SECTION II. To multiply by a number consisting of a single figure followed by one or more noughts. Multiply 734 by 10. Proceeding as in the last chapter, we obtain the product 7340, and comparing this with the multiplicand we see that the same result may be obtained by merely writing o after 734; and the reason for this is evident from the nature of notation, for in the multiplicand there are 7 hundreds 3 tens 4 units, whereas in the product there are 7 thousands 3 hundreds 4 tens; that is, the denomination of each figure in the product is ten times greater than before, and therefore the given number is multiplied by 10. Similarly we may multiply by 100, 1000, &c., by placing on the right of the number as many noughts as there are in the multiplier. SECTION III. If we multiply by the factors of a number the product is the same as if we multiply by the number itself. For 8 multiplied by 6 is 48; and since the factors of 6 are 2 and 3, we get the same product by saying twice 8 is 16 and 3 times 16 is 48; or 3 times 8 is 24 and twice 24 is 48. We are now in a position to multiply by such numbers as 20, 30, 40, &c.; 200, 300, 400, &c. ; that is by any number consisting of a single figure followed by one or more noughts. Multiply 768 by 20. Here 2 and 10 are factors of 20, multiplying by 2 and putting o after this partial product, as it is called, we get 15360, which is twenty times the given number. This may be conveniently done in practice thus; set down the 2 (having o after it) under the units' figure of the multiplicand and say bring down the 768 o; then twice 8 is 16, set down 6 and carry 1, and so on, as before. 20 15360 Multiply 3876 by 4000. Bring down the three noughts and say 4 times 6, &c. Examples.-Multiply— 3876 4000 15504000 D. (1) 346 by 20. 879 by 50. (#) 578 by 30. 2764 by 60. (8) 7968 by 90. (3) 657 by 40. (6) 4796 by 70. Multiplication of any two numbers whatever. We will first explain the principle by working the following example and then deduce from the operation the general rule for multiplication. Multiply 38529 by 6407. Here we must repeat 38529 six thousand four hundred and seven times, that is six thousand times, four hundred times more, and seven times in addition, or, which is the same thing, seven times, four hundred times more, and then six thousand times more. Multiplying 38529 separately by each of the numbers 7, 400, 6000. we obtain three partial products, and these added together as under will give the entire product required. 269703 15411600 231174000 246855303 If we omit the noughts at the end of the second and third lines and write the multiplicand and multiplier over the partial products thus: 38529 269703 We observe that the multiplicand is multiplied by each of the figures of the multiplier, and that the right hand figure in each product is put in the same column as the figure of the multi246855393 plier from which it arises. 154116 231174 The operation is generally performed as follows: -First multiply by 7: 7 times 9 is 63; put 3 under the 7 and carry 6; 7 times 2 is 14 and 6 are 20; o and carry 2 ; &c. Since there are no tens in the multiplier we multiply next by the hundreds' figure 4; 4 times 9 is 36; put 6 in the column under 4 and carry 3; 4 times 2 is 8 and 3 are 11; 1 and carry 1 ; &c. Next multiply by the thousands' figure 6; 6 times 9 is 54; put 4 in the column under 6 and carry 5; 6 times 2 is 12 and 5 are 17; 7 and carry 1 ; &c. Add these products together and the same will be the entire product required. If the multiplicand or multiplier, or both of them, end with noughts, we may omit them in the working, and place on the right hand of the product as many as we omitted from the multiplicand or multiplier, or both of them. From the preceding illustrations we deduce the following rule for the multiplication of any two numbers. Omit the noughts, if any, at the end of the numbers to be multiplied; place the multiplier under the multiplicand so that units may be under units, &c.; draw a line under the multiplier, and multiply the multiplicand by each figure of the multiplier beginning with the units' figure. Place the first figure of each product under its multiplying figure; add the partial products together, and if any noughts were omitted from the multiplicand, or multiplier, or both of them, annex the same number of noughts to the product. The accuracy of the multiplication may be tested by the following method.. Find the remainders arising from dividing the multiplicand, multiplier, and product by 9; multiply together the remainders from the multiplicand and multiplier and divide the result by 9, if the remainder is equal to that obtained from the product the sum is probably right, but if different it is certainly wrong. Instead of actually dividing a number by 9 we |