Cyphers on the right of decimals do not alter their value. But cyphers before decimal figures, and after the separating point, diminish the value in a tenfold proportion for every cypher. So that, in any mixed or fractional number, if the separating point be moved one, two, three, &c. places to the right, every figure will be 10, 100, 1000, &c. times greater than before. But if the point be moved toward the left, then every figure will be diminished in the same manner, or the whole quantity will be divided by 10, 100, 1000, &c. ADDITION OF DECIMALS. RULE. 1. Set the numbers under each other according to the val ue of their places, as in whole numbers, or so that the decimal points may stand each directly under the preceding. 2. Then add as in whole numbers, placing the decimal point in the sum directly under the other points. 2. What is the sum of 276, 39°213, 72014 9, 417, 5032, and 2214 298? Ans. 79993'411. 3. What is the sum of '014, 9816, 32, 15914, 72913, and '0047? Ans. 2'20857. 4. What is the sum of 27'148, 918'73, 14016, 294304, *7138, and 221°7? Ans. 309488'2918. 5. Required the sum of 312'984, 21'3918, 2700 ̊42, 3′153, 27'2, and 581'06. And. 3646 2088. SUBTRACTION OF DECIMALS. RULE. 1. Set the less number under the greater in the same manner as in Addition. 2. Then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points. EXAMPLES. (1) 214.81 4.90142 209.90858 2. From 9173 subtract •2138. 3. From 2.73 subtract 1.9185. 4. What is the difference between 91.713 and 407? Ans. •7035. Ans. 0.8115. Ans. 315-287. Ans. 783 715. 5. What is the difference between 16 37 and 800·135? MULTIPLICATION OF DECIMALS. RULE.* 1. Set down the factors under each other, and multiply them as in whole numbers. 2. And from the product, toward the right point off as many figures for decimals, as there are decimal places in both the factors. But if there be not so many figures in the product as there ought to be decimals, prefix the proper number of cyphers to supply the defect. * To prove the truth of the rule, let 9776 and 823 be the numbers to be multiplied; now these are equivalent to 97766 and 823 1000, 9776 whence 823 10000 X1000 8045648 10000000 10000 =8045648 by the nature of Notation, and consisting of as many places, as there are cyphers, that is, of as many places as are in both the num bers; and the same is true of any two numbers whatever. EXAMPLES. (1) 91.78 •381 9178 73424 27534 34.96818 2. What is the product of 520-3 and •417? Ans. 216.9651. 3. What is the product of 51.6 and 21 ? Ans. 1083 6. Ans. 0093527. 5. What is the product of ⚫051 and ⚫0091 ? Ans. 0004641. NOTE. When decimals are to be multiplied by 10, or 100, or 1000, &c. that is, by 1 with any number of cyphers, it is done by only moving the decimal point so many places farther to the right, as there are cyphers in the said multiplier; subjoining cyphers, if there be not so many figures. CONTRACTION. When the product would contain several more decimals than are necessary for the purpose in hand, the work may be much contracted, and only the proper number of decimals retained. RULE. 1. Set the unit figure of the multiplier under such decimal place of the multiplicand as you intend the last of your product shall be, writing the other figures of the multiplier in an inverted order. 2. Then in multiplying reject all the figures in the multiplicand, which are on the right of the figure you are multiplying by; setting down the products so that their figures on the right may fall each in a straight line under the preceding; and carrying to such figures on the right from the product of the two preceding figures in the multiplicand thus, namely, 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c. inclusively; and the sum of the lines will be the product to the number of decimals required, and will commonly be the nearest unit in the last figure, EXAMPLES. 1. Multiply 27.14986 by 92-41035, so as to retain only four places of decimals in the product. |