of the sign x: thus, 3.7 is the same as 3 x7, or 21, while 3.7, is 3 and 7 tenths ; or, as it is usually read, 3 decimal 7, or 3 point 7: in like manner, 24.36, means 24 x 36, while 24:36, is 24 decimal 36, or 24 point 36; that is, 24 and 36 hundredths, and so on. It is necessary that should keep in mind this new sign for multiplication, as it is very frequently used ; so that, whenever you meet with such an expression as 2.3.7.4, you may know that it means 2x3x7x4. There is another useful little sign of abbreviation, which it is high time you should be made acquainted with: it is the sign :., which is the mark, not for an operation, but for a word—the word therefore (or, consequently), a word of such frequent occurrence, in numerical inquiries, as to render a short sign for it very acceptable. Henceforth, I shall use this sign :: for therefore, rather than introduce the word in the midst of arithmetical work. (79.) To reduce a Common Fraction to a Decimal. RULE 1. Annex a zero to the numerator, which then take for a dividend, the denominator being the divisor: if the dividend be sufficiently large, find the first figure of the quotient; but if it be too small to give a significant first figure, put zero for the first figure, and annex another zero to the dividend; if this be still too small, put a second zero in the quotient, and annex another zero to the dividend; and so on, till the dividend be large enough to give a significant figure in the quotient. 2. If there be a remainder, conceive another zero to be annexed; and continue the division, still annexing a zero to every remainder, till the work terminates of itself, or till the quotient has been carried to as many places as may be required: this quotient, with the decimal point before it, will be the value of the proposed fraction in decimals. Note. It will be enough if we imagine the zeros to be annexed as above, without actually inserting them. Should the division not terminate of itself, but admit of being carried on to any extent, then, at whatever point we stop the work, an unused remainder will be left ; so that the decimal quotient will not, in strictness, be the complete value of the fraction ; but the process may always be extended so far as to render the correction of the quotient too minute to be worth notice. It may indeed be made to become as small as we please. If the fraction be an improper fraction, the quotient will of course be partly integral and partly decimal, the decimal point occurring as soon as we begin to add zeros. Ex. 1. Reduce ž to a decimal. 8) 7000 = .875. 2. The value of 64 is 12.8. .875 3. In like manner, 1 = 1.875, and so on. 4. Reduce 10% to a decimal. From the operation in the margin, 256) 1500('05859375 it appears that 150 is too small to 1280 give a significant figure, so that the first figure of the quotient is 0. We 2200 see also that the remainders become 2048 exhausted only after eight decimal places are obtained in the quotient: the 1520 value of the proposed fraction, which 1280 might have been got by short division, is therefore, accurately, •05859375. 2400 (80.) The value of the last decimal is 2304 so small a part of unity, namely, the part 100000000; as to be in most practical 960 matters quite unworthy of consideration ; 768 we might therefore have stopped the process before arriving at this place, without 1920 troubling ourselves to see whether the 1792 work would spontaneously terminate or not. When an end is in this manner put 1280 to the operation, it is customary and 1280 proper to notice what the next figure would be if another step in it were to be made. Should this next figure prove to be a 5, or a figure still greater, then the figure at which we stop is increased by 1, because by so doing we secure the greatest possible accuracy for our result, as far as the operation has been carried. Thus, if we had stopped at the 7, in the present instance, we should have changed the 7 into 8, foreseeing, as we might, that the next figure would be 5, and knowing that if this 5 had been brought out, a 5 added to it would have converted the 7 into 8, while a 5 taken away from it,—that is, the suppression of the 5, would have left the 7 as it is. The error of adding a 5 or taking it away is, of course, the same; only in the one case it is an error in excess, and in the other case an error in defect. So far as accuracy, or the nearest approach to the truth, is concerned, it is here matter of indifference which plan we adopt; but it seldom happens that the work terminates at the figure next to that at which we stop; more figures would in general follow ; so that the probability is, that by increasing our last figure by unit, when we foresee that the next figure that would arise is a 5, our error in excess is less than our error in defect would be if we were to suppress this 5, and all the following figures with it, and leave our last figure unaltered. When the next figure is greater than 5, the propriety of the alteration is obvious ; so that by increasing the figure at which we stop by 1, whenever the next figure is foreseen to be 5 or greater, we generally attain a closer approach to the truth than we should do by leaving our last figure unaltered, and can never be farther from the truth. If the preceding result had been restricted to six places of decimals, it would have been •058594; if it had been restricted to five places it would have been •05859; if to four places, .0586; and so on. And such is always the plan whenever superfluous decimals are suppressed. You will have to observe it in working the examples below. (81.) The reason of the operation just performed is easily explained: the fraction is equal to 1000000000; that is, dividing numerator and denominator by 256, it is equal to which, in the decimal notation, is ·05859375; thus the equivalent decimal is obtained by dividing the 15, with the requisite number of zeros annexed, by the 256. And a similar explanation applies to every case. 5 8 5 9 3 7 5 100000000" Exercises. Reduce to decimals the following fractions, namely, 1. 1 4. 100 5. 6. 145 7. 8. 20 7589 9. of. 11 12. of of 6. 37" 2. 4 3. 67 1 2 5 93 250 625 (82.) ADDITION AND SUBTRACTION OF DECIMALS. Very little need be said as to the addition and subtraction of numbers involving decimals. Just as in integers, we must be careful in these operations to place units under units, tens under tens, and so on; so here we must, in addition to this, be also careful to place tenths under tenths, hundredths under hundredths, and so on : that is, we must keep the decimal points all in the same vertical line or column, as in the examples following Exercises. 1. Find the value of 27.62+.358+17:3+61 +.007 +173:1. 2. 5862:93+38:041 +1:01 +176.4+.0004 +265.04. 3. 385.02 +18.176—7.03—11•11-21.625+5•328 +.061. 4. 1.0628+123:51—26.04+13-18.261+12.403—.082. 5. •623+.0042 +079—:31 - .002+:11+.08—.0003. 6. 246 + 187–5.613–19·148—7.03–104.6+:0018. (83.) MULTIPLICATION. RULE. Place the multiplier under the multiplicand, regardless of the decimal points, and proceed as you would with integers. Having thus got the product, mark off from it as many decimal places as there are decimal places in both the factors together, and the correct product will be obtained. If the product terminate in zeros, these need not be inserted, but they must be taken into account in pointing off the decimal places; and if there should happen to be fewer figures in the product than there are decimals in the factors, zeros must be prefixed to the product to make up the deficiency; and the decimal point is to be placed before them. Ex. 1. Multiply 325.201 by 2.43. 325.201 Here the product, disregarding the decimal 2:43 points, being found, as in the margin, five figures of it are to be pointed off as decimals, because the 975603 number of decimals in both factors amounts to 1300804 five. 650402 From looking at the foot-note in next page, you will see that the work may be abridged, by 790.23843 simply multiplying the first partial product by 8. 2. Multiply 4132•65 by •346. 4132.65 Here, proceeding as in the margin, the ter •346 minating zero of the product is suppressed as useless ; but as five decimals are to be pointed 2479590 off, of which one has already been removed, we 1653060 point off but four. 1239795 The work of the next example following may be shortened by multiplying the first partial pro- 1429.8969 duct by 3, as explained in the foot-note below. 3. Multiply .217 and .0431 together. 0431 Here the product consists of but five figures, •217 while there are seven decimals in the factors; therefore two zeros must be prefixed, and the 3017 decimal point placed before them. To show 431 the correctness of the operation, it will be 862 sufficient to examine what is done in an example. Thus, the factors in Ex. 1 are 325 2001 .0093527 and 24%; that is, they are 325201 and 448; 325201 x 243 the product of these is ; the five zeros in the 100000 denominator implying that when the multiplication in the numerator is performed, five places must be pointed off for decimals. And it is plain, that in all cases by treating the factors in this way, the divisor will be 1, with as many zeros as there are decimal places in both factors. Exercises. 1. 32.605 X 6.417. 8. 24000 x .0016 X:35. 2. 183:52 X 734. 9. 2.016 X 3.004 x .0756. 3. 43.92 X 2600. 10. 273.4 x .036 x .004. 4. •038 x .072. 11. 21000 x 1.02 X:0268. 5. •0037 x .00021. 12. 1:4 X.04 X •4 x .004. 6. 2.46 x 321 X :07. 13. 71.380164 x 2.7354.* 7. 1073 X.032 x .0105. 14. 138.6147 x 5•2575 X.03. (84.) CONTRACTED MULTIPLICATION. I have now to introduce to your notice some considerations of especial importance in the multiplication of decimals, to * It may not be amiss to point out to the learner here, that whenever a multiplier has figures side by side, which, when taken by themselves, form a number that is a multiple of some other figure, or of a number |