DECIMAL FRACTIONS. 164. 1. What is 1 of the ten equal parts of anything? 2. What is 1 of the ten equal parts of? 3 parts? 3. What is 1 of the ten equal parts of? 35 parts? 4. What is 1 of the ten equal parts of Too? 43 parts? 5. What part of is? Of is Too? говор? Of 1000 is 6. What part of is 10? Of 180 is 1000? Of 1000 is 10000? 165. The divisions of anything into tenths, hundredths, thousandths, etc., are called Decimal Divisions. 166. One or more of the decimal divisions of a unit are called a Decimal Fraction. The word decimal is derived from the Latin word decem, ten. 167. Since tenths are equal to ten times as many hundredths, hundredths equal to ten times as many thousandths, etc., decimals have the same law of increase and decrease as integers, and the denominator may be indicated by the position of the figures. In the decimal system of notation, the representative value of a figure is decreased tenfold by each removal one place to the right; hence: The figure at the right of units expresses tenths. The figure at the right of tenths expresses hundredths. The figure at the right of hundredths expresses thousandths. The figure at the right of thousandths expresses tenthousandths. The figure at the right of ten-thousandths expresses hundred-thousandths, etc. 168. A period, called the Decimal Point, is placed before the decimal. The number is read, 3 million, 415 thousand, 673 and 413 thousand 207 millionths. The orders of decimals below millionths are ten-millionths, hundredmillionths, billionths, ten-billionths, hundred-billionths, trillionths, etc. EXERCISES IN NUMERATION. 169. 1. Read the expression 5.239. The whole expression is read, 5 and 239 thousandths. RULE. Read the decimal as an integral number, and give it the denomination of the right-hand figure. - EXERCISES IN NOTATION. 170. 1. Express decimally seventy-nine thousandths. EXPLANATION. Since thousandths occupy the third place, three figures are required to express the decimal. Hence, the number seventy-nine is written, and a cipher prefixed to cause the figures to occupy their proper position. Hence, the decimal is written .079. RULE. Write the numerator of the decimal, prefix ciphers if necessary to indicate the denominator, and place the decimal point before tenths. Express decimally: 2. Eight tenths. Three tenths. Five tenths. Four hundredths. Eight hundredths. Six hundredths. 3. Five thousandths. Three hundred four thousandths. Seven ten-thousandths. Eight hundred sixty-five tenthousandths. Sixty-eight thousandths. 4. Fifteen hundredths. Twelve hundred-thousandths. Forty-eight thousandths. Four hundred-millionths. 5. Ninety-five ten-thousandths. Ninety billionths. Sixtytwo hundred-thousandths. Fifty-five thousandths. 6. 210 millionths. 403 thousandths. 15 hundredths. 4256 hundred-thousandths. 1268 hundred-millionths. 7. 56345 millionths. 389 ten-thousandths. 3854 hundred-thousandths. 518 ten-millionths. 8. Sixty-seven, and three hundred forty-nine ten-thousandths. Fifty, and fifty-five millionths. 9. Eighty-eight, and five thousand five hundred-thousandths. Nine, and nine ten-thousandths. 22. How do the fractions in 16, 17, and 18 compare in value? How do the decimals compare? effect of annexing ciphers to decimals? What is the 23. How do the fractions in 19, 20, and 21 compare in value? How do the decimals compare? What is the effect of prefixing decimal ciphers to a decimal ? 24. How does the number of places in a decimal compare with the number of ciphers in the denominator. 171. PRINCIPLES. not alter its value. -1. Annexing ciphers to a decimal does 2. Each decimal cipher, prefixed to a decimal, diminishes the value of the decimal tenfold. 3. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as there are figures in the decimal. 172. In expressions of the currency of the United States, the cents, mills, etc., may be read as decimals of a dollar : Thus, $5.375 may be read 5 dollars and 375 thousandths, or 5 dollars 375 cents. Read the following as dollars and decimals of a dollar : 1. $6.495. 2. $5.083. 3. $7.394. 4. $5.865. 5. $4.004. 6. $8.056. REDUCTION. 173. To reduce dissimilar to similar decimals. 1. Reduce .5, .25, .046, and 2.0506 to similar decimals. .5 = .25 = .5000 .2500 .0460 .046 = 2.0506 =2.0506 EXPLANATION. Since the lowest order of decimals in the given numbers is ten-thousandths, all the decimals must be changed to ten-thousandths. This may be done by annexing ciphers (Prin. 1, Art. 171). RULE. - Give all the decimals the same number of places by annexing ciphers. Reduce to similar decimals: 2. .4, .65, .175. 3. .05, .015, .75, .0104. 4. .045, .476, .00055. 8. .3460, .17, .4, 17.6. 9. 4.08, .7, .0004. 10. 9, .9, .009, 90. 11. 6.1, 1.054, 12.36876. 12. 75, 4.1, .268, .0057. 13. 100, .001, 1000, .000001. 174. To reduce decimals to common fractions. 1. Reduce .75 to a common fraction. EXPLANATION. -.75 expressed as a common frac.75=75% tion is, or, when it is reduced to its lowest 100 = terms. RULE. Omit the decimal point, supply the proper denomi nator, and reduce the fraction to its lowest terms. |