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product of the preceding 3 by the 2 is so great as 6, we carry 1 on that account, and the insertion of this 1 completes the inultiplications.
2. Find the product of .339377 and 325, to as many places of decimals as can be depended upon.
Here, as the multiplier, which is without error, .339377 has three figures, and as there are six decimals 5230 in the factors, only three decimals are to be preserved in the product, which is, therefore, 101813 110-297. If we had computed to four places of 6787 decimals, we should have got 110.2975. As
1697 already noticed, we may always compute to one place more than the number of places to 110.297 be preserved, and may increase the last of the preserved figures by unit, if the additional figure be so great as 5: in the present case, 110.297 and 110.298, may be considered to be about equally correct.
Exercises. 1. Multiply 480.14936 by 2.72416, and retain in the pro
duct only four decimals. 2. Multiply 15.917127 by 30-31667, retaining as many
decimals as may safely be depended upon. 3. Multiply 1•7958563 by 30.31667, to four places of de
cimals. 4. Multiply •62311052 by 170, to six places of decimals,
which is one more than can be strictly depended upon. 5. Multiply 1.628894 by 214.87, retaining no decimals that
cannot be relied on. 6. Multiply 81•4632 by 7.24651, retaining only three de
cimals. 7. Multiply 3.7719214 by •4471618, retaining all the de
cimals to be depended upon, namely, six. 8. Multiply ·053407 by .047126, retaining all the decimals
that are likely to be correct. 9. Multiply 325•701428 by •7218393, preserving only three
decimals in the product. 10. Multiply ·63942, &c. by •53217, &c.
Note. It is proper to state here, that from our ignorance of the true value of the decimals suppressed in our factors, and compensated for by a modification of the last decimal that is retained, and from the like modi. fication of the last decimal in certain of our partial products, we cannot always be quite sure that the last decimal in our contracted product is invariably true to the nearest unit. It may in unfavourable cases err to the
extent of a unit; but it may generally be relied on as the true product to within this amount of error. An error to the extent of two units in the last figure is highly improbable. In thus speaking of the occasional departure from strict accuracy in the final decimal of our contracted product, it is to be understood, that the accuracy adverted to is that which the result would have if the suppressed decimals in the factors were restored. If it be of consequence, in any particular calculation in which we may be engaged, that the final decimal preserved in the product should be strictly correct, the safest way will be to compute, by the contracted method, to one or two decimals beyond those which are to be preserved, and then to dismiss them from the product. Many important money calculations are performed by decimals; and it is necessary that the computer should be cautioned against the very prevalent mistake of supposing that his accuracy is increased as he increases the number of the retained decimals. The contractions in this article are recommended, not on the score of brevity, but with a view to the securing of strict truthfulness as far as it is attainable. An error of a unit in the second decimal of a result expressing pounds is only about 21d.: a like error in the third decimal is less than one farthing. In any inquiry in which it is of consequence to secure accuracy in the decimals or integers of a product, beyond the places furnished by contracted multiplication, we may pronounce such accuracy to be unattainable, till our factors, erring as they do in the final decimal, he rendered more correct by the insertion of additional places. It is most remarkable, that in many of the modern books on arithmetic, in extensive use in instruction, not a word is said about contracted multiplication and division of decimals; the time and labour of the learner is occupied in working out long strings of figures, which the authors ought to know are all worthless, because all wrong ; and what is worse, the pupil is all the while under the delusion that this useless labour is essential to the accuracy of his result.
(88.) DIVISION OF DECIMALS.
RULE 1. If the divisor and dividend have not the same number of decimals, annex zeros to make the number of decimal places equal.
2. Imagine the decimal points to be suppressed, and proceed as in division of integers, only with this difference, namely, if the divisor be greater than the dividend, annex a zero to the dividend : if the divisor be still the greater, put zero for the first figure of the quotient, and annex another to the dividend ; and so on, putting a zero in the quotient for every zero annexed to the dividend, after the first.
3. Having thus made the dividend sufficiently great (disregarding the decimal points) to contain the divisor, carry on the work as with whole numbers, annexing a zero to every remainder that arises after the figures of the dividend have been brought down, till as many decimals as are wanted are obtained in the quotient, or till the operation ends of itself. The number of zeros, employed in this latter way, together with whatever zeros the quotient may have commenced with, will be the same as the number of decimal places to be pointed off in the quotient.
Or you may count all the decimals used in the dividend, including, of course, every zero annexed to a remainder: the difference between this number of decimals and the number of decimals in the given divisor will be the number of decimals in the quotient.
NOTE. You will often find this latter to be the most convenient way of finding the places to be pointed off in the quotient, as you may then dispense with adding zeros to the divisor when it has fewer decimals than the dividend.
Ex. 1. Divide 721:17562 by 2.257432.
Here the operation in the margin has been car- 2.257432)721•17562(319.467, &c. ried on till nine decimal
6772296 places of the dividend have been used, namely,
4394602 the five decimal figures
2257432 originally given, and four zeros besides. And since
21371700 there are six decimal
20316888 places in the divisor, three places must be
10548120 pointed off in the quo
9029728 tient, agreeably to the note above. If we had
15183920 proceeded strictly by the
13544592 rule, and had written the dividend 721.175620, in
16393280 order to make the num
15802024 ber of decimal places the same as in the divisor,
591256 we might have intro
&c. duced the decimal point in the quotient as soon as the dividend thus written had been exhausted ; that is, as soon as the third figure 9 was found.
2. Divide 50714 by 8275.
Here, it is easy to see 8275.000)5•7140000(.0006905 that the quotient will com
49650 mence with zero; therefore annexing three zero deci
74900 mals to the divisor, since
74495 there are three decimal places in the dividend, and
42500 imagining the decimal
41375 points removed, we proceed by the rule, as in the
1125 margin, adding four zeros to the dividend, in order to make it large enough for the divisor, and consequently putting three zeros in the quotient. It is usual, however, to omit the zeros in divisor and dividend, and to proceed as if they were inserted; or, having no regard to the terminating zeros in the divisor at all, to perform the work agreeably to the note, leaving a gap at the beginning of the quotient, if a zero is foreseen to commence it, and then to complete the decimals as the note directs.
3. Divide •079085 by •83497.
The work of this example, •83497) ·079085 (094716 freed from unnecessary zeros, is
751473 as in the margin: the last decimal 6 is a little too great, but
39377 is much nearer the truth than 5.
333988 The zero with which the quotient commences is put in last.
59782 We see that the eleven decimal
584479 places used in the dividend, diminished by the five in the
13341 divisor, leaves six for the num
83497 ber of places in the quotient, so that a zero must be prefixed to
49913 the decimal figures to make up
500982 the requisite number.
(89.) The reason of the directions given in the rule is obvious, when the decimal places in dividend and divisor are made the same in number, by the addition of zeros should need be; the suppression of the decimal point is merely equivalent to multiplying each by unity, followed by as many zeros as there are decimal places in each ; so that the value of the quotient is undisturbed by these changes. It follows also from multiplication, that the decimal places in both divisor and quotient must make up the number of places in the dividend.
The quotients obtained in the preceding examples are true in all the decimals only on the supposition that the final decimal in both dividend and divisor is strictly correct. This however is not generally the case; as it usually happens that a number consisting of several places of decimals is only an abridgment of a number with more decimals, as already explained. The last decimal of such a number is therefore always affected with some error,—some fractional part of the unit to which it belongs; and to prevent the influence of this error in the quotient of two such numbers, all that part of the work which the error affects should be lopped off, and the operation confined within trustworthy limits, as in contracted multiplication. In the following examples the decimals are supposed to be unaffected with error.
Exercises. Find each quotient to three places of decimals. 1. 267•15975;1.25.
4. 519.62; 7849. 2. 5.474558= .0325.
5. 47•298-6.029. 3. 325:1.0125.
6. 3176.82 : :09317.
(90.) CONTRACTED DIVISION. Contracted Division, like Contracted Multiplication, is a method of finding the result sought to as many figures as can be safely depended upon, without introducing into the operation any more work than what contributes to this object. The rule is as follows:
Rule 1. Find the first figure of the quotient as in the uncontracted method, and thence the first remainder.
2. Instead of annexing a new figure from the dividend, or a zero, to this remainder, keep it as it is, and employ the divisor with its final figure cut off for the next step.
3. In like manner use the second remainder, and the divisor with two figures, cut off for the next step; and so on till all the figures of the divisor have been dismissed. It must be observed, that what would be carried from the figure cut off step after step, is still to be carried in multiplying the quotient figure by the curtailed divisor.