EXAMPLES. 1. Required the continued product of 2,, of and 2. Prepare the fractions as in multiplication; then invert the divisor, and proceed exactly as in multiplication. * The reason of the rule may be shewn thus: Suppose it were required to divide by 3. Now 2÷2 is manifestly of 4, or 3 4X2 ; but}=} of 2,..of 2, or must be contained 5 times as often in as 2 is; that is 3X5 4X2 the answer; which is ac cording 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 de nominator. 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. Thus, o'5 is equal to or TO 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 a single repetend, and a point over the first and last figures 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 1cths, or increasing toward the left by roths, the same as whole or integral numbers do. As in the following SCALE OF NOTATION. ∞ Hundreds of thousands. co Tens of thousands. ∞o Thousands. ∞ Hundred thousandth parts. ∞ Hundreds. ∞ Hundredth parts.' Tenth parts. ∞ Ten thousandth parts. ∞ Thousandth parts. Millionth parts. ∞ Units, ∞ Tens. &c. 8 8 8 8 8 8 8 8 8 8 8 Cyphers on the right of decimals do not alter their value. And 500 or I O 50 I O 500 ΙΟ is is is But cyphers before decimal figures, and after the separating point, diminish the value in a tenfold proportion for every cypher. So $5 is 7 or So that, in any mixed or fractional number, if the separating point be moved one, two, three, &c. places to the right hand, every figure will be 10, 100, 1000, &c. times greater than before. But if the point be moved toward the left hand, then every figure will be diminished in the same manner, or the whole quantity will be divided by 10, 100, 1000, &c. ADDITION 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 pre ceding. 2. Then add as in whole numbers, placing the decimal point, in the sum, directly under the other points. EXAMPLES. (1) 7530 3'0142 *03014 8513.10353 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 208574. What is the sum of 27:148, 91873, 14016, 294304, 7138 and 2217? Ans. 309488 2918. 5. Required the sum of 312984, 213918, 2700°42, 3153, 272 and 58106. 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. EXAMPLES. 4. What is the difference between 91713 and 407 ? Ans. 315 287. 5. What is the difference between 16:37 and 800*135,? Ans. 783.765. 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 hand, 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. EXAMPLES. * To prove the truth of the rule, let 9776 and 823 be the numbers to be multiplied; now these are equivalent to and 9776 823 9776 ; whence 0000X100010056488045648 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. |