persons. The first steamboat was put into successful operation, A. D. 1807, by Robert Fulton. How long since the date of each occurrence? 6. From $127, 66 cts. 5 mills, take $37, 38 cts. 6 mills. 7. From $1049, 66 cts. 6 mills, take $999, 77 cts. 7 mills. Thus, { 1049.666 Or, 1049.666-999.777-$49, 8 dimes, 8 cts. 9 mills, Ans. The sign between the numbers, in the operation above, signifies that the number which precedes, is the minuend; and the number which follows, is the subtrahend. Therefore we may, when this sign (-) is used, call its name minus or less. Ans. $49.889 SIMPLE MULTIPLICATION. It is adding a number to itself a certain number of times. RULE. The number which is to be multiplied is wrote down first, which term the multiplicand. The number by which the multiplicand is multiplied, you will next write down under it, units under units, tens under tens, &c. which term the multiplier. Begin with the units' figure of the multiplier, and multiply each figure of the multiplicand by it, carrying one for every ten as in simple addition, placing the result of this operation under the figure of the multiplicand from which it arose, calling it the product. Then multiply, in the same manner, by each of the remaining figures, if there be any, placing the first figure of each product under its own multiplier. If there be ciphers on the right hand of either or both of the factors,* totally neglect such, but annex as many ciphers to the total product as are thus neglected. Also, disregard ciphers standing between figures in the multiplier. Add up the several products, which are placed as the preceding directs, as you would any column of places, carrying one for every ten. This sum you may term the total product. Demonstration. When the multiplier is a single digit; it is plain that we find the product; for, by multiplying every part of the multiplicand, it is evident we multiply the whole; and in writing down the products which are less than ten, or the excess of tens, under the place of the figures multiplied, and carrying the tens to the product of the next place, is only gathering together By factors are meant the multiplicand and multiplier. L. factor, an * agent. the similar parts of the respective products, and is, therefore, the same in effect, as if we wrote down the multiplicand as often as the multiplier expresses, and added such together. When the multiplier consists of several figures; we find the product of the multiplicand, by the unit figure, and then suppose the multiplier divided into parts, and, after the same manner, find the product of the multiplicand by the second figure of the multiplier; but, as the figure by which we are multiplying, stands in the place of tens, the product must be ten times its simple value; and, therefore, the first figure in this product must be noted in the place of tens, or, which is the same, directly under the figure by which we are multiplying. And proceeding in the same manner with all the figures in the multiplier separately, it is evident, we shall multiply all the parts of the multiplicand by all the parts of the multiplier: Therefore, these several products being added together, will be equal to the whole required product. In order to further illustrate the rule, let 568 be multiplied by 476. These numbers may be decomposed and multiplied thus: Multiplier, 400706= 476 3000+360+48 3408 35000+ 4200+ 560 = 3976 200000+24000+ 3200 =2272 Ans., 200000+59000+10400 +920+48=270368 In the preceding process, we multiplied through, first by 6 units, then through by 7 tens, or 70, and then by 4 hundreds, or 400, placing the several products underneath, and adding the respective columns. Example. A merchant sold 2060 bales of cotton, each weighing 204 pounds. How many pounds did he sell? OPERATION. Multiplicand, 204 EXPLANATION. Here the quantity is made the number multiplied, which, strictly speaking, should be; but, the larger number, whether price or quantity, could be used as the multiplicand. And you will observe, as by direction of the rule, the Total product, 420240 cipher annexed to the total product, making 0 units, which was neglected. Also, the multiplicand is multiplied only by the 6 in the multiplier first, producing 1224, the first product; then, again, as by the rule, by the 2 in the multiplier secondly; also observing to "disregard" the cipher in the multiplier. Again-when multiplying by the 6 in the multiplier, we first say, 6 times 4 units are 24, which are wrote down as you observe in the first product, because the cipher in the multiplier, is disregarded, and also the 0X6-0, which would cause the 2 in the 24 to occupy a place under the column of ciphers, made by the present situation of the factors.* TO PROVE MULTIPLICATION. 204 Example. RULE. Having performed the multiplication, cast out the 9s from the multiplicand, say of the preceding example, also from its multiplier, placing the excesses, one under the other. Multiply these excesses together, and cast the 9s from the product, putting the excess underneath. 2060/83 excesses. 48, product. Total, 420240 | 3 Cast out the 9s from the total product, and if the work is right, the two excesses will be alike. Reason. The excess of the total product is the same as the excess of the amount in Addition-hence the excess of the excesses multiplied, is the same as the excess of 9s in the column of excesses in adding.' NOTE 1.-Also another method of proof depends upon this proposition, that if two numbers are to be multiplied together, either of the numbers may be made the multiplier or multiplicand, and the product will be the same, NOTE 2.-In multiplication the product bears the same ratio to the multiplicand that the multiplier does to unity or 1. Questions. What is Simple Multiplication ?-Which number is wrote down first ?-What is it termed ?-How are the numbers which are to be multiplied together, to be wrote down?-Why? How wrote down, if ciphers are at the right hand of both or either? Why?-Where do you begin to multiply?-What do you multiply? How do you proceed before writing down the result? Where do you place the result?-How do you proceed if there be any other figures in the multiplier?-If there are ciphers, what? When do you annex ciphers to the total product?-How?What is the total product?-Why do you find the product when the mul tiplier is a single figure?-If more, why?-When multiplying by other than a unit figure, why do we place the product of the multiplier and unit *The sign X signifies the number preceding is to be multiplied by the number which follows; thus 6X424, viz. 6 times 4 are equal to 24. 22 ABBREVIATIONS IN MULTIPLICATION. figure of the multiplicand under the figure by which you multiply?-How do you prove multiplication?-Explain it on the blackboard. Give the reason. What other method of proof depends on what proposition?Repeat the last note. ABBREVIATIONS* IN MULTIPLICATION. Reasons to the six following rules, one to each, may be drawn from the preceding rule and its demonstration. 1. To multiply by a composite number,† viz:- A number which is the product of two other numbers, either being more than unity or 1. RULE.-Multiply first by one of those. figures, and that product by the other. The last product is the answer. Example. 59375,X35 7,X5-35 415625 5,X7-35 2078125, Ans. Example. 75964×13 227892 2. When the multiplier is 13, 14, &c. to 19. RULE. Place the multiplier at the right of the multiplicand, placing the sign of multiplication between-multiply the multiplicand by the unit figure of the multiplier, removing the product one place to the right of the multiplicand, and both, added, make the answer. 3. When the multiplier is 101, 102, &c., to 109. 987532, A. RULE.-Multiply by the unit figure of the multiplier, and remove the product two places to the right of the multiplicand. Add together as before for the product. Example. 9417 119 4. When the multiplier is 111, 112, &c., to 119. RULE.-Multiply by the unit figure only of the multiplier; add to each multiplication the two figures which stand next at the right hand of that which you multiplied, and to the two last figures, Ans. 1120623 separately, add what you carry. 5. When the multiplier is 21, 31, &c., to 91. RULE.-Multiply every figure of the given multiplicand by the tens' figure only of the multiplier, and write the unit figure of the first product under the tens' place of the multiplicand; and both, added, make the answer. * Abbreviate signifies to shorten. Example. Or, 91×4793 43137 Ans. 436163 Composite, [L. compositus,] composed; formed from several others Component, [composing.] ABBREVIATIONS IN MULTIPLICATION. 6. When the multiplier is 22, 23, &c., to 29. &c., to 92, &c., to 99. 23 Or, 32, &c., 42, Example. 7657 29 Ans. 222053 RULE.-Multiply every figure of the multiplicand by the unit figure of the multiplier, and add to each product twice, thrice, &c., to 9 times, (as may be,) that figure which stands next at the right hand of the figure, you multiplied; and to twice, or thrice, &c., to 9 times (as may be,) the last figure, add what you carry. 7. When the multiplier is any number of 98. RULE. To the right of the multiplicand write as many Os as there are 9s in the multiplier-then under this new multiplicand write the given one, units under units, tens under tens, &c. Subtract, and the remainder is the answer. Example.-Multiply 98765 by 999. Operation. 98765000 98765 A., 98666235 Reason. Had the multiplier been 1000, it would have been but a unit or 1, larger than now; and adding the three ciphers in the 1000 is by the rule, the same as multiplying; therefore, subtracting that once the multiplicand too many, from the multiplicand thus multiplied, at once brings the answer. Operation 8. When the multiplier is nearly 10, 100, &c. RULE. Write as many ciphers at the right hand of the multiplicand, as there are figures in the multiplier. From this subtract the product, arising from the given multiplicand being multiplied by the excess of 10, or 100, &c., over the multiplier. Example.-Multiply 378 by 8. 10-8-2 756 3780, multiplicand 756 twice too many. 3024, Ans. NOTE 1.-It is presumed that a familiarity with the preceding, and what other examples which may be supplied by the teacher, or pupil, will be sufficient without further explanations, or more examples. NOTE 2.-The top line and left hand column of figures in the following table are the factors in producing the products, to be found at the point of intersecting lines, which products are also the dividends, those factors being, each, in their turn, the divisor, or quotient. Questions. Give case 1 of the preceding abbreviations. Give the rule. Give the reason. The teacher may apply these three questions to the seven preceding abbreviations. |