where the figures express 3 hundred-thousands no tenthousands 6 thousands 5 hundreds 7 tens 4 units, or three hundred six thousand five hundred and seventyfour.

31. In general, in Writing a Number by figures, we write each order of figures expressing its orders of units, and note the absence of any order of units, by a cipher. Thus,

Five thousand twenty, or 5 thousands no hundreds 2 tens no units, is written 5,020.

One hundred-thousand forms a unit of what order? Give the names of the first six orders. How are figures separated for convenience in reading? From what order does each period take its name? Name the first two periods. In general, how do we write figures in expressing numbers?

30. The first SIX ORDERS OF UNITS, beginning with the primary units, are named: UNITS, tens, hundreds, THOUSANDS, ten-thousands, hundred-thousands.

For convenience, in reading numbers expressed by figures, places, or orders of figures, are separated by a comma (,) into groups, of three figures each, called Periods. Each period takes its name from its righthand order. Thus, we have,

Second Period,

THOUSANDS.

First Period,

UNITS.

Decimal Point

Fifty thousand two hundred two, or 5 ten-thousands 2 hundreds 2 units, is written 50,202.

Five hundred fifty-five thousand two hundred twentytwo, or 5 hundred-thousands 5 ten-thousands 5 thousands 2 hundreds 2 tens 2 units, is written 555,222.

EXERCISES.

Write in figures and read : —

1. Six units of the fifth order, with seven units of the third order.

Ans. 60,700; read, sixty thousand seven hundred.

2. 3 hundred-thousands 4 ten-thousands 4 thousands. Ans. 344,000; read, three hundred forty-four thousand.

3. Two units of the fourth order, with three units of the second order.

4. Eight units of the fifth order, with three units of each of the lower orders.

5. 9 units of the sixth order, with no units of the fifth order, and six units of each of the lower orders.

6. 316 in the second period, 0 in each order of the first period.

7. 21 thousands 2 tens and 1 unit.

8. 2 hundred-thousands 5 ten-thousands with five units of the third order.

9. The greatest number that can be expressed by three figures.

10. The greatest number that can be expressed by six figures.

REVIEW. What are ten tens called? (25.) Ten hundreds? (27.) Ten thousands? (28.) Ten ten-thousands? (29.) Name the first six orders. (30.) The first two periods. (30.)

NOTATION.

32. Notation is the method of writing numbers.

The method of expressing numbers by figures is called the Arabic, because it was used by the Arabs.

33. The Periods are named from the right; thus, Units, thousands, millions, billions, trillions, etc. The Orders are named from the right; thus,

UNITS, tens, hundreds; THOUSANDS, ten-thousands, hundred-thousands; MILLIONS, ten-millions, hundred-millions; BILLIONS, ten-billions, hundred-billions; TRILLIONS, tentrillions, hundred-trillions, etc. Hence the following

79 3, 1 5 4, 2 0 5,

6

where the figures express five hundred sixty-two TRILLION, seven hundred ninety-three BILLION, one hundred fifty-four MILLION, two hundred five THOUSAND, six hundred and thirty-eight.

What is Notation? What is the method of expressing numbers by figures called? Why so called? Name the Periods from the right. Name the Orders. State the Orders in the table.

The periods above TRILLIONS, in their order, are QUADRILLIONS, QUINTILLIONS, SEXTILLIONS, SEPTILLIONS, etc.

PRINCIPLES OF NOTATION.

34. 1. Ten units of any order are always equal to one of the next higher.

For ten units make one ten, ten tens make one hundred, ten hundreds make one thousand, and so on. Hence,

2. Each removal of a figure an order towards the left makes the value expressed ten-fold.

For the value of the units expressed by a figure is determined by the order occupied by the figure.

WRITTEN EXERCISES.

35. 1. Write in figures, seven million six hundred ninety-three thousand two hundred four.

Writing the 7 millions as the only order of the third period, 693 thousands as the orders of the second period, and 204 as the orders of the first period, we have 7,693,204. Hence, the following

Rule. - Beginning with the highest period to be expressed, write the figures belonging to each period, in their orders, and observe to mark the omission of any order of units with a cipher.

What periods are above Trillions? How many units of any order are always equal to one of the next higher? How is the value expressed increased by each removal of a figure toward the left? Recite the Rule for Notation.

Examples.

Write in figures the following numbers:

2. Seven hundred seventy.

3. One thousand eight hundred eighty-five.

4. Three thousand five hundred fifty-three.
5. Eleven thousand one.

6. Eleven hundred eleven.

7. Seventy-three thousand five hundred ninety-two. 8. Eighty-four thousand nine hundred nine.

9. Two hundred thirty thousand five hundred six. 10. Forty-one thousand nineteen.

11. Nine thousand nine hundred seven.

12. Eighty-nine thousand ninety-seven.

13. Twenty-one thousand one hundred twenty-one. 14. Three hundred thousand six.

15. Fifteen thousand one hundred fifteen.

16. Seventy-nine thousand nine hundred seven. 17. Sixty-seven thousand three hundred six.

18. Six hundred thirty-five thousand four hundred thirty-eight.

19. Forty-two thousand four hundred forty-four.

20. Ninety-eight thousand six hundred nine.

21. Nineteen thousand three hundred fifty-one. 22. One hundred thousand forty-seven.

23. One million ten thousand ten.

24. Sixty-one million sixteen thousand six hundred five.

25. Eight hundred twelve thousand three hundred forty-seven.

What is an Example? Ans. A question, or something to be done, under a rule.