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PROOF OF ADDITION.
The mode adopted by the best accountants for proving this rule, is, after you have completed your sum, to go carefully over it again, varying it a little, that is to say, commencing at the top instead of the bottom of the column, when, if the amount of each column is found to be the same, they conclude the addition is correct.
PROOF OF SUBTRACTION. When your sum is complete, you have three lines of figures, the larger first; then the lesser or second line, which
you have subtracted from the first; and the difference, or third line, which is the difference between the lesser and the greater. It is therefore evident that if you add the difference to the lesser, it will produce the greater,
work has been correct.
191 lesser adding the remainder 79 to
79 remainder the lesser, 191, that it produces the greater, 270.
MULTIPLICATION. Multiplication is a compendious method of performing addition. To multiply means to increase. Multiplication is the art of increasing; so also is addition. But as addition increases by adding several sums differing in amount together, multiplication increases by joining the same sum several times, as six times six are 36, six times seven are 42 ; these are two sums in multiplication, either of which could be done by addition ; if you write six sixes in a column, or six sevens in a column, and add them, you will have the same amount as is given above; but you at once perceive how inconvenient and troublesome such a process would be, and also how much such trouble would be increased, when you would have a line of eight or ten figures to be multiplied by another line of five or six figures.
Such being the case, and it being utterly impossible to understand or work this rule without a perfect knowledge of the Multiplication Table, placed at the beginning of the book, if you are not already able to say it by rote, you will now return and learn it. The best method, I think, of doing so is, by writing down the figures as you speak them, and in the same order as they are written in the table; thus you acquire practice in writing figures, and when you multiply any two figures, the product will rise in your mind, be breathed by the lips, and traced by the fingers at the same moment.
This proficiency may be attained in a week; and when I tell you that a knowledge of this table facilitates almost all the operations of arithmetic, I feel confident that you will not proceed further until it is attained.
In multiplying any two numbers together they produce a third, the correctness of which third number is not affected by whichever of the given numbers we multiply: for instance, 4 times 6 are twenty-four, and 6 times 4 are twenty-four; in both instances the product is the same, and so it would be with any
other two numbers. But when it takes several figures to express the sum to be multiplied, we find it easier to take the smaller number for the multiplier. For instance, if we add another figure to either of these mentioned, say the 4, let us write a 9 upon the right, and it will be 49; here it would be easier to multiply the 49 by 6 than 6 by 49; hence we always choose the larger number for the multiplicand, or, as some call it, the passive factor; the smaller is the multiplier, or active factor; the sum produced is called the product.
Let us set down the three 49 Multiplicand figures for multiplication. You 6 Multiplier see the 49 written for a mul
294 Product. tiplicand; 6, the multiplier, is written in the units place under the 9. Having so placed it, and drawn a line underneath, we proceed with our multiplication, saying six times 9 are 54, which you know to be five tens and four; you write the 4 under the 6, or in the units place, and proceed with the next figure, which is in the place of tens, saying six times 4 are 24, and 5 you carry make 29, which is written in full, there being no more figures to multiply. In this manner can any sum, however large, be multiplied by a single figure, that is, by writing the units in the product of each figure underneath itself, and carrying the tens and adding them to the product of the next figure we multiply. For example, let us increase the above multiplicand to thousands, by adding two more figures, thus : we say 6 times 5 are 30, write a 0 under the 6 and
6 three; 6 times why are 42 and 3 make 45, write a 5 and carry 4;
29850 6 times 9 are 54 and 4 make 58, write 8 and carry 5; 6 times 4 are 24 and 5 make 29, which is written in full, there being no more figures to multiply.
After doing the foregoing examples on your slate, as you read the instructions, work for practice the following sums. The multiplier of any of them does not exceed twelve, all are therefore done in a single line.
x Sign of multiplication, expresses the word by, Example (1), thus : 246 X 6, multiply 246 by 6.
(2.) 5247121 3 (7.) 4940325 x 7
24351 x 6 (10.) 1475263 x 10
Judging now that you perfectly understand how you are to multiply with any number not exceeding twelve, we will proceed to multiply with larger numbers, which is seldom done in one line; for although it may be done, it is attended with more difficulty, with greater liability to error, and is therefore but seldom resorted to.
But there is another method adopted, equally simple and almost as expeditious, and it is this : take the second sum in the example for practice, and add another figure, say 6, to its multiplier, and it will stand thus :
I need not remind you that the figure 5247121 multiplicand. six represents units;
36 multiplier. with it we commence to multiply, saying,
31482726 product of six. 6 times 1 are 6, which
15741363 product of thirty. 6 we write down in 188896356 product of thirty-six. the place of units, and say 6 times 2 are 12, write 2 and carry l;
six times 1 are 6 and 1 makes 7 ; write 7, and say, six times 7 are 42, write 2 and carry 4; six times 4 are 24, and 4 make 28, write 8 and carry 2 ; six times 2 are 12 and 2 are 14, write 4 and carry 1; six times 5 are 30 and 1 makes 31, which is written in full. The next figure of the multiplier is 3, and represents three tens, or thirty; you therefore write the right figure of its product under the second figure of the former product, and proceed as before directed, writing down the units and carrying the tens until you multiply the last figure, whose product you write in full, and find the left figure is one step in advance of the product above it. You then add the two lines of figures together, and the total will be the product of 5247121 multiplied by 36.
The most important thing to be observed, when you multiply by any large sum comprising three or four figures, is the situation of each figure in the product. For this purpose the multiplier offers a guide; if you write the unit figure of each product under the figure of the multiplier which produces it, you cannot go wrong.
684 4 2 3 product by 9. 3041 8 8 product by 40. 15209 4 product by 200.
18935 7 0 3 product by 249
Again, let us see each of the above products if the sum was multiplied by 9, by 40, and by 200, separately. You can now see clearly the
Add the reason that the product of 76047
products, each figure in the multiplier
684423 is placed one step in advance,
684423 3041880 and also what you must do
15209400 whenever a cipher appears in the units place in your
multi- 76047 18935703
40 plier, that is, write a cipher under it, or in the units place,
3041880 and go on multiplying by the next figure. Should there be a third figure in your mul
76047 tiplier, you will recollect to
200 write its product under itself, which will bring it in the place 15209400 of hundreds. See the following examples.