hundred and ninety-eight,

66

66

66

66

203

680

998

14. The simple name of any significant figure is always the same, but the number indicated by it depends upon the place the figure occupies; for example, 6 is always six, and never seven. So in each of the following numbers, 2, 20, and 200, the left hand figure is two, but in the first it is two units; in the second, two tens or twenty; and in the third, two hundreds. Thus each significant figure has a simple or name value, and a local or place value.

15. When two or more figures are used together they are said to express different ORDERS of units. The right hand figure represents simple units, or units of the first order; the second figure represents tens, or units of the second order; the third represents hundreds, or units of the third order; thus, in the number 426 the 6 is simply six, while the 2 is two tens or twenty, and the 4 is four hundreds; and the number expressed by the three figures taken together is four hundred and twenty-six.

16. The figures of large numbers, for convenience in reading, are often separated by commas into groups or periods of three figures each, beginning at the right. The first or right-hand group contains units, tens, and hundreds,

13. Other numbers to one thousand? 14. Does the name of a figure ever change? Does its value change? How many values has a figure? The names of those values? 15. What is said of orders of units?

is said of groups or periods of figures?

16. What

Tens of Trillions,
Trillions,

and is called the period of units; the second period contains thousands, tens of thousands, and hundreds of thousands, and is called the period of thousands, etc., as in the following

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17. The value of the figures in this table, expressed in words, is sixty-nine trillion, five hundred and forty billion, seven hundred and six million, four hundred and seventy-six thousand, eight hundred and forty-three.

NOTE. The READING of a number consists of two distinct processes: First, reading the order of the places, beginning at the right hand; thus, units, tens, hundreds, thousands, etc., as in the Numeration Table; and, second, reading the value of the figures, beginning at the left, as in Article 17, above. To distinguish these processes, the first may be called numerating, and the second reading, the number.

18. The value of a figure is increased ten fold by removing

Name the periods in the Numeration Table, beginning at the right. Name the figures in each group. 17. Read the value of the figures in the Numeration Table. NOTE. How many processes in reading a number? Describe them, and tell what they are called. 18. How is the value of a figure affected by changing its place? Illustrate. What general law?

it one place toward the left; a hundred fold by removing it two places, etc., that is, ten units of the first order make one ten, ten tens make one hundred, ten hundreds make one thousand, and, in short, ten units of any order make one unit of the next higher order.

19. The cipher, when used with other figures, fills a place that would otherwise be vacant; thus, in 206 the cipher occupies the place of tens, because there are no tens expressed in the given number.

20. From the foregoing, to numerate and read a number expressed by figures, we have the following

RULE 1. Beginning at the right, numerate, and point off the number into periods of THREE figures each.

2. Beginning at the left, read each period separately, giving the name of each period except that of units.

EXERCISES IN NUMERATION.

Let the learner read the following numbers:

21.

1.

8

11.

4,683

21.

300,006

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NOTE. The teacher should give examples similar to the above upon the blackboard or slate, sometimes inserting and sometimes omitting the commas, until the pupil can readily group, numerate,

19. The cipher, for what used? 20. Rule for numerating and reading. a number?

and read all numbers likely to occur in his lessons or general reading. A like remark applies to all the following parts of the book. The teacher should give many original examples, varying in difficulty according to the abilities of his classes, and should encourage his pupils to make examples for themselves and for each other.

22. To write numbers, we have this

RULE 1. Beginning at the left, write the figures belonging to the highest period.

2. Write the figures of each successive period in their order, filling each vacant place with a cipher.

EXERCISES IN NOTATION.

23. Let the learner write the following numbers in figures, and read them:

1. Five units of the third order and six of the first. Ans. 506.

NOTE. A cipher is written in the second place, because no unit of the second order is given.

2. Three units of the fourth order, six of the second, and four of the first. Ans. 3,064.

3. Two units of the seventh order, one of the sixth, three of the third, and five of the second. Ans. 2,100,350. 4. Eight units of the fifth order, two of the third, and six of the first.

5. Six units of the eighth order, four of the sixth, two of the fourth, and five of the third.

6. Nine units of the sixth order, two of the fourth, and eight of the first.

22. Rule for writing a number. NOTE. In Notation where should 0 be written?

7. What orders of units are there in the number 3,462,895 ? How many units in each order?

8. What orders of units in the number 62,304,500? *How many units in each order?

9. How many tens in 46? How many units beside the tens? How many units in the whole of the number?

10. In 347 how many hundreds? How many tens in the tens' place? How many units in the units' place? How many tens in the number? How many units in the number?

24.

Write the following numbers in figures:

1. Two hundred and fifty-six.

2. Fifty-four.

3. Six thousand and nineteen.

Ans. 256.

Ans. 54.

Ans. 6,019.

4. One thousand eight hundred and sixty-five.

5. Four hundred and forty.

6. Twenty-five thousand two hundred and forty-nine. 7. Two hundred and forty-five thousand six hundred and fifty-four.

8. Five million six hundred thousand eight hundred and sixteen.

9. Twenty-two million two hundred and twenty-two thousand two hundred and twenty-two.

10. Five hundred and six million forty thousand two hundred and four.

11. Four billion eight million six thousand eight hundred and ten.

12. Thirty-five trillion four hundred and six billion eight hundred and twenty million two hundred and eighteen thousand four hundred and sixty-seven.

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