numbers are to be added; but the name of a smaller number coming before a larger, or before an equal number, signifies that the second is to be multiplied by the first.

17. Read the numbers:

764,123,897; 40080; 795,013; 103,547,020; 71,003,054; 3,125,476,890; 79,501,346,081; 3,001,574.

18. Write in order the numbers from 1 to 25; from 95 to 115; from 195 to 215; from 985 to 1011; from 9995 to 11111; from 99999 to 100,011.

19. Write in figures, then read from the figures:

One hundred ten; one hundred one; two hundred seven; three hundred seventy; six hundred forty-one; seven hundred thirty-two; eight hundred eight; eight hundred eighty; nine hundred sixty-four; nine hundred ninety-three; one hundred seven; one thousand seven; one thousand ten.

Six million one thousand one; three million five; five million six; four million three hundred thousand three hundred three; seven million six hundred thousand eight hundred twenty-nine; eighty-one thousand ninetyfive; seventeen hundred thousand millions; twenty-one thousand thousands; eighty-three million millions. One hundred ten million two hundred seventy-nine; nineteen trillion four million three hundred three; one quadrillion one hundred twelve trillion three hundred thirtyfour million two hundred eleven; ten quintillion two trillion three hundred billion five thousand seven.

CHAPTER II.

DECIMAL FRACTIONS.

20. THOSE things of which we do not naturally ask, How many? but, How much? we endeavor to measure; and we answer the question, How much? by answering, How many measures?

1

21. The measure of any kind of quantity is readily conceived as divided into ten smaller measures. A dollar, for example, as a measure of value, is divided into ten dimes; each dime into ten cents; each cent into ten mills.

22. Dollars are signified by the mark $ written at the left of the figures.

When part of a dollar is to be written, a full point is written after the dollars, the dimes are written to the right, then the cents and mills.

For example: $17 is seventeen dollars; $18.20 is eighteen dollars, two dimes, or eighteen dollars, twenty cents; $35.875 is 35 dollars, 87 cents, 5 mills; $0.08 is 8

cents.

23. Read as dollars, cents, and mills: $76.375; $163.58; $241.185; $357.34; $12.50; $0.875; $0.125; $1.01; $10.10.

24. A dime is the tenth of a dollar, a cent the hundredth of a dollar, a mill the thousandth of a dollar.

Parts of other measures than those of value may be written in the same way; with tenths, hundredths, etc., to the right of a point. Thus, if we omit the mark $ from $5.375, it may stand for 5 quarts, yards, bushels, or any other full measures, and 375 thousandths of another measure.

25. Parts thus written are called Decimal Fractions. We write and number to the right of the units' place, precisely as we do to the left, first carefully marking the units' place with a decimal point to its right. Thus, in the figures

9,876,543,210.123,456,789

the full point after 0 shows that O stands in the units' place. The 1 to the left is 1 ten, the one to the right is 1 tenth; the 2 to the left is 2 hundreds, the 2 to the right is 2 hundredths; the 3 to the left is 3 thousands, the 3 to the right is 3 thousandths; the 4 to the left is 4 ten-thousands, the 4 to the right is 4 ten-thousandths; the 5 to the left is 5 hundred-thousands, the 5 to the right is 5 hundred-thousandths; the 6 to the left is 6 millions, the 6 to the right is 6 millionths; and so on.

In like manner, the 210 is the units' period; the 543 is the thousands', the 123 the thousandths' period, etc.; so that the number may be read 9 billions, 876 millions, 543 thousands, 210, and 123 thousandths, 456 millionths, 789 billionths.

26. In reading decimal fractions in this manner, we are obliged, if the right-hand period contains less than three places, to fill the missing places mentally with naughts. For example, .0004 would be read 400 millionths. This is sometimes objectionable (for reasons which will hereafter be plain to the student), and therefore the more usual way of reading is:

Read the decimal precisely as if a whole number, and add the fractional name of the lowest place.

For example, 5.17 is read 5 and 17 hundredths; 5.0017, five and 17 ten-thousandths; 6.0203107, six and 203 thousand 107 ten-millionths.

In either of these ways of reading decimals, the word "and" is distinctly pronounced at the decimal point and carefully omitted in all other places. Thus, one hundred forty-seven means 147; but one hundred and forty-seven thousandths means 100.047; and .147 must be read one hundred forty-seven thousandths.

Another ambiguity in this way of reading can be avoided only by a pause; thus, .300 is three hundred . . . thousandths, while .00003 is three . . . hundred-thousandths.

27. To avoid these ambiguities, practical computers introduce the word "decimal" at the place of the point, and then pronounce the digits in succession to the right. Thus, 203.07051 is read two hundred three, decimal, naught, seven, naught, five, one.

28. Read the mixed numbers:

17.23; 18.41; 27.49; 341.07; 1.52; 0.52; .52; .1357; 201.106; 11.111; 13.013; 17,000.017; 6132.0173; .0609; .00613; 26.7; 2.67; .267; .00267; 195.123; 0.83; 0.0087; .00091; 3.1416; 3.14159; 3.14159265.

29. The necessity for putting the decimal point in its right place may be seen on comparing $312.50, $31.25, $3.125, and $0.3125. To move the point one place is to multiply or divide by ten.

30. Write in figures, then read from the figures:

75 hundredths; 8 thousandths; 7 tenths; sixty hundredths; 77 thousandths; 83 ten-thousandths; eight and three ten

thousandths; six, decimal, naught, naught, one, naught, three; five, decimal, six, naught, seven, three, one; Nine and 43 millionths; 143 millionths; one hundred and 43 millionths; one hundred forty and three millionths; nine hundred forty-three thousand and nine hundred forty-three thousandths; 722 ten millionths; thirteen, decimal, naught, one, four, six, eight.

31. Point 6753241 in eight different ways, and read each. Point 8957 in eight different ways, prefixing or annexing naughts, and read each.

32. Write in figures:

Three hundred seven; three hundred and seven thousandths; three hundred seven thousandths; three and one hundred seven thousandths; three hundred seven and fourteen thousandths.

33. Read as dollars, cents, and mills:

$24.073; $16.187; $35.625.

Read the same figures as whole numbers and decimal fractions without naming any unit.

34. Write in figures:

81 thousand and 345 thousandths; 37 hundred 41 and 675 thousandths; 4 hundred 13 and 8 hundredths; 96 and 96 thousandths.

Read them as dollars, cents, and mills.

35. Write as money, then read as whole numbers and fractions without naming any unit:

Five and a half dollars; thirteen dollars and twenty-five cents; 83 dollars and 14 cents; 60 dollars and 12 and a half cents; 6 dollars, 9 cents, and 9 tenths of a mill.