which I must urge your particular attention, because although commonly adverted to in books on Arithmetic, they are usually presented in a way very much calculated to mislead. I have already told you, that decimals are frequently-far more frequently than otherwise-given in an incomplete form, because the complete form would often require an interminable extent of figures. I have told you (page 116), that it is customary, in such cases, to write down only a limited number of decimals, and to compensate for the figures omitted, by adding a unit to the final figure we retain, whenever the row of omitted figures commences with 5, or a figure greater than 5; but to make no compensation when this commencing figure is less than 5. In the comparatively few cases that may occur, in which the decimals we work with are really complete, we may multiply such decimals together, as above, and may be sure that our products are correct in every figure but in the great majority of instances actually presented to us in practice, the decimals are curtailed, as just noticed; and, consequently, the last figure of every such decimal is not strictly correct: the error may be to the extent of half a unit in the place that last figure occupies. The last figure of our multiplier being thus, in general, incorrect, it is plain, that the entire row of figures produced from it must be incorrect; or, at least, except under very favourable circumstances, that is, except when the error in our multiplier is very minute, the whole row, after one or two of the leading figures of it, must be incorrect. In like manner, the final figure of our multiplicand being erroneous, in multiplying it by the successive figures of our multiplier, the products which arise from it must need correction; so that when all the partial multiplications are executed, and we proceed to add up, we must feel that, for formed by any of the adjacent figures, as in the present example, the work may be abbreviated into a sort of short multiplication, thus : 71.380164 3854528856 = the product by 3, or 300. = the foregoing prod. by 9, for the 27, or 27000. the last product by 2, for the 54. 214140492 1927264428 The work of Examples 9 and 14 may be abridged in the same way. several steps of this addition, we are really adding up wrong figures; and, consequently, that, as far as the influence of these reaches, our final product must be erroneous. I will give you an example. Suppose we have to multiply 27-14986 by 92-41035; and suppose, moreover, that these decimals had resulted from curtailing others of greater extent; as, for instance, 27.149855213, &c., and 92.41034604, &c. we should obtain the product of our proposed factors, as in the margin: but if, instead of five places of decimals in our factors, we had taken six, we should have had 27.14986 92.41035 13574930 8144958 271 4986 10859 944 54299 72 2443487 4 2508 928 065051 27.149855 × 92·410346 2508-927 49439983. It is very plain, therefore, that the small error introduced into the fifth decimal of the factors, employed in the margin, has sufficient influence on the product, to render all the decimals of it, after the first three, widely erroneous. Even the third decimal differs by a unit from the more correct product above; but this latter is itself not strictly accurate, because advanced decimals have still been omitted in the factors: if the 213, &c., had been included in our multiplicand, all the decimals, beyond the vertical line, that is, beyond the 7, would have been affected; so that the 4, at present next to the 7, would have been a 5: we may conclude, therefore, that the product found in the margin is true, to the nearest unit, as far as three places of decimals; but that all the figures beyond these three should be expunged, as necessarily erroneous. In most books of Arithmetic, you are told, that these advanced decimals should be omitted, because they are superfluous, giving to the result a degree of minute accuracy not usually requisite in practical matters: but you see, from what is here shown, that they should be omitted, because no confidence can be placed in them, because, in fact, they are all wrong, and are no more worthy of being retained in our result than any row of figures written down at random in their place. (85.) The practical conclusion you are to draw from what has now been said is this: when you multiply two factors together, the decimals in which have been curtailed, as here supposed, in adding up the partial products, disregard the G sums of all the columns up to that column, inclusive, in which the final figure of the last partial product is placed, and retain only the decimals furnished by the remaining columns. The last of the decimals thus retained should be increased by unit, if the first of the dismissed figures be a 5, or a greater number. In the marginal work above, a vertical line is drawn, cutting off the columns, of which the sums contribute nothing but inaccuracy to the result. It will, of course, occur to you, that it would save much waste labour if we could be spared the work of these inaccurate columns ; and you will be glad to find that this may be done by a very simple contrivance. It was easy to foresee, before commencing the operation above, that seven decimals would have to be suppressed in the result; and, therefore, that three decimals only were to be retained: our object, then, would be to limit our operation to just so much work as would be necessary to furnish us with these three decimals; but as it is desirable that we should know what the fourth decimal would be, in order that the third may be as correct as we can make it; that is, in order that the third may be increased by a unit, should the fourth be a high figure, we ought to be able to get four decimals in our result, and then to limit it to three, which may be presumed to be correct in the last figure to the nearest unit: we have, therefore, to multiply 27-14986 by 92-41035, so as to give only four places of decimals in the product: this is done as follows: place the units figure of the multiplier under the fourth decimal of the multiplicand, and then write all the other figures of the multiplier, so as that the entire row may be reversed: then, in proceeding with this inverted multiplier, observe the following caution: reject all the figures of the multiplicand which lie to the right of the figure by which you multiply, carrying, however, from these rejected figures, whatever would have been carried if they had been retained; and write the first figure you get, in each partial product, in the same vertical line, as in the margin, and you will thus find the product, 2508 928, true to three places of decimals. The figure in the units place of the given multiplier being 2, this 2 is first put under the fourth decimal figure of the multiplicand; after which the inverted multiplier is completed, and the work 27.14986 5301429 24434874 542997 108599 2715 81 14 2508.9280 carried on agreeably to the preceding directions. By comparing it with the more lengthy operation before given, you will see that the partial products, as far as they are required, arise in reverse order, and are correct, as far as they go, to the nearest unit: you will, of course, observe, that, in the carryings from the rejected figures of the multiplicand, the uniform principle of compensating for a rejected 5, or greater figure, by adding a unit to the figure on the left of it, is to be attended to, and applied: thus, in multiplying by the 1, the product, arising from the 9 in the multiplicand, on the right, is rejected, but a compensating 1 is carried to the next product; that is, we say, once 4 is 4, and 1 carried makes 5. In like manner, when we reach the last figure, 5, we say, 5 times 7 are 35; carry 4: 5 times 2 are 10, and 4 are 14. This example, with the explanations that have accompanied it, will sufficiently prepare you for the following rule. (86.) To find the Product of Two Factors, containing Decimals, to a proposed Number of Places. RULE 1. Count, from the decimal point in the multiplicand, as many decimals, annexing zeros if the decimals are too few, as you wish to secure decimal places in the product. 2. Under the last of these, put the unit-figure of the multiplier, or a zero, if there be no unit-figure, and then introduce all the other figures of it, so that the entire multiplier may appear with its figures in reverse order. 3. Multiply by the several figures of this reversed multiplier, neglecting, however, all those in the multiplicand to the right of the figure you are using, but, at the same time, carrying what would be carried, if nothing were neglected, and, moreover, carrying an additional unit, if 5 or a greater figure be rejected from the product. 4. Let each terminating figure of the partial products thus found, form one vertical column; the first column to be summed up in the addition process: then, when this process is completed, you will have the product required, the decimal point being so placed, as to mark off the proposed number of decimals. (87.) When the decimals in each of your factors are strictly true to the last figure, and your product is to be applied to a purpose, for which so many exact decimals as would make up the number in both the factors are not necessary, you may, by this rule, limit the number brought out to as few as you please. It is, therefore, matter of choice with you, whether, in such a case as this, you take the trouble to work your example in full, and thus give to your result a needless degree of minute accuracy, or content yourself with only the amount of accuracy really wanted; and use, for this end, the contracted method: but remember, you have no choice when the decimals in your factors are not thus each of them complete and accurate in the final figure; you must then use the contracted method, not to dispense with needless accuracy, as above, but in order to preclude absolute error. In this case, you should count all the figures of that factor which contains the greater number, and so many figures of the uncontracted product, cut off from the right hand, should be expunged, not as merely useless, but as erroneous. You must, therefore, so apply the preceding rule, as to exclude, from the product, just this number of figures. NOTE, If one of the factors be quite correct, then only so many figures as this contains are to be rejected. The following examples will sufficiently illustrate the application of the rule. 1. Multiply 348 8414 by 51.30742, so as to preserve only four decimals in the product. Here, reversing the multiplier, after having taken care to put the 1 in the multiplier, under the fourth decimal of the multiplicand, we see, that a vacant place occurs in the multiplicand over the final figure 5 of the reversed multiplier we therefore supply this vacant place, by putting in it a 0, and then multiply, as in the margin. The result may be considered as correct, as far as it goes, provided the factors producing it, have no error in their last decimals; but, if we are not assured of this, then we cannot depend upon more than the first two decimals, for since the complete product would have nine decimals, and that each factor has seven figures, 9-7= 2, expresses the greatest number of decimal places in the product that can be relied upon, with any confidence. The work would, therefore, be as here annexed: the 1 in the multiplier being now placed under the second decimal of the multiplicand. As the 2 in the multiplier has no figure above it, the product by this 2 is, of course, 0, but as the 348.84140 2470315 174420700 3488414 1046524 24419 1395 70 17898.1522 348.8414 24703150 1744207 34884 10465 244 14 1 17898.15 |