= where the sign - between 7 and 3 denotes the subtraction of the latter from the former, and the sign between 3 and 4 shews the equality of the excess to 4. III. MULTIPLICATION. 31. DEF. Multiplication consists in finding the amount of a number, when repeated any number of times, and this amount is termed the Product. The former of these numbers is called the Multiplicand, and the latter the Multiplier. Ex. 1. To multiply 7 and 42 by 4 and 5 respectively, being to find the sums arising from the numbers 7 and 42, four and five times repeated, we may determine the products as underneath; but the operations are expressed more briefly, as follows: Ex. 2. Find the products arising from the multiplication of 256 by 10, 11 and 12 respectively. By Article (10) we know that 256 will become ten times as great by merely affixing to the right of it the auxiliary digit 0, and thus we have the following operation: 256 the multiplicand: 10 the multiplier : 2560 the product. To multiply 256 by 11, we consider that 11 being equal to 1 and 10 together, the product will be equal to the sum of 256 taken once and ten times, 256 1 1 2 5 6=256 taken once, 2 8 1 6=256 taken eleven times : that is, 2816 is the product of 256 by 11: and the omission of the O on the right of the fourth line in the operation, can cause no inconvenience, as the places of the succeeding figures adequately determine their values. To find the product of 256 by 12, 256 must be taken twice and ten times together, and we have whence, the product of 256 by 12 is 3072, the observation above made holding good with respect to the omission of the 0 at the end of the fifth line. 32. From the mode in which the results above have been obtained, it is manifest that Multiplication is merely a compendious method of performing the addition of two or more equal numbers: and the following scheme, which is termed the Multiplication Table, presents at one view the product arising from the multiplication of any two numbers not exceeding 12; and though the products of the nine digits form the basis of those of all numbers whatever, it is here extended for the sake of practical convenience, and should be carefully committed to memory. 2 4 6 8 10 12 14 16 18 20 22 24 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 11 22 33 44 55 66 77 88 99 110 121 132 In this table, the first horizontal line consists of the first twelve numbers in order: the second consists of the products of the same numbers when multiplied by 2: the third contains their products when multiplied by 3: the fourth when multiplied by 4, and so on: and the table is repeated in the following manner. Thus, to make use of the second line of figures, we say Ex. 1. Let it be required to multiply 854 by 6: then, since the product of 854 by 6 is evidently equal to the sum of the products of all its parts, namely, 800 and 50 and 4, by 6, we have In practice, we mentally combine into one sum, the figures of all these products as they arise: thus, first multiplying 4 by 6, we find the product to be 24 by the table; and having placed the 4 units under those of the quantity proposed, we carry the 2 tens to the product of 5 by 6, which is here 30 tens, and thus obtain 32 tens; whereof the 2 being put under the tens' place, and the 3 being carried to the product of 8 by 6, or to 48 hundreds, the entire number of hundreds is 51; and the whole product is 5124. Ex. 2. Multiply 486 by 357. Here, proceeding with each of the figures 7, 5 and 3, according to the last example, we have 48 6 357 3 4 0 2 product of 486 by 7: 2430 1 4 5 8 = product of 486 by 50: = product of 486 by 300: 1 7 3 5 0 2 = product of 486 by 357. In this instance, the situations of the figures in the fourth and fifth lines render them equivalent to the products of 486 by 50 and 300 respectively, without supplying the auxiliary digits 0. If one or more of the figures of the multiplier be 0, it is evident that the corresponding partial product will be 0, and the lines may be entirely omitted after placing down each O once, to give the proper value to the product arising from the next figure. 33. The reasoning here employed being independent of the examples made use of to illustrate it, we are enabled to lay down a Rule in the following words. Rule for performing Multiplication. Place the multiplier under the multiplicand, as before, and draw a line under the whole: multiply every figure in the multiplicand by the figure in the units' place of the multiplier, observing to carry to the next product the number of tens in that arising from the multiplication of any of the digits in the multiplicand, and to place down the units under the figure multiplied, till the last product is obtained, which place down in full: proceed in the same manner with the figure of the multiplier in the tens' place, the figure on the right of this product being placed under the said figure; then with the figures in the succeeding places; add these products together, and the sum will be the entire product. 34. If the multiplicand and multiplier change places, the product must be the same as before, otherwise the same numbers would have more products than one; and if the products be the same, we have some proof that the operation has been correctly performed in each case. Thus, The best practical proof of this operation by "Casting out the Nines," depends upon that of the next subdivision: but we will enunciate the Rule, and apply it to this example, in the form usually adopted. "Find the sums of the figures in the Multiplicand and Multiplier, |