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To illustrate the plan which is here recommended, the working of an example is subjoined in detail :
23186 All that need be repeated is as follows : 2 and 6, 3 and 8, 8 and 1, 5 and 3, 4 and 2.
It is never easy or agreeable to change our habits of calculation, but the trouble is in many cases well repaid.
Multiplication.-Every one who has had the perseverance to learn by rote, a more extended multiplication table than is commonly taught at schools, must soon have felt the practical benefit of his attainment. To be able to multiply at one step, by any number under fifty or even under twenty, is of great advantage to all classes of computers, and to none more so than to officers of excise. A person of determined spirit, will thus acquire, in a few spare hours, a power of ready operation, that is scarcely to be over-valued, and will never desert him if it be occasionally exercised. The most convenient way of gaining the desired knowledge, is to carry about a card on which the products are written down, and to commit them gradually to memory, whenever a leisure moment occurs. An effort to get them off all at once, may only lead to the task being abandoned in despair.
Rapid multiplication is effected after a little practice, with quite as much certainty as rapid addition or subtraction, provided we trust implicitly to our first impressions. It cannot be too often repeated, that the anxiety to secure exactness at the sacrifice of despatch, by cautious, retrospective, working, defeats its own object by quickly wearing out the attention, and leads at last to such confusion, that we are unable to satisfy ourselves which of our many impressions of a result is really the correct one. The ordinary multiplication table, it need bardly be observed, should be so familiar to the mind, as to furnish the true product of any two numbers contained in it, the instant its factors are named or seen in connexion.
In some cases of multiplication, it may be useful to complete each part of the process, as we go along, adding the products by each figure of the multiplier to the line immediately above, instead of deferring the act of addition according to the usual method, until all the products are obtained. Thus
465930 5125230 42399630
415143630 This plan, however, is open to the objection that it involves the labour of several additions, when one final addition would suffice. If the multiplier consist of only two figures, it has certainly the advantage of shortening the operation. In multiplying by any number that lies between 10 and 20, it is merely an idle formality and a loss of time, to bring down the figures of the multiplicand, as the result of its multiplication by 1, and then to add it to the products by the unit
figure of the multiplier. All that is necessary, is to add to the multiplicand itself, each of these products, except the first, as soon as it is formed. Example
1307772 Operation :-(3)2, 43 and 4 are (4)7, 52 and 5 are (5)7, 21 and 6 are (2)7, 58 and 2 are (6)0, 6 and 7 are 13. The parenthesis denotes that the figure so enclosed, is the one to be carried, and, to be repeated therefore with emphasis.
The computer should never neglect an opportunity of applying those devices by which the labour of direct multiplication is in many cases greatly abridged. The substitution of some easier process is generally the best expedient. For instance, instead of multiplying by 25, he should annex two ciphers, and divide by 4. Instead of multiplying by 1.25, he should annex one cipher, and divide by 8. Thus, to reduce 48312 gallons of spirits at the strength of 25 per cent. O.P., to an equivalent at proof.
60390 Galls. Answer. To annex two ciphers, and subtract the original number, produces the same effect as to multiply by 99. Most treatises on arithmetic afford examples of numerous abbreviations of this kind, all worthy of attention.
As a ready means of testing the accuracy of a product, the method of « casting out the nines" may be adopted on all occasions with advantage. Although it does not supply a decisive proof of the correctness of the work, we are at least justified in concluding that if it fails to answer, our result must be wrong. Those who have the time and the patience to perform their multiplications in the reverse order, making that the multiplicand which was the multiplier, employ, of course, a much better check. Few persons, however, are willing to take the trouble of verifying every process in this tedious though effectual manner, especially when it is known that the calculations will be revised by another hand.
The whole of the work required in checking a product by the system of casting out the nines, is shown in the subjoined example.
74258 Remainder after rejecting nines 8
Proof. 267848606 Operation-Add together the successive figures of the multiplicand, reject 9 as fast as it arises, carrying each remainder to the next figure, and set out the last remainder : thus 7, 11, (9 and 2 over) 2, 4, (9), 8; remainder 8. Do the same with the multiplier ; multiply together the two remainders 7 and 8; cast out the nines from their product, and again set out the remainder; in this case, 2. Then cast out the nines from the sum of the figures in the product of the given num. bers, and compare the remainder with that obtained from the multiplicand and multiplier. When these results are alike, as in the present instance, it is highly probable that the work has been rightly performed. But if, in any part of the original process, one figure had been made too small, and another as much too great, or if any of the figures had been misplaced, the work would still appear to be correct, on casting out the nines. For which reason, this test, although extremely useful and easy of application, cannot be regarded as affording more than a strong presumption of accuracy.
When the remain ler from the multiplicand or first line, proves to be 0, it is only necessary to try whether the remainder from the product will also be 0; since whatever remainder may be furnished by the multiplier, the multiplication of that by O can produce only 0.
The principle of the method of casting out the nines, depends on a property of numbers, which the reader will find explained in almost every modern treatise on arithmetic.
Division.—Division by a large number is always a more laborious and protracted task than a multiplication of equal extent—for in such divisions, the exercise of some judgment and reflection is required before the proper figure can be placed in the quotient, by even the most expert computer; whereas the process of multiplying may be entered on at once, and pursued to the end, as rapidly as the small, partial products present themselves to the mind. There is no need of thought or trial, as in the case of division.
The actual work of long division, where whole numbers are concerned, admits of only one abridgment that it is worth while as a general rule to practise. This consists in performing multiplication and subtraction at one step. Half the figures used in the ordinary method are thus saved, the drawing of lines is dispensed with, and the computer soon learns to carry on the double operation with as much ease and precision as if he had been trained to it from the beginning. A single example will be sufficient to illustrate the superior neatness and compactness of this mod of division
35147) 22816394285 (649170
16295 In obtaining the first remainder, 17281, we say~42 and 1 are (4)3 ; 24 and 4 are 28, and 8 are (3)6; 6 and 3 are 9, and 2 are (1)1; 30 and 1 are 31, and 7 are (3)8;. 18 and 3 are 21, and I are 22; and so on for each figure of the quotient. The chief difficulty is to bear in mind the increased carriage which is sometimes given by the subtraction.
The plan of separating a divisor into its factors and dividing by each of these successively, instead of by the whole number at one operation, answers very well, when the factors are obvious, and not any of them greater than 12. But if there are more than two, or at the most three, such factors, it is quicker to divide by the whole number. No time would be gained in dividing, for instance, by 9, 8, 7, and 3, one after another, rather than by their product 1512, especially if it were necessary to take exact account of the several residual fractions.
In division, as well as in multiplication, the computer should be always on the alert to avail himself of every opportunity of shortening and simplifying his work by the employment of some device that may lead to the result more readily than the direct process. He should never divide by 5, when it is so much easier to point off one decimal place, and multiply by 2; or divide by 125, when he has the alternative of pointing off three decimal places and multiplying by 8, &c.
The method of casting out the mines, is also a good check on the correctness of a division. First, cast out the nines from the sum of the figures in the divisor : then, from the quotient ; multiply the remainders together; cast out the nines from the product; carry the remainder, to the final remainder of the division, if any, and cast out the nines from this. If the remainder now left, differ from the remainder left on casting out the nines from the dividend, the work is undoubtedly wrong. To take the example of division given on page 63 :
35147) 22816394285 (649170
16295 Remainder. Romainder from divisor . . . 2
» quotient .
„ remainder . . . } Proof.
» » dividend . . When the dividend exactly contains the divisor, the mode of applying the check of casting out the nines, is the same as in the case of multiplication, considering the divisor and the quotient as the numbers multiplied together, and the dividend as their product.
The subject of mental calculations will be treated of in the article on Practice.
Signs of operation. The signs or characters which are used to denote the various operations in arithmetic, or the relation between numerical expressions, should be fully understood and regularly employed by officers who wish to exhibit their own work compendiously. These signs afford so much facility in the statement of processes and results, and are so generally introduced into modern treatises on every branch of science, that a knowledge of them cannot be dispensed with by any person desirous of self-improvement. An explanation of the signs chiefly needed in elementary arithmetic is here subjoined for convenience of reference : Signe.
Meaning and Uses. + Called plus, and, or more, is the sign of addition. When placed between two
or more numbers it signifies that these numbers are to be added together.
Thus, 5+7 +9, implies that the sum of 5, 7, and 9 is to be taken.
numbers, it signifies that the difference of these numbers is to be taken.
Thus, 10 — 4, implies the deduction of 4 from 10. X Called into or multiplied by, is the sign of multiplication, and signifies that
the numbers between which it stands are to be multiplied together. Thus, 6x5, signifies that 6 is to be multiplied by 5; that the product of the numbers is to be taken.
Meaning and Uses. • Divided by, is the sign of division, and signifies that the former of the numbers
between which it is placed is to be divided by the latter. Thus, 852 signifies the division of 8 by 2 ; that the quotient of the numbers is to be taken. But it is more usual to express the operation of division by placing the dividend over the divisor, with a bar between them, as in the
form of a vulgar fraction ; for example, s.
is a method of stating that 12 and 9 together are equal to 21; 7 -2 = 5,
& Signs of the square root and the cube root respectively,
means that the square root of 16 is to be extracted.
directs the extraction of the cube root of 16. Sometimes the sign of the square root is written simply thus,
Called brackets. These are used to enclose two or more numbers or
Instead of employing brackets, a bar or line, called a vinculum is sometimes drawn over quantities when meant to be taken collectively;
thus, 9+6x7 — 4 is the same as (9+6) x (7 — 4.) 2. VULGAR FRACTIONS.—It has already been remarked that vulgar fractions, excepting those of sums of money, are very little used in excise accounts, the decimal divisions of the unit being adopted for almost every calculation. Still, since there are occasions on which vulgar fractions are employed, officers require to have some knowledge of their proper management, as well as for the reason that an acquaintance with this part of arithmetic is made one of the tests of fitness for promotion to the rank of Examiner. Indeed, independently of its practical applications, there is hardly a more useful means of mental exercise than the study