In this table the arrangement is the same as in the tables of Addition and Subtraction-the product of any two numbers being placed in the square formed by the crossing of the vertical and horizontal columns in which these numbers are placed. 35. If one factor contain fewer figures than the other, it is convenient to make the former the multiplier. Hence it becomes necessary to prove that the product of two factors is the same whichever be made multiplier. Now the multiplication of one number by another may be made by repeating every simple unit of the multiplicand as often as the multiplier contains unity; and the sum of the whole assemblage of simple units forms the product of the two numbers. Conceiving the ones contained in the multiplicand to be written in a horizontal line, and as many such lines as the multiplier contains unity to be placed under each other, thus1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, then beginning with the first horizontal line, combining the ones in it under one name; combining the successive ones of the second horizontal line with the sum of those in the first, and so proceeding with a third, a fourth, an mth line, the sum of all the ones in the table, that is the product of the two numbers, is obtained. But the sum of all the ones in the table may be also obtained by beginning with the left vertical column, adding together the ones in it; adding to this and the successive sums, the ones in a second, a third, . . . . an nth column, until the whole are again combined into one number. Now, in the first instance, the horizontal line is multiplicand, and the vertical column multiplier; and in the second, the vertical column is multiplicand and the horizontal line multiplier. In each case the result or product is the same, being made up of all the ones in the table. This reasoning is independent of any particular values of the factors whence a×b=b×a; a and b denoting any whole numbers. a. The product of two numbers may be multiplied by a third number, this product by a fourth number, &c.; and in these cases also the value of the last result is not affected by the order in which the factors are multiplied together. The proof, as in the instance of two factors, depends on the principles, that multiplication is repeated addition, that all the factors admit of decomposition into ones, and that the sum of these ones is the same in whatever order they are combined into one number. When the number of factors exceeds two, the last result is termed the continual product of these factors. 36. To apply these principles, let it be required, first, to find the product of 5879 and 6, by addition, and also by multiplication. The operation by addition requires no explanation. That by multiplication is performed as follows:— 9 × 6=54. The 4 is written and the 5 carried, as in addition. The 7 is written and the 4 carried. 7 × 6=42, and 42 + 5 carried=47. 8 × 6=48, and 48+ 4 carried=52. 5 × 630, and 30+ 5 carried=35. both figures are set down as in addition. It will be observed that the only difference between the operations, in this instance, is that by addition it is necessary to say 9 and 9 are 18, and 9 are 27, and 9 are 36, and 9 are 45, and 9 are 54: while by multiplication the table gives at once 9×6=54; and so on with the other columns in addition and the corresponding figures in multiplication. To multiply 5879 by 60 it is only necessary to annex 0 to 35274: Thus, 5879 × 60=352740. In like manner, 5879 x 6003527400. 2nd. Ex. Multiply 57496 by 378. The product of 57496 by 378 is composed of the following partial products; 57496 x 8, 57496 x 70, and 57496 x 300. Also, 57496 x 8=459968; 57496 x 70-574960 x 7=4024720; 57496 x 300-5749600 x 3=17248800. If the factors and the partial products are arranged as in (1), (2), and the partial products are added together, the calculation is made to appear in the usual form. . In (1.) the zeros arising from the multiplication of the multiplicand by the tens, the hundreds, &c. of the multiplier are retained, and in (2.) omitted. If the last figure of the product of the multiplicand by a significant figure of the multiplier is written directly under that figure of the multiplier, the zeros may be omitted. For the product of simple units by simple units being simple units, of simple units by tens being tens, &c., it is evident that the last figure of each partial product thus obtains its proper relative value. 37. In Multiplication, as in Addition, the calculation is, for reasons of convenience and brevity, made from right to left. But the partial product of every figure of the multiplicand by every figure of the multiplier may (regard being had to the relative as well as the absolute value of all the figures) be formed separately, in any order, and the whole product found by taking the sum of these partial products. 38. The product of any two factors is expressed by as many figures as are contained in both factors, or by that number less one. Let the highest figure of the multiplicand and multiplier be each replaced by 10. With these relative values, the tens are respectively greater than the multiplicand and multiplier ; and therefore the product of these tens followed by zeros, equal in number to the remaining figures of the multiplicand and multiplier, must exceed the product of the multiplicand and multiplier. Now the product 10 x 10=100. To this annexing as many zeros less one as there are figures in the multiplicand, and as many less one as there are figures in the multiplier, the whole number of figures expressing the product is one more than the number contained in both factors. But 1 followed by zeros is the least number which can be expressed by the number of figures contained in it. Therefore the product of the multiplicand and multiplier, which is a smaller number than 1 followed by these zeros, cannot be expressed by more figures than are contained in both factors. Again, let the highest figures of the multiplicand and multi plier be each replaced by 1. The multiplicand and multiplier cannot respectively be less than the ones with these relative values, and therefore the product of the multiplicand and multiplier cannot be less than the product of the ones followed by zeros equal in number to the remaining figures of the multiplicand and multiplier. The product 1x 1=1. To this annexing as many zeros less one as there are figures in the multiplicand, and as many less one as there are figures in the multiplier, the whole number of figures expressing the product is one less than the number contained in both factors. But the product of the multiplicand and multiplier cannot be expressed by fewer than this number of figures. Therefore the number of figures in the product cannot fall short of the number in both factors by more than one figure. The proposition with which this article begins is consequently established. 39. General Rule for Multiplication : Having written the figures of the multiplier under the figures of the same order of the multiplicand, take, as a partial factor, the last figure of the multiplier. Multiply the last figure, or simple units, of the multiplicand by this factor; write the units of the product in the vertical column of the partial multiplier, and reserve the tens (if any) for combination with the product of the tens of the multiplicand by this multiplier. Multiply the tens of the multiplicand by the partial multiplier, and to the product add the number of tens reserved from the product of the simple units; write the units of this sum on the left of the figure which expresses the simple units of this partial product, and reserve the tens for combination with the product of the units of the third order, or hundreds of the multiplicand by this multiplier. Proceed in this manner until the product of every figure of the multiplicand by this multiplier is obtained. In the same manner form the partial product of the multiplicand by the figure which expresses the tens of the multiplier; write the last figure of this product in the same vertical column with the partial multiplier, and the other figures in order, from right to left, according to their relative values. Form, similarly, the partial product of the multiplicand by the figure expressing the hundreds, the figure expressing the thousands, &c. of the multiplier, and write these successive products, one below another, in horizontal lines, placing always the lowest figure of each partial product in the vertical column of its own partial multiplier. The sum of all the partial products is the product of the proposed numbers. 40.. Examples in Multiplication of whole numbers : 1. Multiply 3546 by 2, and verify the result by addition. 28. Multiply 42587 by 247 29. 147635 by 358. |