to the right. Thus .2 expresses 2 tenth parts of the integer, .02, 2 hundredths parts. Unity may be considered as a fixed point, from whence whole numbers proceed, infinitely increasing toward the left hand, and decimals infinitely decreasing toward the right hand, which is exemplified in the following TABLE. Hundreds of millions. Hundredths. Hundred millionths. 765 Integers. 3 2 1.2 3 4 5 6 7 8 9 Write the following numbers in words. Q. What is a decimal fraction? Q. How are decimals distinguished from integers? Q. How are decimals distinguished? Q. What does a mixed number consist of? ADDITION OF DECIMALS. $91. RULE 1. Write down the numbers so that the decimal points, and like places, may be under each other. 2. Find their sum as in whole numbers, and point off as many places for decimals, as are equal to the greatest number of decimal places in any of the given numbers. EXAMPLES. 1. Find the sum of 24.025+0.8724+136.+.00053+ 1843.213. 24.025 0.8724 136. .00053 1843.213 2004.11093 the sum. 2. Add .0084+30.67+.4085+2.081+3.15. 3. Add 40.40+5.862+307.5+.4075+37.+.8. 4. Add 9.99+.415+.00001+.847+.0083+375.375. 5. Add 3+.3+.7+7+.9841+81.18+189.432+.009. 6. Add .0001+23.1817+84621.5+38.472+8.816. 7. Add 500.5001+7.903+46.731+73.143+16.32. Q. How do you write decimals to be added? SUBTRACTION OF DECIMALS. § 92. RULE. Place the less number under the greater, so that like places stand under each other; then subtract as in whole numbers, and point off the decimals as in addition. Q. How do you place decimals to be subtracted? MULTIPLICATION OF DECIMALS. § 93. RULE 1. Place the factors, and multiply them, as in whole numbers. 2. Point off as many figures from the product as there are decimal places in both the factors; and if there are not so many places in the product, supply the deficiency by prefixing ciphers. 3. To multiply by 10, 100, 1000, &c., we only have to remove the point as many places to the right, as there are ciphers in the multiplier. The reason of pointing off as many decimal places in the product, as there are in both factors, is that the operation is the same as in multiplication of vulgar fractions; for since each decimal place in either factor diminishes its value ten times, it must equally diminish the value of the product. Thus, .75×.33=.2475, and 1756×13=24756, which is the same. 33 100 Q. How do you multiply decimals? Q. How do you proceed if the product has not so many places as are necessary to be pointed off? Q. Why do you point off as many decimal places in the product as there are in both factors? Q. How do you multiply by 10, 100, 1000, or the like? DIVISION OF DECIMALS. § 94. RULE. 1. Divide as in whole numbers, and from the right hand of the quotient point off as many places for decimals as the decimal places in the dividend exceed those in the divisor. 2. If the places in the quotient are not so many as the rule requires, supply the defect by prefixing ciphers. 3. If at any time there be a remainder, or the decimal places in the divisor be more than those in the dividend, ciphers may be affixed to the dividend, and the quotient carried on to any degree of exactness. 4. To divide by 10, 100, 1000 or the like, move the point as many places to the left as there are ciphers in the divisor. The quotient figure is always of the same value with that figure of the dividend, under which the units place of its product stands. Or, the decimal parts in the divisor and quotient must always be equal in number to those of the dividend. When the decimal places in the divisor and dividend are equal, and there is no remainder after dividing, the quotient will be whole numbers. When the places of decimals in the dividend exceed those of the divisor, the decimal parts in the quotient must be equal to that excess. If the divisor exceed the dividend in decimal places, annex ciphers to make them equal, and if there is no remainder after dividing, then will the quotient be integers. 5. The reason of pointing off as many decimal places in the quotient as those in the dividend exceed the divisor, will easily appear; for since the number of decimal places in the dividend is equal to those in the divisor and quotient taken together, by the nature of multiplication it therefore follows that the quotient contains as many as the dividend exceeds the divisor. Q. How do you fix the decimal point in the quotient? Q. How do you proceed, if, after the division is finished, the quotient has not so many decimals as the rule requires? Q. How do you proceed when there are more decimals in the divisor than in the dividend? Q. Why do you point off as many places in the quotient, as the number in the dividend exceeds that in the divisor? Q. How do you divide by 10, 100, 1000, or the like? REDUCTION OF DECIMALS. § 95. CASE 1. To reduce a vulgar fraction to its equivalent decimal. RULE. Annex ciphers to the numerator, and divide by the denominator, and the quotient will be the decimal required. Write down the given decimal as a numerator, and for a denominator write 1 with as many ciphers as there are places in the decimal for a denominator, and then reduce the fraction to its lowest terms. M |