OF MULTIPLICATION. MULTIPLICATION is a compendious method of Addition, teaching how to find the amount of any given number when repeated a certain number of times ; as, 4 times 6, which is 24. The number to be multiplied, or repeated, is called the Multiplicand.-The number you multiply by, or the number of repetitions, is the Multiplier.-And the number found, being the total amount, is called the Product.--Also, both the multiplier and multiplicand are, in general named the Terms or Factors. Before proceeding to any operations in this rule, it is necessary to learn off very perfectly the following Table, of all the products of the first 12 numbers, commonly called the Multiplication Table, or sometimes Pythagoras's Table, from its inventor. 9118 10 20 30 40 50 60 70 80 90 100 110 120 To multiply any Given Number by a Single Figure, or by any Nurnber not more than 12.. * Set the multiplier under the units figure, or right-hand place, of the multiplicand, and draw a line below it. Then beginning at the right-hand, multiply every figure in this by the multiplier.-Count how many tens there are in the product of every single figure, and set down the remainder directly under the tigure that is multiplied; and if nothing remains, set down a cipher.-Carry as many units or ones as there are tens counted, to the product of the next figures ; and proceed in the same manner till the whole is finished. EXAMPLE Multiply 9876543210 the Multiplicand. 2 the Multiplier 19753086420 the Product. To multiply by a Number consisting of Several Figures. | Set the multiplier below the multiplicand, placing them as in Addition, namely, units under units, tens under tens, &c. drawing a line below it. -- Multiply the whole of the multiplicand by each figure of the multiplier, as in the last article ; 5678 4 * The reason of this rule is the same as for the process in Addition, in which 1 is carried for every 10, to the next place, gradually as the several products are produced one after another, instead of setting them all down one below cach other, as in the annexed example. 32 230 XXX 2400 20000 70 X 600 5000 x 4 22712 = 5678 X 4 † After having found the produce of the multiplicand by the first figure of the multiplier, as in the former case, the multiplier is supposed to be divided into parts, and the product is found for the second figure in the same manner: but as this figure stands in the place of tens, the product must be ten times its simple value; and therefore the first figure of this product must be set in the place of setting setting down a line of products for each figure in the multiplier, so as that the first figure of each line may stand straight under the figure multiplying by.--Add all the lines of products together in the order as they stand, and their sum will be the answer or whole product required. TO PROVE MULTIPLICATION. There are three different ways of proving Multiplication, which are as below. First Method.-Make the multiplicand and multiplier change places, and multiply the latter by the former in the same manner as before. Then if the product found in this way be the same as the former, the number is right. Second Method.—*Cast all the 9's out of the sum of the figures in each of the two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9's out of the product, as also out of tens; or, which is the same thing, directly under the figure multiplied by. And proceeding in this manner separately with all the figures of the multiplier, 1234567 the multiplicand. it is evident that we shall multiply all 4567 the parts of the multiplicand by all the parts of the multiplier, or the 8641969 7 times the mult. whole of the multiplicand, by the 7407402 60 times ditto. whole of the multiplier : therefore, 6172835 500 times ditto. these several products being added 4938268 =4000 times ditto. together, will be equal to the whole required product; as in the example 563827489 = 4567 times ditte. annexed. * This method of proof is derived from the peculiar property of the number 9, mentioned in the proof of Addition, and the reason for the one may serve for that of the other. Another more ample demonstration of this rule may be as follows: - Let P and Q denote the number of 9's in the factors to be multiplied, and a and b what remain; then 9P+ a and 9Q + b will be the numbers themselves, and their product is (9 P X9Q) + (9PX6). + (9Q X a) + (a X 6); but the first three of these products are each a precise number of 9's because their factors are so, either one or both: these therefore being cast away, there remains only a Xb; and if the Y's also be cast out of this, the excess is the excess of 9 in the total product: but a and b are the excesses in the factors themselves, and a xb is their product; therefore the rule is true. the the whole product or answer of the question, reserving the remainders of these last two, which remainders must be equal when the work is right.*--Note, It is common to set the four remainders within the four angular spaces of a cross, as in the example below. Third Method.-Multiplication is also very naturally proved by Division; for the product divided by either of the factors, will evidently give the other. But this cannot be practised till the rule of Division is learned. Multiply 123456789 by 3. Multiply 123456789 by 4. Multiply 123456789 by 5. Multiply 123456789 by 6. Multiply 123456789 by 7. Multiply 123456789 by 8.. Multiply 123456789 by 9. Multiply 123456789 by 11. Multiply 123456789 by 12. Multiply 302914603 by 16. Multiply 273580961 by 23. Multiply 402097316 by 195. Multiply €2164973 by 3027. Multiply 7564900 by 579. Multiply 8496427 by 874359. Multiply 2760325 by 37072. Ans. 370370367. * If the two remainders be equal, it by no means follows that the answer is correct. Thus, if we multiply 13 by 12, the product is 156, and the remainders are each 3. but if we take for the answer or product any of the numbers 165, 561, 516, 246, &c, the remainders are the same as before, and therefore the rule is defective. Ed. CONTRAC CONTRACTIONS IN MULTIPLICATION. I. When there are Ciphers in the Factors, If the ciphers be at the right-hand of the numbers ; multiply the other figures only, and annex as many ciphers to the right-hand of the whole product, as are in both the factors. When the ciphers are in the middle parts of the multiplier; neglect them as before, only taking care to place the first figure of every line of products exactly under the figure multiplying with. EXAMPLES 1. 2 Mult. 9001635 Molt. 390720400 by. 70100 by 406000 9001635 63011445 23443224 15628816 631014613500 Products 158632482400000 3. Multiply 81503600 by 7030. Ans. 572970308000. 4. Multiply 9030100 by 2100. Ans. 18963210000. 5. Multiply 8057069 by 70050. Ans. 564397683450. II. When the multiplier is the Product of two or more Numbers in the Table; then, * Multiply by each of those parts separately, instead of the whole number at once. EXAMPLES 1. Multiply 51307298 by 56, or 7 times 8. 51307298 7 359151086 8 2873208688 * The reason of this rule is obvious enough ; for any number multiplied by the component parts of another, must give the same product as if it were multi. plied by that number at once. Thus, in the 1st example, 7 times the product of 8 by the given number, makes 56 times the same number, as plainly as 7 times 3 makes 56. 4 2. Mail VOL. 1. |