What is Case 8th? What is the rule for it? What is the note? What must be done before fractions can be added? What else requires to be done? What is the rule for the addition of fractions? What note follows? What is note 2d? What preparations are necessary before fractions can be subtracted? What is the rule for subtracting fractions? How is a fraction multiplied by a whole number? How is a fraction multiplied into a quantity equal to its denominator. What is the rule for multiplying fractions by fractions? What is the rule for canceling? Into what three kinds is division of fractions naturally divided? How is a fraction divided by a whole number? What is the rule for dividing a whole number by a fraction? How are fractions divided by fractions? What is the rule for canceling? What note follows the rule? DECIMAL FRACTIONS. In the preceding rule we have contemplated the unit as divided into any number of equal parts. We are now to regard it as divided first into ten equal parts, then each of these into ten other equal parts, or the whole unit into one hundred equal parts; and these parts again, each into ten other parts; or the whole into a thousand equal parts, &c. The expressions obtained by these several divisions, therefore, decrease in value in the constant ratio of ten, from the left to the right, and are called decimals. Whole numbers, as was shown in Numeration, increase in the same ratio from the right to the left, and both commence their enumeration with the unit figure. The connection between them is therefore so intimate as to render them susceptible of being written together and subjected to the same operations. The only important consideration in writing them, in addition to what has already been explained, is to distinguish the one from the other. This is effected by the period, called in decimals, the point of separation, which is always placed between them. In the expression, 23.56, the 23 is the whole number, and the .56 the decimal. It will be observed that decimals, although they express parts of units, do not, like vulgar fractions, require two terms to express them. The given decimal may however be regarded, as in truth it is, a numerator, with a denominator always under stood. What this denominator is, it shall be our next object to illustrate. The scholar may therefore, in the first place, carefully examine the following table of whole numbers and decimals. The decimals, it will be observed, are read or numerated from the left to the right. 100 8 7 6 5 4 3 2.3 4 5 6 7 8 30 34 By an examination of the preceding table, the scholar will see that the 3 on the right of the separatrix is so many tenths. In the preceding rule, this would be thus expressed,, which, by section 8th of the introductory remarks of the same rule, equals 3. He will also see, that the 4 on the same side of the separatrix, is so many hundredths, or T. Now it is obvious, that these two fractions united, would make 34 The same process of reasoning will show, that the next figure, or 5, is so many thousandths; and since 100=1000' 340 if the 5 be added, the amount will be 345 To From the preceding, we therefore learn, that if the decimal consist of one figure only, it is so many tenths; if it consist of two figures, it is so many hundredths, and if of three, it is so many thousandths; and from this we derive the following conclusion, viz. that the denominator of a decimal fraction always consists of a figure 1, with as many cyphers annexed to it, as there are figures in the given decimal. The scholar may write a denominator to each of the following decimals, viz. .6; .356; .26; .7426 ; .98654; .71639. From the preceding explanation of the nature of decimals, it is obvious, that cyphers added to the right of a decimal do not effect its value, while those placed on the left diminish its value in a ten-fold ratio. For .5 is the same in value as, or. Now if a cypher be added to the right of the .5, it becomes .50, which is of equal value with 50, and this also equals. (See Case 1st, Vulgar Fractions.) If, then, .5 and .50 are each equal to, it is obvious, that the value of a decimal is not affected by cyphers placed on the right of it. But if they be placed on the left, they diminish the value of the decimal in a ten-fold ratio. The decimals .5, .05, and .005, will serve as an illustration. From the explanation given above, the denominators of these several decimals are 10, 100, and 1000; or the decimals may be thus written: 005 But five hundredths equal only one tenth part of five tenths; and five thousandths, one tenth part of five hundredths. Hence, cyphers placed on the left of a decimal diminish its value as above specified. 5 05 10 100 1000 The following numbers may now be expressed by figures, and then read. 1. Seventy-six and six tenths. Ans. 76.6. 2. One and three hundredths. Ans. 1.03. 3. Eighty and fifty-eight thousandths. 80.058. 4. One hundred and fifty-six, and thirty-nine thousandths. Ans. 156.039. 5. One hundred and one, and five thousandths. 101.005. Ans. The scholar will observe, that if there is but one decimal figure, and that tenths, it requires the point only to be placed at the left of it, to express its true value; if it be hundredths, it requires a cypher to be placed at the left of it, and if it be thousandths, it requires two cyphers to be thus placed, with the decimal point on the left of the cyphers; and so on according to the denomination. 6. Write down nine, and three hundred thousandths. 9.00003. Ans. 7. Write down twelve and one millionth. Ans. 12.000001. 8. Three hundred and seventy-five, and seven tens of thousandths. Ans. 375.0007. 9. Ninety-five hundredths. 10. Three hundred and sixteen thousandths. 11. Forty-five millionths. 12. Sixty-nine, and nine hundred and three thousandths. 13. Four hundred and fifty-six, and seventeen millionths. 14. Five hundred, and three tens of millionths. 15. One, and six hundred of millionths. 16. Eleven, and seven billionths. 17. Seven hundred and sixty-two billionths. 18. Four hundred and twenty-one, and nineteen thousandths. 19. Seven hundred and six, and one hundred and three millionths. 20. Twelve hundred and six trillionths. The scholar is now requested to turn back to Federal Money, and compare the denominations there given, with those here brought to view. He will there find the dollar given as the unit money; the dime as the tenth part of the dollar; the cent as the tenth part of the dime, or the hundredth part of the dollar; and the mill as the tenth part of the cent, the hundredth part of the dime, and the thousandth part of the dollar. It is therefore obvious, that Federal Money and decimals are operated upon by the same general principles. Ans. 1. Reduce $21, 8 dimes, and 6 mills, to mills. Ans. 21806. 2. Reduce 21806 mills to dollars, cents, and mills. $21.806. 3. Reduce $12, 3 dimes, 4 cents, and 9 mills, to mills. Ans. 12349. 4. Reduce 12349 mills to dollars, cents, and mills. Ans. $12.349. 5. Reduce $25 to cents. Ans. 2500. 6. Reduce $9 to mills. Ans. 9000. To reduce dollars to cents, we therefore add two cyphers, and to reduce them to mills, we add three. 7. Reduce 2567 cents to dollars. Ans. $25.67, or $25 and 67 cents. 8. Reduce 38679 mills to dollars, &c. Ans. $38.679, or $38, 67 cents, and 9 mills. To reduce cents to dollars, we therefore cut off two figures, and to reduce mills to the same, we cut off three figures from the right of the given number. 9. Reduce $2 to mills. 10. Reduce 99 cents to mills. 12. Reduce 467 cents to dollars. 13 ADDITION OF DECIMAL FRACTIONS. The scholar must here exercise a good degree of caution in writing the numbers to be added. He will recollect that, in adding vulgar fractions, it was necessary to reduce them all to the same name or denominator, before the numerators could be added; and that in Simple and Compound Addition, the same denominations only could be united. The same is true of Decimal Fractions. Hence, we have the following rule : RULE. Place the whole numbers as in Simple Addition, with units under units, and tens under tens, &c. Also place the decimals on the right of the whole numbers, with tenths under tenths, hundredths under hundredths, and thousandths under thousandths, &c.; then, beginning with the lowest denomination, add up and carry as in Simple Addition. Lastly, from the amount, point off as many decimals as are equal to the greatest number of decimals in any one of the given numbers. Ex. 1. What is the amount of 3.56; 42.923; 125.6; 4.32; and 59.365. ADDED. 3.56 42.923 125.6 4.32 59.36.5 235.768 The greatest number of decimals in either of the numbers given, is three; therefore, three decimals are to be cut off from the sum. It will always be found correct to place the decimal point in the amount directly below those of the given numbers. 2. Add the following numbers, 325.63; 275.215; 1.02; 17.653; 136.1. Amount, 755.618. 3. What is the amount of 72.5; 32.071; 2.1574; 371.4; 2.75? Ans. 480.8784. |