to the upper figure, and then subtract, putting down the remainder as before, and taking care to carry 1 to the next figure of the lower row. For example: let it be required to subtract 27385 from 64927; then, placing the former number below the latter (as in the margin), we proceed thus: 5 from 7, and 2 remain: 8 from-not 2-but 12, and 4 remain; carry 1: 4 from 9, and 5 remain: 7 from 14, and 7 remain; carry 1: 3 from 6, and 3 remain. From 64927 Subt. 27385 Rėm. 37542 All that requires explanation here is the carrying, as in the former rule. In the preceding example we see that the 8 cannot be taken from the figure above it, because this is only 2: we, therefore, add 10 to the 2, converting it into 12; but the adding 10 to any figure is simply putting 1 before it; that is, it is adding 1 to the preceding figure, which 1, by carrying it to the next lower or subtractive figure, is taken away again at the next step. In like manner the 4, in the upper row, is converted into 14, and the 1 thus prefixed to it is afterwards taken away, by 1 being carried to the next lower figure, and 3 subtracted instead of 2. It is plain that in subtraction the carrying can never amount to more than 1. As another example, let 86025704 be subtracted from 130741392: then, having arranged the numbers as in the margin, we proceed thus: 4 from 12, 8; carry 1: 1 from 9, 8: 7 from 13, 6; carry 1: 6 from 11, 5; carry 1: 3 from 4, 1: from 7, 7: 6 from 10, 4; carry 1 9 from 13, 4: therefore the remainder is 44715688. 3. From 130741392 Rem. 44715688 There is a sign for subtraction as well as one for addition: it is the little mark placed before the number to be subtracted; it is called minus: 5-2 is therefore 5 minus 2, that is, 5 diminished by 2; the remainder, or difference, is of course 3. so that 5-2 By help of the plus and minus signs, we can easily connect together in a single row a set of numbers, of which some are to be added, and others to be subtracted; thus, 4+6-3-2 means that 4 and 6 are to be added, and 3 and 2 are to be subtracted; so that 4+6-3-2-5. Instead of subtracting first 3 and then 2, we may, of course, subtract 5 at once; so that the above is the same as 10-5-5; and whenever addition and subtraction operations are indicated in this way, it will always be best to find first the sum of the additive quantities, then the sum of the subtractive quantities, and then, as in the foregoing example, to find the difference of the two results: in this manner the result of 125+427— 684+237—15, is computed as in the margin, and found to be 90. So that 125+427-684+237-15-90. I shall here add a few examples to be worked in a similar manner : (1.) 361+483-246-179-419. (2.) 573-184+602-67-924. (3.) 86243+721-649-70+13-9-86319. (4.) 12064+700628-109641+637-2604-601084. (5.) 23596-625+72311075-13758-3506285+6879=68820882. 125 427 684 15 237 699 789 90 result. In order to prove whether subtraction is correctly performed, add the remainder to the number which has been subtracted,—that is, to the lower of the two proposed numbers; the sum will be the upper number, if the work be correct: thus, in each of the two examples above, we have Subtractive number 27385 86025704 Simple Multiplication.-Multiplication is the method of finding the sum of any number of equal quantities, without the trouble of repeating them, one under another, and adding them up; it is a short way of obtaining the results of addition, when the numbers or quantities to be added are all equal. When the quantities are not only equal, but all of one denomination, the operation is called simple multiplication. To perform this operation readily, a table, called the multiplication table, must first be learnt; and the result which arises from multiplying one number by another, provided neither be greater than 12, must be committed to memory; it is one of the few operations in arithmetic where the memory of rules is indispensable. The number by which another is to be multiplied, is called the multiplier ; the number which is multiplied, the multiplicand; and the result obtained, and which, as just stated, is the same as would be got by writing down the multiplicand as often as there are units in the multiplier, and adding all up,—this result is called the product. The multiplication table shows what the product is in every case in which neither multiplicand nor multiplier exceeds 12; and, by knowing this table, the product may always be found, whatever numbers be proposed as multiplicand and multiplier. It may be as well to mention here that the numbers called by these names, when spoken of together, are generally called factors of the product, as they make or produce it: thus, 2 and 3 are factors of 6, since 3 taken twice, or 2 taken three times, make or produce 6. That the product in any case is really what the table states it to be, the learner can easily prove for himself; he has only to take the multiplicand as often as there are units in the multiplier, and, by addition, to find the sum of all; thus, the table states that 8 times 6 are 48, which is true, because 6, written eight times, and all added, produce 48, that is, 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 48. and so of any other pair of factors within the limits of the table. I. When the Multiplier is not greater than 12. RULE.—Put the multiplier under the multiplicand, units under units; and, by aid of the table, multiply each figure of the multiplicand, commencing at the units' figure, by the multiplier. Set down the right hand figure only of the product when it is a number of more than one figure, and carry as in addition. For example: multiply 6432 by 4. The multiplier 4 being Multiplicand 6432 placed under the multiplicand 6432, as in the margin, we proceed Multiplier thus: 4 times 2 are 8 : 4 times 3 are 12; 2 and carry 1: 4 times 4 are 16, and 1 are 17; 7 and carry 1: 4 times 6 are 24, and 1 Product 25728 A beginner, with the table before him, can easily perform operations of this kind; but he must learn to work them without looking at the table. It is as well to show him the time and trouble saved, by actually exhibiting the work 6432 of such examples by addition, as here annexed. The following, worked like 6432 the example above, require no further explanation : 6432 78326 57387 264135 6432 12 are 25. 8 9 10 25728 626608 81658926 573870 3169620 The multiplication by 10, as in the third of these examples, requires, in fact, no actual work, or reference to the table. You know that a number becomes ten times as great by simply putting a 0 after the figures; this causes each figure to advance a place to the left, so that its local value is increased tenfold. In like manner, a number becomes multiplied by 100 when two O’s are added to it; by 1000 when three are added, and so on, as is evident from numeration. The cipher, though in itself of no value, thus plays an important part in our notation; by filling up what would otherwise be gaps between figures, it keeps them in their proper places, and preserves their local values; and by being put after a number, it has the effect of multiplying that number by 10, 100, &c., according as it is written once, twice, &c. The sign for multiplication is x placed between the factors, thus: (1.) 346 X 7=2422. (2.) 6047 X 5 = 30235. (3.) 246053 x 6 1476318. (4.) 53274 X 800 = 42619200. In working this fourth example, the plan is to consider 8 only as the mul 53274 tiplier, and to put the ciphers to the right of it, as in the margin, annex 800 ing them afterwards to the product by 8. (5.) 470329 X 11 = 5173619. (6.) 375842 X 12 4510104. 42619200 II. When the Multiplier is greater than 12. Rule.—Place the multiplier under the multiplicand, units under units, tens under tens, &c. Commencing with the units' figure, multiply by each in succession, and arrange the several rows of results, so that the first figure on the right in each row may be directly under the multiplying figure that produced it. Add up all these products, and the sum will be the complete product. For example, if we have to multiply 426 by 34, we place the 34 under the 426 26, and proceed thus : 4 times 6 are 24; 4 and carry 2 : 4 times 2 are 8, and 2 are 10; 0 and carry 1: 4 times 4 are 16, and 1 are 17. The first row is now completed, and we begin anew, with the next figure, 3, as multiplier, taking 1704 care to put the first figure we get in the new row directly under this 3. 3 times 6 are 18; 8 and carry 1: 3 times 2 are 6, and 1 are 7: 3 times 4 are 12. The 14484 rows are now completed, so that, drawing a line and adding up, we find the product to be 14484. 34 1278 You see from the local position of our second multiplier, 3, that it is in 1704 reality 30, and 426 X 30=12780: adding this product to the former product, 12780 that given by the 4, as in the margin, the whole product by 34 must neces 14484 sarily be the result; and you see that it agrees with that above. If our multiplier had been a number of three figures, as 534, then, to the partial products above, we must have added the product due to the 5; which, having 426 regard to its local value, is 500; and if we retain the noughts, the whole opera 534 tion would be as here annexed. And it is plain that we may always omit the noughts, provided we take care, as the rule directs, to put the first figure 1704 of each partial product directly under the multiplying figure, which supplies 12780 that product. It is worthy of notice, too, that the product will always be the 213000 same, whichever of the two numbers be regarded as the multiplier: you 217484 may easily satisfy yourself that 426 multiplied by 534, is the same as 534 multiplied by 426. To be convinced that this principle is perfectly general, you have only to assure yourself of the fact within the limits of the multiplication table, which you may do by replacing multiplication by 2647 356 356 addition, as shown in the first example, p. 11; that is, proving 2647 to yourself that 3 times 7 is the same as 7 times 3; that 5 times 15882 2492 8 is the same as 8 times 5, and so on, as the table declares : be- 13235 1424 cause, whatever be the two factors, the multiplication of one by 7941 2136 the other is made up only of multiplications within the limits 712 of the table. It is in general most convenient to take that for 942332 942332 the multiplier which gives the fewer partial products, or rows of figures. (See the operations in the margin.) The learner may now exercise himself in the process, by showing that the following statements are true : (1.) 4214 X 24 =101136. (2.) 658 X 243=159894. (3.) Show that 243 X 616=9 X 9 X11 X 8 X 7 X 3. (6.) Show that 2048 X 1936=64X 121 X 32 X 16. When the multiplier consists of two figures, forming a number greater than 12, there are two partial products, or rows of figures, to add up; but, with a little address, the product may be written down at once, whenever the multi 2378 plier does not exceed 20. Suppose, for instance, it were 16, then, if we 16 multiply by the 6, and, as we go on, add in not only what we carry 38048 from any figure of the multiplicand, but also the immediately pre This is the ceding figure of the multiplicand, the complete product will be obtained in one line, as in the margin; the operation being carried on thus : 2378 6 times 8 are 48; 8 and carry 4: 6 times 7 are 42 and 4 are 46 and 8 are 54; 4 and carry 5: 6 times 3 are 18 and 5 are 23 and 7 are 30; O and carry 3: 6 times 2 are 12 and 3 are 15 and 3 are 18: 14268 2378 8 and carry 1: 1 and 2 are 3. It will be advisable for the learner to practise this short way with the multipliers, 13, 14, 15, 16, 17, 18, 38048 19. Multiplication, which is thus performed in one line, is called short multiplication ; when there are more lines, it is long multiplication. a same 8.8 16 Method of proving multiplication by casting out nines. I shall here mention a useful method of trying whether the product of two numbers is correct; bat I must postpone the explanation of the principle of the method till you arrive at Algebra. I can only mention here, that if any number be divided by 9, the remainder will be the same as would arise from dividing the sum of the figures in that number by 9: for 10 is equal to once 9+1; 100 is equal to 11 times 9+1; 1000 to 111 times 9+1; and so on : that is, the remainder arising from dividing 1, followed by any number of noughts, by 9, is always 1. Consequently the remainder arising from dividing 2, or 3, or 4, &c., followed by any number of noughts, is 2, or 3, or 4—the same as the figure preceding the noughts. It therefore follows, that whether we divide a number, such as 4326, which is of course 4000+300+20+6, by 9, or simply divide 4+3+2+6, that is 15—the sum of the figures — by 9, we must, in each case, get the same remainder. This property, taken in connexion with the principle referred to above, and to be proved in Algebra (see the multiplication of compound quantities in Algebra), suggests the following rule: Rulb.-Add together the figures of the multiplicand, not counting any 9 that may occur, rejecting also 9, whenever, in adding up, the sum amounts to 9 or more: when all the figures are added, the result will therefore be less than 9 : note this result. Proceed in like manner with the figures of the multiplier ; noting the result. Multiply the two results together; retaining, as before, only what is left after the rejection of all the nines the new result contains. Do the same thing with the figures of the product; and compare this third result with that just found : if the two be the same, the work may be presumed to be correct; if they differ, it is certainly wrong. The usual way of noting the four results is to make a cross, to put the first in the left hand opening; the second in the opposite opening; the third above, and the fourth below. If the upper and lower results are the same, the work is most likely correct, but otherwise it is wrong. Let iis proceed in this way to test the accuracy of the work at page 12. Commencing at the right of the multiplicand, we say 7 and 4 are 11, therefore rejecting 9, 2 and 6 are 8 and 2 are 10: the first result, therefore, rejecting 9 from this 10, is 1, which we place in the opening of the cross to the left. Taking now the multiplier, we say 6 and 6 5, 11; 2 and 3, 5, the second result, which we place opposite the former. The product 5 of the two is 5, with no 9 to reject: this is the third result, to be placed above. Lastly, taking the product, we say 2 and 3 are 5 and 3 are 8 and 2 are 10 : 1 and 4 are 5; which is the fourth result, and, as it agrees with the preceding, we conclude the work to be correct. It is plain, however, that if any of the figures in the product were made to exchange places, the agreement of the third and fourth results would remain, though the product would be wrong; as would also be the case if one figure of it were increased and another diminished, by the same number : all, therefore, that we can safely infer, is, that the agreement spoken of must have place if the work be correct; so that if it fail the work is wrong. Suppose, for instance, that 3 we had made 73084163x7584=554270392192: then, applying the test, we get, from the first factor, the result 5; from the second, the result 6; and from the product of these, 6 the result 3 : but, from the above-stated product of the two numbers, the result is 4: this product, therefore, is incorrect; and, upon revising the multiplication, we find that the 3, after the nought, should have been a 2. 5 5 5 Simple Division.-The operation by which we find how many times one number or quantity is contained in another number or quantity of the same kind, is called division. It is also the operation by which we find the 4th part, the 5th part, &c. of a number or quantity. The number or quantity divided is called the dividend ; that by which we divide it, the divisor; and the result obtained, the quotient. You must not fall into the common mistake of considering the quotient to express always how many times the dividend contains the divisor: the 4th part of a mere number tells us how many times that number contains 4; but the fourth part of a quantityma sum of money, for instance—is just the fourth part, and nothing else :-it is itself also a sum of money. The division is called simple when the quantities concerned are of but one denomination; when you come to the division of compound quantities, you will find some further remarks on the true nature of division in general; at present both dividend and divisor, and therefore the quotient, are to be regarded as mere numbers. I. When the Divisor is not greater than 12. Rule.—Place the divisor to the left of the dividend, with a mark of separation, thus ), between the two. a |