significant figures, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged while the denominator will be increased ten times for every cipher which is annexed, and the value of the fraction will be decreased in the same proportion (see 82.) Hence, Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed. Take the fraction ,2=l as an example. ,02= 10% by prefixing one cipher : ,0002=10 % by prefixing three ciphers : in which the fraction is diminished ten times for every cipher prefixed. Also, ,03 becomes ,003 by prefixing one cipher; and ,0003 by prefixing two. Q. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the d nator ? 'What effect then has it on the value of the fraction? What does ,5 become by prefixing a cipher? By prefixing two ciphers ? By prefixing three? What does ,07 become by prefixing a cipher? By prefixing two? By prefixing three ? By prefixing four ? ADDITION OF DECIMAL FRACTIONS. $ 125. It must be recollected that only like parts of unity can be added together, and therefore in setting down the numbers for addition the figures occupying places of the same value must be placed directly under each other. The addition of decimal fractions is then made in the same manner as that of whole numbers. Add 37,04, 704,3 and ,0376 together. In this example, we place the tenths 1 OPERATION. under tenths, the hundredths under 37,04 hundredths, and this brings the decimal 704,3 points and the like parts of the unit ,0376 directly under each other. We then 741,3776 add as in whole numbers. Ans. Hence, for addition of decimals we have the following RULE. I. Set down the numbers to be added so that tenths shall fall under tenths, hundredths under hundredths, Sc. This will bring all the decimal points directly under each other. JI. Then add as in simple numbers and point off in the sum, from the right hand, so many places for decimals as are equal to the greatest number of places in any of the given numbers. Q. What parts of unity may be added together? How do you set down the numbers for addition? How will the decimal points fall ? How do you then add ? How many decimal places do you point off in the sum ? EXAMPLES 1. Add 4,035, 763,196, 445,3741 and 91,3754 together. 2. Add 365,103113, ,76012, 1,34976, 93549 and 61,11 together. Ans. 428,677893. 3. 67,407+97,004+4+,6+,06+,3=169,371. 4. ,0007+1,0436+,4+,05+,047=1,5413. 5. ,0049+47,0426+37,0410+360,0039=444,0924. 6. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. 7. Required the sum of two hundred dollars one dime three cents and nine mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill. Ans. $641,249, or 641 dollars 2 dimes 4 cents 9 mills. 8. What is the sum of one tenth, one hundredth, and one thousandth ? Ans. 9. What is the sum of 4, and 6 ten thousandths ? Ans. 4,0006. 10. Required in dollars and decimals, the sum of one dollar one dime one cent one mill, six dollars three mills, four dollars eight cents, nine dollars six mills, one hundred dollars six dimes, nine dimes one mill, and eight dollars six cents. Ans. $ 11. What is the sum of 4 dollars 6 cents, 9 dollars 3 mills, 14 dollars 3 dimes 9 cents 1 mill, 104 dollars 9 dimes 9 cents 9 mills, 999 dollars 9 dimes 1 mill, 4 mills, 6 mills, and 1 mill? Ans. $1132,365. SUBTRACTION OF DECIMAL FRACTIONS. § 126. Subtraction of Decimal Fractions teaches how to find the difference between two decimal numbers. EXAMPLES. 1. From 3,275 take ,0879. In this example a cipher is annexed to / OPERATION. the minuend to make the number of deci 3,2750 mal places equal to the number in the ,0879 subtrahend. This does not alter the value 3,1871 of the minuend (see § 123.) Hence, we have the following RULE. I. Set down the less number under the greater, so that figures occupying places of the same value shall fall directly under each other. II. Then subtract as in simple numbers, and point off in the remainder as many places for decimals as are equal to the greatest number of places in either of the given numbers. Q. What does subtraction teach? How do you set down the numbers for subtraction? How do you then subtract? How many decimal places do you point off in the remainder ? 2. From 3295 take ,0879. Ans. 3294,9121. 3. From 291,10001 take 41,375. Ans. 249,72501. 4. From 10,000001 take ,111111. Ans. 9,888890. 5. From three hundred and ninety-six, take 8 ten thousandths. : Ans. 6. From 1 take one thousandth. Ans. ,999. 7. From 6378 take one tenth. Ans. 6377,9. 8. From 365,0075 take 3 millionths. Ans. 365,007497. 9. From 21,004 take 97 ten thousandths. Ans. 10. From 260,4709 take 47 ten millionths. Ans. 260,4708953. 11. From 10,0302 take 19 millionths. Ans. 10,030181. 12. From 2,01 take 6 ten thousandths. Ans. MULTIPLICATION OF DECIMAL FRACTIONS. EXAMPLES. OPERATION, 296 1000: and § 127. 1. Multiply ,37 by ,8. If we multiply the fraction by to we find the product to be generally, the number of ciphers in the denominator of the product, will be equal to the number of decimal places in the two factors. ,37=100 2. Multiply ,3 by ,02. ,8= 8 ,296=i 17 296 1000 OPERATION. ,3x,02=i_x10=1600=,006 answer. To express the 6 thousandths decimally we have to prefix two ciphers the 6, and this makes as many decimal places in the product as there are in both multiplicand and multiplier. Therefore, to multiply one decimal by another, we have the following RULE. Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for decimals as are equal to the number of decimal places in the multiplicand and multiplier ; and if there be not so many in the product, supply the deficiency by prefixing ciphers. Q. After multiplying, how many decimal places will you point off in the product? When there are not so many in the product, what do you do? Give the rule for the multiplication of decimals. EXAMPLES. 1. Multiply 3,049 by ,012. Ans. ,036588 (2.) (3.) Multiply 365,491 Multiply 496,0135 by ,001 by 1,496 Ans. ,365491 Ans. 742,0361960 4. Multiply one and one millionth by one thousandth. Ans. ,001000001. 5. Multiply one hundred and forty-seven millionths, by one millionth. Ans. 6. Multiply three hundred, and twenty-seven hundredths by 31. Ans. 9308,37 7. Multiply 31,00467 by 10,03962. Ans. 311,2751050254. 8. What is the product of five-tenths by five-tenths. Ans. 9. What is the product of five-tenths by five thousandths. Ans. ,0025. 10. Multiply 596,04 by 0,00004. Ans. 11. Multiply 38049,079 by 0,00008. Ans. 3,04392632. § 128. Note. When a decimal number is to be multiplied by 10, 100, 1000, &c., the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier, and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers 10 67,9 679 100 Thus, 6,79 multiplied by < 1000 6790, 679000, 3700,36 100 37003,6 Also, 370,036 multiplied by 1000 370036, 10000 3700360, 100000 37003600, 67900, } Q. How do you multiply a decimal number by 10, 100, 1000, &c. ? If there are not as many decimal figures as there are ciphers in the multiplier, what do you do? DIVISION OF DECIMAL FRACTIONS. § 129. Division of Decimal Fractions is similar to that of simple numbers. We have just seen, that, if one decimal fraction be multiplied by another, the product will contain as many places of decimals as there were in both the factors. |