And the number found, after the work is finished, is called the Product. Both the multiplier and multiplicand are, also, in general, called Terms, or Factors. Multiplication, Pence, and Shilling Table 1. s. d.l. s. d.l. s. d. l. s. d.l. s. d. l. s. d. l. s. d. l. s. d. l. s. d. l. s. d.l. s. d. Pence 0.0.20.0.30.0.40.0.50.0.60.0.70.0.80.0.90.0.100.0.11 0.1.0 1 2 3 4 5 6 7 8 9 10 11 12 Shills. 0.2.00.3.00.4.00.5.00.6.00.7.00.8.00.9.00.10.00.11.00.12.0 0.0.20.0.40.0.60.0.80.0.100.1.00.1.20.1.40.1.60.1.80.1.100.2.0 14 16 18 20 22 24 2 4 6 8 10 12 0.0.30.0.60.0.90.1.00.1.30.1.60.1.90.2.00.2.30.2.60.2.90.3.0 3 6 9 12 15 18 21 24 27 30 33 36 0.3.00.6.00.9.00.12.00.15.00.18.01.1.01.4.01.7.01.10.01.13.01.16.0 60 5 10 15 20 25 30 35 40 45 50 0.0.60.1.00 1.60.2.00.2.60.3.00.3.60.4.00.4.60,5.00.5.60.6.0 0.0.70.1.20.1.90.2.40.2.11 0.3.60.4.10.4.80.5.30.5.100.6.50.7.C 84 0.0.80.1.40.2.00.2.80.3.40.4.00.4.80.5.40.6.00.6.80.7.40.8.0 80 10 20 30 40 50 60 70 0.0.11 0.1.100.2.90.3.80.4.70.5.60.6.50.7.40.8.30.9.20.10.10.11.0 RULE.* 1. Place the multiplier under the multiplicand, so that units may stand under units, tens under tens, &c., and draw a line below them. * 1. When the multiplier, in this rule, is a single digit, it is plain that the product will be rightly determined by the method here given; for by multiplying every figure or part of the multiplicand by this digit, 2. Begin at the right hand, and multiply every figure in the multiplicand by the unit's figure in the multiplier, setting down the whole of such products as are less than ten directly under the figures that are multiplied. 3. But for those that surpass ten, or a number of tens, write down the excess only; or, if there be no excess, a cipher, and carry a unit for every ten that was borrowed to the product of the next two figures. 4. Proceed in the same manner with each of the other figures of the multiplier, observing to place the first figure of every line immediately under the figure multiplied by. we in effect multiply the whole; and writing down the products which are less than ten, or the excess of tens, in the place of the figures multiplied, and carrying the number of tens to the product of the next place, is only collecting the similar parts of the respective products properly together, according to their values; whence the result so obtained must be evidently equal to the whole required product. 2. If the multiplier be a number consisting of more than one digit; after having found the product of the multiplicand by the first figure of the multiplier, as above, we suppose the multiplier to be divided into parts, and find, after the same manner, the product of the multiplicand by the second figure of the multiplier; but as the figure we are now multiplying by stands in the place of tens, the product will be ten times its simple value; and therefore the first figure of this product must be put in the place of tens, or, which is the same thing, directly under the figure we are multiplying by. And by proceeding in this manner, separately, with all the figures of the multiplier, it is evident that we shall multiply all the parts of the multiplicand by all the parts of the multiplier, or the whole of the multiplicand by the whole of the multiplier; whence these several products, being added together, will give the whole required product. To this, the following examples are subjoined, in order to render the reason of the rule, and the method of proof above given, as obvious as possible. It may here, also, be observed, that besides the method of proving the 5. Then add all the lines of products together, according to the order in which they stand, and their sum will be the answer or whole product required. NOTE. When the multiplier does not exceed twelve, the product can be written down as it arises, at one operation. truth of the operation, as before given, there is another very convenient and easy one, by means of that peculiar property of the number 9, mentioned in Addition; which is performed thus: RULE. Cast the nines out of each of the factors, as in Addition, and multiply the two remainders together; then, if the excess of nines in their product be equal to the excess of nines in the total product, the work is right. which is equal to the excess of 9's in 3 × 5, or 15, which is 6. Demon. of the Rule.- Let 9n+r and 9n' +r' be the two factors to be multiplied; then will their product, or (9n+ r) × (9n' +r'), be equal to 9onn' + 9n'r + 9n'r+rr'; and since the first three terms of this are, each, an exact number of 9's, it is evident that the excess of 9's in the fourth term rr', will be the same as the excess of 9's in the whole product; but r and r' are the excess of 9's in the factors themselves, and rr' is their product; whence the truth of the rule is obvious. This method of proof, which is usually ascribed to DR. WALLIS, is of a much earlier date, being given by LUCAS DE BURGO, in his work entitled Summa de Arithmetica, &c. printed in folio at Venice, 1494; and though it is, in many respects, a very convenient rule, there are circumstances in which it may fail. Thus, if two or more figures should be transposed in the work, or the value of one figure be too great, and that of another as much too little, or if a 9 be set down instead of a 0, or the contrary, the excess of nines in these cases will evidently be the same as if the work was right. METHOD OF PROOF. Make the former multiplicand the multiplier, and the multiplier the multiplicand; and if the product found from this operation be the same as before, the work is right. EXAMPLE WITH THE METHOD OF PROOF. EXAMPLES FOR PRACTICE. (3.) Multiply 24031042 by 2. (4.) Multiply 51420034 by 2. (5.) Multiply 32745675 by 2. (6.) Multiply 374328756 by 3. (7.) Multiply 5806342748 by 4. (8.) Multiply 8435674567 by 5. (9.) Multiply 2745675464 by 6. (10.) Multiply 54328432847 by 8. (11.) Multiply 8643597307 by 9. (12.) Multiply 796534289 by 11. (13.) Multiply 900909099 by 12. (14.) Multiply 732468756 by 15. (15.) Multiply 947137610 by 18. (16.) Multiply 273580961 by 23. (17.) Multiply 27501976 by 271. (18.) Multiply 82164973 by 3027. (19.) Multiply 62473864 by 27356. (20.) Multiply thirty-three millions three hundred thousand and sixteen, by one hundred and twenty thousand and four. Ans. 3996135120064. CASE II. Ans. 10810909188. Ans. 6292362103. Ans. 7453035496. Ans. 248713373271. Ans. 1709035023584. When there are ciphers at the right of either or both factors. RULE. Multiply the other figures only, and place as many ciphers to the right of the product as are in both the factors. And if, instead of being at the end, they are in any part of the multiplier, neglect them as before, observing to place the first figure of every product exactly under the figure you are multiplying by. Ans. 2400355079025000. (8.) Multiply 27156084900000 by 90060573000. Ans. 2445692566530647700000000. CASE III. When the multiplier is the product of two or more numbers, each of which does not exceed 12. RULE. Multiply the first by one of these numbers, and then the product thus arising by the other, and so on for the rest; and the result will be the answer required. *The reason of this method is obvious; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once. Thus in Example the First, on next page, 5 times the given number multiplied by 5, make 25 times that number, as plainly as 5 times 5 make 25. |