The fourth order to the right is ten-thousandths; the fifth, hundred-thousandths, and so on.

174. Since decimals of the first order are tenths, of the second order hundredths, of the third order thousandths, etc., it follows that

The denominator of a decimal fraction is 1 with as many ciphers annexed as the numerator has decimal places.

175. Since each lower order is one tenth of the next higher, it follows, in decimals, as in simple whole numbers, that

Ten of any lower order are always equal to one of the next higher.

Hence integers and decimals may be expressed together as a mixed number, by placing the decimal point between them. Thus,

17 and 25 hundredths may be written 17.25.

An integer may be regarded as a decimal by placing the decimal point on the right of the order of units; and the expression may be read, according to the decimal orders annexed, as a number of tenths, hundredths, etc. Thus,

17 may be written 17.0, 17.00, etc., and read 170 tenths, 1700 hundredths, etc.

176. The orders of decimals are named from the decimal point to the right; thus,

Tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, etc. Hence, the

What is the fourth order to the right of the point? How are the orders named?

co Hundred-thousands.

3

4

6

7

∞ Tens.

~ Units.

8 2

5 0

3 2, where the mixed number is read, three hundred forty-six thousand seven hundred eighty-two units, and five hundred nine thousand one hundred thirty-two millionths.

177. In United States Money, the dollar is the unit, and cents and mills are decimal parts of a dollar.

Thus, $5.25 is the same as 5 dollars and 25 hundredths of a dollar; and $7.125 is the same as 7 dollars and 125 thousandths of a dollar.

Case I.

Reading Decimals expressed by Figures.

178. 1. Write and read the decimal .035.

SOLUTION. The figures are 0 tenth, 3 hundredths, and 5 thousandths; or, since 3 hundredths equal 30 thousandths, and 30 thousandths plus 5 thousandths, equal 35 thousandths, we have, as the value expressed, 35 thousandths. Therefore, .035 is read thirty-five thousandths.

Give the mixed number expressed in the Table. How may cents and mills be regarded?

Rule. Read the decimal as an integer, giving it the name of the right-hand order.

179. When the decimal is part of a mixed number, we may read the integral part as expressing units. Thus,

16.45 may be read, sixteen units and forty-five hundredths.

Writing Decimals in Figures.

180. 1. Express in figures thirteen thousandths.

SOLUTION. Since thirteen thousandths equal 1 hundredth and 3 thousandths, we write 3 in the order of thousandths, 1 in the order of hundredths, and, as there are no tenths, O in the order of tenths; and, prefixing the decimal point, we have .013.

Recite the Rule.

Rule. Write the figures at the right of the decimal point, each in its proper order, noting vacant orders, if any, by ciphers.

7

3. 18. Ans. .03. 7. 10. Ans. .007. 11. 191. Ans. 19.009.

Ans. 101.65.

15. Twenty-three ten-thousandths.

16. Thirty-six units, and five tenths.

17. One hundred one units, and sixty-five hundredths.

18. Sixteen units, and sixteen thousandths.

Ans. 16.016.

19. Three hundred twenty-five ten-thousandths. 20. Five units, and three hundred thousandths. 21. Nineteen units, and six hundred thirty-one tenthousandths.

REDUCTION.

181. Annexing a cipher to a decimal does not alter the value of the decimal.

For, the order of the significant figures of the decimal is not changed. Thus, .3, or .30, is the same as

182. Hence, to change decimals having different denominators, to equivalent decimals having a common denominator,

Recite the Rule. What effect has the annexing of a cipher to a decimal? How do you change decimals having different denominators to equivalent decimals having a common denominator?

Make each decimal have the same number of decimal orders, by annexing ciphers.

EXERCISES.

Change to equivalent fractions having a common denominator,

A Decimal to a Common Fraction.

183. 1. Change .35 to a common fraction.

OPERATION.

.35%, Ans.

is. Therefore, .35 is equal to

Removing the decimal point and writing the denominator, we have, which, reduced,

Rule. Write the denominator to the decimal, omit the decimal point, and reduce the common fraction to its lowest terms.

A Common Fraction to a Decimal.

184. 1. Change & to a decimal fraction.

Recite the Rule for reducing a decimal to a common fraction.