6. Multiply 44, of, and 18, continually together. Ans. 940 DIVISION OF VULGAR FRACTIONS. RULE.* Prepare the fractions as in multiplication; then invert the divisor, and proceed exactly as in multiplication. * The reason of the rule may be shown thus. Suppose it were required to divide by. Now 2÷2 is manifestly of or -3; but }=} of 2,. . } of 2, or 3 must be contained 5 times 4X3 as often in as 2 is; that is 3X5 the answer; which is according to the rule; and will be so in all cases. NOTE.-A fraction is multiplied by an integer, by dividing the denominator by it, or multiplying the numerator. And divided by an integer, by dividing the numerator, or multiplying the denominator. DECIMAL FRACTIONS. A DECIMAL is a fraction, whose denominator is an unit, or 1, with as many cyphers annexed, as the numerator has places; and is commonly expressed by writing the numerator only, with a point before it, called the separatrix. A finite decimal is that, which ends at a certain number of places. But an infinite decimal is that, which is understood to be indefinitely continued. A repeating decimal has one figure, or several figures, continually repeated, as far as it is found. As 33, &c. which is a single repetend. And 20-2424, &c. or 20 246246, &c. which are compound repetends. Repeating decimals are also called circulates, or circulating decimals. A point is set over a single repetend, and a point over the first and last fig ures of a compound repetend. The first place, next after the decimal mark, is 10th parts, the second is 100th parts, the third is 1000th parts, and so on, decreasing toward the right by 10ths, or increasing toward the left by 10ths, the same as whole or integral numbers do. As in the following SCALE OF NOTATION. &c. ∞ Hundreds of thousands. ∞ Millions. Tens of Thousands. ∞Thousands. ∞ Hundreds. ∞ Tenth parts. ∞ Tens. ∞ Units. Hundredth parts. Ten thousandth parts. ∞Thousandth parts. &c. parts. 8 8 8 8 8 8 8.8 8 8 8 8 8 8 Cyphers on the right of decimals do not alter their value. 500 is 1000 But cyphers before decimal figures, and after the separating point, diminish the value in a tenfold proportion for every cypher. Tooo or 200 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 value 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. Ans. 2 20857. 3. What is the sum of 014, 9816, 32, 15914, 72913, and ⚫0047 ? 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. Ans. 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. 4. What is the difference between 91'713 and 407? Ans. 315 2 87. 5. What is the difference between 16°37 and 800*135? Ans. 783 7 55. 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 mumber of cyphers to supply the defect. EXAMPLES. 1). 91.78 381 9178 73424 27534 34 96818 2. What is the product of 520 ̊3 and '417? 3. What is the product of 51'6 and 21 ? Ans. 216 9651. Ans. 1083 6. Ans. 0093527. * To prove the truth of the rule, let 9776 and 823 be the numbers to be multiplied; now these are equivalent to 9778 and 823 1600; whence 9776 823 10000X1 1000 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 numbers; and the same is true of any two numbers whatever. |