These groups are called the units' group, the thousands' group, the millions' group, etc., from the lowest order of units which they express.

NOTE. The units themselves are grouped as the figures are. (Arts. 4 and 10.)

24. In writing large numbers it will be found convenient to think chiefly of the groups as above described. Thus, let it be required to write the number Forty-nine billion, three hundred seven million, seventy thousand, six hundred forty-three. The groups are

49 billion, 307 million, 70 thousand, 643.

and the number itself is written

49,307,070,643.

25. Exercises.

I. Beginning with the units' group, repeat the names of the groups to trillions; repeat the names from trillions to units. II. Write the groups of figures required to express the following numbers, with the names of the groups:

17. Forty-six thousand, five hundred twenty.

18. Four hundred six thousand, five hundred two. 19. One million, one thousand, one hundred ten.

26. Exercises.

I. Write in figures the following numbers:

20. Eighty-five million, five hundred three thousand, seven. 21. Nine hundred six million, two hundred eighteen thousand, twenty-eight.

22. Three billion, thirty-seven million, nine hundred thousand, two hundred.

23. Eighteen billion, four.

24. Forty million, seven hundred thousand.

25. Thirty-seven trillion, ninety-nine billion, nine million.

II. Write in figures as many of the numbers named on page 62 as the teacher may indicate.

Reading Numbers.

27. Let it be required to read the number 53869214. To prepare this expression for reading, we begin at the right, and point off three figures for the units' group, three more for the thousands' group, leaving two for the millions' group, thus:

53,869,214.

Now beginning at the left, we name the number expressed by each group, adding the name of the group, thus: Fifty-three million, eight hundred sixty-nine thousand, two hundred fourteen [units].

NOTE. The name of the units' group is usually omitted in reading.

II. Read across the page as many of the numbers expressed on page 60 as the teacher may indicate.

29. It is frequently convenient to separate a number into parts, each part containing only the units of a single order. Thus, the number 734 may be separated into 7 hundreds, 3 tens, and 4 units. Such parts are called the terms of a number.

Decimal Fractions.

30. As a hundred is made up of ten equal parts, each of which is a ten, and as a ten is made up of ten equal

parts, each of which is one, so we may consider one to be made up of ten equal parts, each of which is a tenth; a tenth to be made up of ten equal parts, each of which is a hundredth; a hundredth to be made up of ten equal parts, each of which is a thousandth; and so on.

Now a hundred is written 100; the tenth part of a hundred (ten) is written 10, the figure 1 being moved one place to the right; and the tenth part of ten (one) is written 1, the figure 1 being moved one place further to the right; so, following the same plan, the tenth part of one (one tenth) is written 0.1; the tenth part of a tenth (one hundredth) is written 0.01; the tenth part of a hundredth (one thousandth) is written 0.001; and so on.

Tenths, hundredths, thousandths, etc., are fractional units, or fractions; and, as they form a decimal scale (Art. 7), collections of such units are called decimal fractions.

31. The dot put at the right of the units' place is called the decimal point.

32. The relations of these fractional units to the higher units are shown by the following table, which may be extended both ways as far as we please :

33. We see then that decimal fractions may be written on the principle stated in Art. 21. Thus, we write

Two tenths.

Five hundredths.

. 0.05 Thirty-two thousandths.

Twenty-five hundredths 0.25 Three hundred sixteen thousandths 0.316

34. The method of writing decimal fractions is shown by the following table, which is merely an extension of the table given in Art. 22.

NOTE.

Places.

Figures. O

70 8 9 6 3 4 3 2

In writing decimal fractions it is well to fill the units' place with

a zero when there is no other figure to be written there.

35. To read a decimal fraction, name the number expressed by the figures, and then add the name of the units expressed by the right-hand figure.

Thus, 0.0739 is read "seven hundred thirty-nine tenthousandths." See Appendix, p. 300.

When a whole number and a decimal fraction are written together, read first the whole number and then the fraction.

Thus, 56076.028 is read "fifty-six thousand seventy-six, and twenty-eight thousandths."

II. Write in figures the following numbers:

(53.) Seven tenths.

(54.) Seven hundredths.

(55.) Seven thousandths.

(58.) 7 units and 5 thousandths. (59.) 25 units and 49 ten-thousandths.

(56.) Twenty-five hundredths. (60.) 306 hundred-thousandths. (57.) Thirty-nine thousandths. (61.) 5047 hundred-thousandths.

Let the teacher dictate other numbers between units and millionths for the pupil to write.

37. Questions for Review.

What is a number? How are numbers reckoned? (Art. 4.) What general name do you give to one, a ten, a hundred, a thousand, etc.? How do you distinguish the different units? What kind of a scale do they form? What system of numbers is in common use? Why is it so called?

What is the meaning of the word thirteen? eleven? twelve? twenty? thirty-seven? (Appendix, page 299.)

How many units make a thousand?

How many thousands make a million? How many millions make a billion?

What is the use of figures? How are numbers higher than nine written ? On what principle are all numbers written? (Art. 21.) What is the use of zeros?

How are the figures used to express a number grouped? Name the first five groups. How do you write large numbers? (Art. 24.) Illustrate. How do you read a number? (Art. 27.) Illustrate. What are the terms of a number? Name the terms of the number 6725.

What is the largest number that can be expressed by one figure? by two figures? by three figures?

What is the least number of figures that will express units? thousands? millions?

In 100, how many tens? how many units? In 15000, how many hundreds? units? tens? In 18462, how many tens, and how many units remain? how many hundreds, and how many units remain?

What is the effect of placing zeros at the right of an expression for whole numbers? at the left?