rules of decimals, which has been given at length in explaining the Rules for Simple Addition, Simple Subtraction, and the other fundamental rules in whole numbers. Reason for the above process. If we convert the decimals into fractions, and add them together as such, we obtain (or reducing the fractions to a common denominator), (1) 234, 14.3812, 01, 32:47, and '00075. (2) 232-15, 3.225, 21, 0001, 34.005, and 001304. (3) 14.94, 00857, 1.5, 5607-25, 530, and 0057. 2. Express in one sum: (1) '08+165+1·327+0003+2760'1+9. (2) 346+0027+25+·186+72·505+0014+00004. (5) 67-8125+27·105+17·5+000375 +255 +3·0125. 3. Add together: (1) 2.0068, 04137, 987641, 1·0000009, 57, and 1·5; and prove the result. (2) ⚫0003025, 29′99987, 143′2, 5·000025, 9000, and 3′4073; and verify the result. result. (3) 21-74, 075, 103.00375, 0005495, and 4957-5; and verify the (4) Five hundred, and nine-hundredths; three hundred and seventy-five; twenty thousand and eighty-four, and seventy-eight hundred-thousandths; eleven millions, two thousand, and two hundred and nine millionths; eleven thousand-millionths; one billion, and one billionth. SUBTRACTION OF DECIMALS. 89. RULE. Place the less number under the greater, units under units, tens under tens, &c., one-tenths under one-tenths, &c.; suppose cyphers to be supplied if necessary in the upper line to the right of the decimal: then proceed as in Simple Subtraction of whole numbers, and place the decimal point under the decimal point above. Ex. Subtract 5:473 from 6.23. Proceeding by the Rule given above, 6.23 5.473 ⚫757 Reason for the above process. If we convert the decimals into fractions, and subtract the one from the other as such, we obtain 1. Find the difference between 2.1354 and 1.0436; 7.835 and 2.0005; 15.67 and 156·7; 001 and 0009; 305 and 000683. 2. Find the value of (1) 2135-18125. (3) 603-6584003. (5) 582-09647. (2) 0516-0094187. (4) 17.5-13.0046. (6) 9.233-0536. 3. Take 01 from 1; 57-704 from 713-00683; 35.009876 from 56-078; 27·148 from 9816; and prove the truth of each result. 4. Required the difference between seven and seven tenths; also between seven tenths and seven millionths; also between seventy-four+ three hundred and four thousandths and one hundred and seventy-four+ one hundredths; and verify each result. MULTIPLICATION OF DECIMALS. 90. RULE. Multiply the numbers together as if they were whole numbers, and point off in the product as many decimal places as there are decimal places in both the multiplicand and the multiplier; if there are not figures enough, supply the deficiency by prefixing cyphers. Ex. 1. Multiply 5·34 by 21. Proceeding by the Rule given above, 5.34 •21 534 1068 11214 Now the number of decimal places in the multiplicand+ the number of those in the multiplier=2+2=4; therefore product=1∙1214. Ex. 2. Multiply 5·34 by 0021. 5.34 ⚫0021 534 1068 11214 We must have 6 decimal places in the product; but there are only 5 figures; and therefore we must prefix one zero, and place a point before it thus 011214. (1) 3.8 and 42; 38 and 42; 3·8 and 4.2; '038 and '0042. (2) 417 and 417; 417 and 417; 71956 and 000025. 2. Multiply (proving the truth of the result in each case) 4. Find the continued product of 1, 01, 001, and 100; also of 12, 1.2, 012, and 120; and prove the truth of the results. 91. First. When the number of decimal places in the dividend exceeds the number of decimal places in the divisor. RULE. Divide as in whole numbers, and mark off in the quotient a number of decimal places equal to the excess of the number of decimal places in the dividend over the number of decimal places in the divisor; if there are not figures sufficient, prefix cyphers as in Multiplication. Ex. 1. Divide 1.1214 by 5.34. Proceeding by the Rule given above, 5·34) 1·1214 (21 1068 534 534 Now the number of decimal places in the dividend decimal places in the divisor=4-2=2; therefore the quotient =•21. Ex. 2. Divide 011214 by 53'4. 53.4) 011214 (21 1068 534 Now the number of decimal places in the dividend decimal places in the divisor therefore we prefix three cyphers, and the quotient is 00021. 92. Secondly. When the number of decimal places in the dividend is less than the number of decimal places in the divisor. RULE. Affix cyphers to the dividend until the number of decimal places in the dividend equals the number of decimal places in the divisor; the quotient up to this point of the division will be a whole number; if there be a remainder, and the division be carried on further, the figures in the quotient after this point will be decimals. |