beginning with the units, by the figure in the tens' place of the multiplier, placing the first figure so obtained under the tens of the line above, the next figure under the hundreds, and so on. Proceed in the same way with each succeeding figure of the multiplier. Then add up all the results thus obtained, by the rule of Simple Addition. Note. If the multiplier does not exceed 12, the multiplication can be effected easily in one line, by means of the Table given above. 28. Ex. Multiply 7654 by 397. Proceeding by the Rule given above, we obtain 7654 397 53578 68886 22962 3038638 The reason for the Rule will appear from the following considerations. When 7654 is to be multiplied by 7, we first take 4 seven times, which by the Table gives 28, i.e. 8 units and 2 tens; we therefore place down 8 in the units' place and carry on the 2 tens: again, 5 tens taken 7 times give 35 tens, to which add 2 tens, and we obtain 37 tens, or 7 tens and 3 hundreds; we put down 7 in the tens' place, and carry on 3 hundreds: again, 6 hundreds taken 7 times give 42 hundreds, to which add 3 hundreds, and we obtain 45 hundreds, or 4 thousands and 5 hundreds ; we put down 5 in the hundreds' place, and carry on the 4 thousands: again,7 thousands taken 7 times give 49 thousands, to which we add the 4 thousands, thus obtaining 53 thousands, which we write down. Next, when we multiply 7654 by the 9, we in fact multiply it by 90; and 4 units taken 90 times give 360 units, or 3 hundreds, 6 tens, and 0 units: therefore, omitting the cypher, we place the 6 under the tens' place, and carry on the 3 to the next figure, and proceed with the operation as in the line above. When we multiply 7654 by the 3, we in fact multiply by 300; and 4 multiplied by 300 gives 1200, or 1 thousand, 2 hundreds, 0 tens, and O units; therefore, omitting the cyphers, we place the first figure 2 under the hundreds' place, and proceed as before. Then adding up the three lines of figures which we have just obtained, we obtain the product of 7654 by 397. 29. The above Example might have been worked thus, putting down at full length the local values of the figures; 7× 1000+ 6x 100+ 5×10+ 4 7654 397= 63 × 10000 + 21 x 100000+18 x 10000 + 49 x 1000+42 × 100 + 35 x 10+28 which = 21 x 100000+81 x 10000+118 × 1000 + 99 × 100 +71 × 10+28 20 x 100000 + 1x 100000 + 8×100000+1 x 10000 + 1× 100000+1x 10000+ 8×1000 +9x 1000+ 9 × 100 +7x100+1x 10 +2× 10+8 2000000+10 x 100000+ 2 x 10000+17 × 1000+16 × 100+ 3 x 10+8 =2000000+1000000 +2 × 10000+10 × 1000+7 × 1000+10 × 100 +6 × 100+3×10 +8 3000000+2× 10000 + 1 x 10000+7 × 1000+1 × 1000+6 × 100+3×10 +8 =3000000 + 3 x 10000+8 x 1000+ 600 + 30+8 = =3000000+30000 +8000+600+30 +8 =3038638 30. If the multiplier or multiplicand, or both, end with cyphers, we may omit them in the working; taking care to affix to the product as many cyphers as we have omitted from the end of the multiplier or multiplicand, or both. Thus, if 263 be multiplied by 6200, and 570 be multiplied by 3200, we have The reason is clear: for in the first case, when we multiply by the 2, in fact we multiply by 200; and 3 multiplied by 200 gives 600: in the second case, the 7 multiplied by the 2 is the same as 70 multiplied by 200; and 70 multiplied by 200 gives 14000. 31. If the MULTIPLIER contain any cypher in any other place, then, in multiplying by the different figures of the multiplier we may pass over the cypher; taking care, however, when we multiply by the next figure, to place the first figure arising from that multiplication under the third figure of the line above instead of the second figure. The reason of this is clear: for, if we were multiplying by 206, when we multiply by the 6 we take the multiplicand 6 times, when we multiply by the 2 we really take the multiplicand, not 20 times, but 200 times. 32. When two numbers are to be multiplied together, it is a matter of indifference, so far as the product is concerned, which of them be taken as the multiplicand or multiplier; in other words, the product of the first multiplied by the second, will be the same as the product of the second multiplied by the first. therefore the results are the same, that is, 2×4=4×2. That the product of one number multiplied by another, will be equal to the product of the latter multiplied by the former, may perhaps appear more clearly from the following mode of shewing this equality in the case of the numbers 3 and 5. 3=1+1+1; ..3x5=(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1) Now, if we regard the ones from left to right, there are 3 ones taken 5 times; if we regard them taken from top to bottom, we have 5 ones repeated 3 times; and the number of ones in each case is the same; i. e. 3×5=5×3: and so in the case of any two other numbers multiplied together. 33. The truth of all results in Multiplication may be proved by using the multiplicand as multiplier, and the multiplier as multiplicand: if the product thus obtained be the same as the product found at first, the results are in all probability true. 34. We have hitherto confined our attention to products formed by the multiplication of two factors only. Products may however arise from the multiplication of three or more factors; this is termed CONTINUED MULTIPLICATION: thus 2×3×4 denotes the continued multiplication of the factors 2, 3, and 4; and means that 2 is to be first multiplied by 3, Ex. V. and the product thus obtained to be then multiplied by 4. The result of such a process would be 24, which is therefore the continued product of 2, 3, and 4: we may express it thus, 2 × 3 × 4=24. (23) Find the product of 234578 by 18, by 29, and also by 53; of 924846 by 67, by 95, and also by 430; 2846067 by 206, by 1008, and also by 907; 8409631 by 21711, by 7009, by 8435, and also by 7980. (24) Find the product of 1754 and 9306; of 47506 and 4500; of 149570 and 15790; of 554768 and 39314; of 815085 and 20048; of 123456789 and 987654321; and of 57298492692 and 700809050321. (25) Multiply 9487352 by 4731246; 4342760 by 599999; 17376872 by 7399078; 38015732 by 400700065; 574585614865 by 2837154309. (26) Multiply six hundred and fifty thousand and ninety, by three thousand and eight; also seventy-six millions, eight thousand, seven hundred and sixty-five, by nine millions, nine thousand and nine. (27) Find the continued product of 12, 17, and 19; of 3781, 3782, and 3783; and of 6565, 6786, and 9898. (28) Multiply 20470 by 1030, and 2958 by 476, explaining the reason of each step in the process. DIVISION. 35. DIVISION is the method of finding how often one number, called the DIVISOR, is contained in another number, called the DIVIDEND. The result is called the QUOTIENT. 36. Division is of two kinds, SIMPLE and COMPOUND. It is called Simple Division, when the dividend and divisor are, both of them, either abstract numbers, or concrete numbers of one and the same denomination. It is called Compound Division, when the dividend, or when both divisor and dividend contain numbers of different denominations, but of one and the same kind. 37. The sign, placed between two numbers, signifies that the first is to be divided by the second. 38. In Division, if the dividend be a concrete number, the divisor may be either a concrete number or an abstract number, and the quotient will be an abstract number or a concrete number, according as the divisor is concrete or abstract. For instance, 5 shillings taken 6 times give 30 shillings, therefore 30 shillings divided by 5 shillings give the abstract number 6 as quotient; and 30 shillings divided by 6 give the concrete number 5 shillings as quotient. SIMPLE DIVISION. 39. RULE. Place the divisor and dividend thus: divisor) dividend (quotient. Take off from the left-hand of the dividend the least number of figures which make a number not less than the divisor; then find by the Multiplication Table, how often the first figure on the left-hand side of the divisor is contained in the first figure, or the first two figures, on the lefthand side of the dividend, and place the figure which denotes this number of times in the quotient: multiply the divisor by this figure, and bring down the product, and subtract it from the number which was taken off |