To read a decimal, first numerate from left to right, and the name of the right hand figure is the name of the denominator. Then numerate from right to left, as in whole numbers, to read the numerator.

18.027

81.0001

75.075

146. A Mixed Decimal Number is a number consisting of integers and decimals; thus, 71.406 consists of the integral part, 71, and the decimal part, .406; it is read the same as 71-406, 71 and 406 thousandths.

1000

.19248

.001385

.1000087

1. Write eighteen, and twenty-seven thousandths.
2. Write four hundred, and nineteen ten-millionths.
3. Write fifty-four, and fifty-four millionths.

4. Write eighty-one, and 1 ten-thousandth.
5. Write one hundred, and 67 ten-thousandths.
6. Read the following numbers:

147. From the foregoing explanations and illustrations we derive the following important principles:

PRINCIPLES OF DECIMAL NOTATION AND NUMERATION.

1. The value of any decimal figure depends upon its place from the decimal point.

Thus, .3. is ten times .03.

2. Prefixing a cipher to a decimal decreases its value the same as dividing it by ten.

Thus, .03 is the value of .3.

3. Annexing a cipher to a decimal does not alter its value, since it does not change the place of the significant figures of the decimal.

Thus, 1%, or .6, is the same as or .60.

60 100

4. Decimals increase from right to left, and decrease from left to right, in a tenfold ratio; and therefore they may be added, subtracted, multiplied, and divided in the same manner as whole numbers.

5. The denominator of a decimal, though never expressed, is always the unit 1, with as many ciphers annexed as there are figures in the decimal.

6. To read decimals requires two numerations; first, from units, to find the name of the denominator, and second, towards units, to find the value of the numerator.

RULE FOR DECIMAL NOTATION. -I. Write the decimal in the same way as a whole number, placing ciphers where necessary to give each significant figure its true local value. II. Place the decimal point before the first figure.

RULE FOR DECIMAL NUMERATION. I. Numerate from the decimal point, to determine the denominator. II. Numerate towards the decimal point, to determine the

numerator.

III. Read the decimal as a whole number, giving it the name or denomination of the right hand figure.

1. Write 425 millionths.

2. Write six thousand ten-thousandths.

3. Write one thousand eight hundred fifty-nine hundred-thousandths.

4. Write 260 thousand 8 billionths.

5. Write 26 thousand and 26 thousandths.

6. Write 1 million and 1 millionth.

7. Read the following decimals:

.6321

.5400027

8. Write five hundred two, and one thousand six millionths.

.748243

.60000000

9. Write thirty-one, and two ten-millionths.

10. Write eleven thousand, and eleven hundredthousandths.

11. Write nine million, and nine billionths.

12. Write one hundred two tenths.

16.0902

Ans. 10.2. 13. Write one hundred twenty-four thousand three hundred fifteenth thousandths.

568.070

14. Write seven hundred thousandths. 15. Write seven hundred-thousandths. 16. Read the following numbers:

12.36

142.847

1.02

305.005

4000.004

.9095

.2962999

.00000006

9.052

32.004

4.0005

14.342

634.0634

5.00005

500.50000

33.33333

62.9999

1858.4583

27.00045

3489.9843

2.000002

80000.567890

3.0802050 18752.5439082

17. Write three thousand two hundred five, and five hundred six hundred-thousandths.

18. Write five twenty-five and five thousand forty-five ten-thousandths.

19. Write seven ninety-seven, and six thousand three forty-nine hundred-thousandths.

20. Write three million six hundred thousand, and nine ten-millionths.

21. Write two, and four million one twenty-three thousand four ninety-two ten-millionths.

OPERATION.

.50000

.37500

3.25401

46.13000

REDUCTION.

EXAMPLES.

148. To reduce decimals to a common denomi

nator.

1. Reduce .5, .375, 3.25401, and 46.13 to their least common decimal denominator.

SOLUTION. The third number contains five decimal places, and hence 100000 must be a common denominator. As annexing ciphers to decimals does not alter their value (147, 3), we give to each number five decimal places by annexing ciphers, and thus reduce the given decimals to a common denominator.

RULE. Give to all the numbers the same number of decimal places, by annexing ciphers if necessary.

1. If the numbers are reduced to the denominator of that one of the given numbers having the greatest number of decimal places, they will have their least common decimal denominator.

2. A whole number may readily be reduced to decimals by placing the decimal point after units, and annexing ciphers; one cipher reducing it to tenths, two ciphers to hundredths, three ciphers to thousandths, and so on.

2. Reduce .17, 24.6, .0003, 84, and 721.8000271 to their least common denominator.

3. Reduce 7 tenths, 24 thousandths, 187 millionths, 5 hundred millionths, and 10845 hundredths to their least common denominator.

4. Reduce to their least common denominator the following decimals: 1000.001, 841.78, 2.6004, 90.000009, and 6000.035.

5. Reduce the following decimals to their least common denominator: 5.05, .006, 200.5632, 5000.9, .352186, .799, and .89432.

6. Reduce .63, 57.56, .297, and .123456789 to their least common denominator.

149. To reduce a decimal to a common fraction.

1. Reduce .75 to its equivalent common fraction. SOLUTION. We omit the decimal point, supOPERATION. ply the proper denominator to the decimal, .75=75%=3 and then reduce the common fraction thus formed to its lowest terms.

100

RULE. Omit the decimal point, and supply the proper denominator.

2. Reduce .125 to a common fraction.
3. Reduce .16 to a common fraction.
4. Reduce .655 to a common fraction.
5. Reduce .9375 to a common fraction.
6. Reduce .0008 to a common fraction.

OPERATION.

40)21.000

.525

150. To reduce a common fraction to a decimal.

1. Reduce to thousandths.

OPERATION.

16)1.0000

.0625, Ans.

2. Reduce to its equivalent decimal.

Ans. 1.

Ans. Ans. 13. Ans. 18.

200

Ans. 1250

2 is of 21. 21-21 thousand of 21.000=21.000÷

SOLUTION. - Dividing as in the former example, we obtain a quotient of 3 figures, 625. But since we annexed 4 ciphers, there must be 4 places in the required decimal; hence we prefix 1 cipher.

RULE. -I. Annex ciphers to the numerator, and divide by the denominator.

II. Point off as many decimal places in the result as are equal to the number of ciphers annexed.

Common fractions in their lowest terms can be reduced to exact or perfect decimals when their denominators contain only the prime factors 2 and 5, and not otherwise. A perfect decimal is called a finite decimal. When a decimal is not exact, the remainder is usually indicated by a common fraction or by the sign +.