of a decimal fraction, compared with another, does not de pend upon the number of its figures, but upon the value of its first left hand figure: for instance, a fraction beginning with any figure less than ,9 such as ,899229, &c. if extended to an infinite number of figures, will not equal ,9. ADDITION OF DECIMALS. RULF.-1. Place the numbers, whether mixed or pure decimals, unaer each other, according to the value of their places. 2. Find their sum as in whole numbers, and point off so many places for the decimals, as are equal to the greatest number of decimal parts in any of the given numbers. EXAMPLES. 1. Find the sum of 41.653+-36,05+24,009 +1,6 141,653 36,05 24,009 1 1,6 Thus. 2 Sum, 103,312, which is 103 integers, and roll parts of I unit. Or, it is 103 units, and 3 tenth parts, 1 hundredth part, and 2 thousandth parts of a unit, or 1. Hence we may observe, that decimals, and FEDERAL Money, are subject to one and the same law of notation, and consequently of operation. For since dollar is the money unit; and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, or unit, it is evident that any number of dollars, dimes, cents and mills, is simply the expression of dollar3, and decimal parts of a dollar: Thus, 11 dollars, 6 dimes, 5 cents,=11,65 or 11 785. dol. &c. 2. Add the following mixed numbers together, (2) (4) Yards. Ounces. Dollars. 46,23456 12,3456 48,9108 24,90400 7,891 1,8191 17,00411 2,34 3,1030 3,01111 ,7012 (3) SUBTRACTION OF DECIMALS.RULE.- Place the numbers according to their value; then subirago 's in whole numbers, and point off the decimals as in Addition. EXAMPLES. Inches 1. From 125,64 2. From 14,674 Take 95,58756 Take 5,91 3. From 761,8109 Take 18,9113 719,10009 7,121 27,15 6. From 480 take 245,0075 7. From 236 dols, take ,549 dols. 8. From ,145 take ,09684 9. From ,2754 take ,2371 1 10. From 271 take 215,7. 11. Frora 270,2 tase 75 4.775 ?. From 107 tase 0001 Ans. 234,9925 Ans. 55,3 self. 13. From a unit, or 1, subtract the millionth part of it Ans. ,999999 MULTIPLICATION OF DECIMALS. Rule.-1. Whether they be mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers. 2. Point off so many figures from the product as there are decimal places in both the factors; and if there be not so many places in the product, supply the defect by prefixing ciphers to the left hand. EXAMPLES. 1. Multiply 5,236 2. Multiply 3,024 by ,008 by 2,23 . Product, ,041888 6,74352 3. Multiply 25,238 by 12,17. Answers. 307,14646 4. Multiply 2461 by ,0529. 130,1869 5. Multiply 7853 by 3,5. 27485,5 6. Multiply ,007853 by ,035. ,000274855 7. Multiply 004 by ,004. ,000016 8. What cost 6,21 yards of cloth, at 2 dols. 32 cents, 5 mills, per yard? Ans. $14, 4d. 3c. 825-m. 9. Multiply 7,02 dollars by 5,27 dollars. Ans. 36,9954 dols. or $36 99 cts. 54.m. 10. Multiply 41 dols. 25 cts. by 120 dollars. Ans. $4950 11. Multiply 3 dols. 15 cts. by 16 cts. Ans. $0,5520–55 cts. 2 mills. 12. Multiply 65 cents, by ,09 or 9 cents. Ans. $0,0585=5 cts. 81 mills. 13. Multiply 10 dols. by 10 cts. Ans. $1 14. Multiply 341,45 dols. by .007 or 7 mills. Ans. $2,39 To multiply by 10, 100, 1000, &c. remove the separating point so many places to the right hand, as the multiplier has ciphers. ( Multiplied by 10, makes 4,25 So ,425 - by 100, makes 42,5 - by 1000, is ,425 For ,425 x 10 is 4,250, &c. DIVISION OF DECIMALS. Rule.-1. The places of the decimal parts of the divisor and quowent counted together, must always be equal to those in the dividend. therefore divide as in whole numbers, and from the right hand of the quotient, puint off so many places for decimals, as the decimal places in the dividend exceed those in the divisor. 2. If the places in the quotient be not so many as the rule requires, supply the defect by prefixing ciphers to the left hand of said quotient. Note.--If the decimal places in the divisor be more than those in the dividend, annex as many ciphers to the dividend as you please, so as to make it equal, (at least,) to tho divisor. Or, if there be a remainder, you may annex cipheri to it, and carry on the quotient to any degree of exactness. 00 00 3. Divide 780,517 by 24,3. Answers. 32,18 4. Divide 4,18 by ,1812. ,23068+ 5. Divide 7,25406 by 957. ,00758 6. Divide ,00078759 by ,525. ,00150+ 7. Divide 14 by 365. ,038356+ 8. Divide $246,1476 by $604,25. ,40736+ 9. Divide $186513,239 by $304,81. 611,9+ 10. Divide $1,28 by $8,31 ,154+ 11. Divide 56 cts. by 1 dol. 12 cts. 12. Divide 1 dollar by 12 cents. 8,333 + 1 13. If 21.1 or 21,75 yards of cloth cost 34,317 dollara what will one yard cost ? $1,577+ NOTE.— When decimals, or whole numbers, are to be di vided by 10, 100, 1000, &c. (viz. unity with ciphers,) it is performed by removing the separatrix in the dividend, s, many places towards the left hand as there are ciphers ir the divisor .. EXAMPLES. ( 10, the quotient, is 57,2 672 divided by < 100, - - - - 5,72 (1000, . . . . ,572 REDUCTION OF DECIMALS. CASE 1. RULE. -Annex ciphers to the numerator, and divide by the deno. minator, and the quotient will be the decimal required. Note:—So many ciphers as you annex to the given nu. merator, so many places must be pointed in the quotient; and if there be not so many places of figures in the quotient make up the deficiency by placing ciphers to the left hand nf the said quotient. EXAMPLES. 1. Reduce 1 to a decimal. 8)1,000 Ans. ,125 2. What decimal is equal to } ? Answers. ,5 3. What decimal is equal to ? - - - - - ,75 4. Reduce to a decimal. • • • • • • ,2 5. Reduce it to a decimal. • • • ,6875 6. Reduce 7 to a decimal. . - - • • - ,85 7. Bring to a deciinal. • - - - ,09375 8. What decimal is equal to ? - - ,037037+ 9. Reduce to a decimal. - - - • ,333333+ 10. Reduce it to its equivalent decimal. - - ,008 11. Reduce into a decimal. • • . .,1923076+ CASEI To reduce quantities of several denominations to a Decimat. RULE.-1. Bring the given denominations first to a vulgar fraction by Problem III. page 71; and reduce said vulgar fraction to its equivalent decimal ; or, 1.2. Place the several denominations above each other. letting the highest denomination stand at the bottom; then divide each utnumin ation (beginning at the top) by its value in the nextd. enominatio last quotient will give the decimal reruired |