RULE. Place the numbers under each other, viz. units under units, tens under tens, &c.; add up the figures in the row of units, and carry as many units to the next row as there are tens contained in the sum: proceed thus till the whole is finished. For the proof.-Divide the numbers to be added into two parts, then add up each part by itself, and collect these sums together for the whole. Note 1. If equal numbers be added to equal numbers, the whole will be equal. 2. If several numbers are to be added together, they will amount to the same sum, when placed regularly one under another, whichever line or row of figures stands uppermost. 3. Dr. Wallis, in his Arithmetic, gives the following rule to prove a simple addition sum. Add the figures in the uppermost row together, reject the nines contained in their sum, and set the excess directly even with the figures in the row. Do the same with each row, and set all the excesses of nine together in a line, and find their sum; then, if the excess of nines in this sum (found as before) be equal to the excess of nines in the total sum, the work is right. (4.) Add 1473, 40734, 371049, 40057, 3471473, 5734, 37492, and 4718375, together. (5.) Collect 371434, 278949375, 67149, 3457143, 714934, 9000987, and 5734747, into one sum. (6.) Add 5714329, 4718714, 34983714, 671493, 74987149, 6777894987, and 19, together. (7.) Add 571493, 40007, 6493497, 4718349, 3714934, 4934938, 174934, and 147349, together. (8.) Suppose the distance from London to Biggleswade be 45 miles, thence to Peterborough 36, thence to Lincoln 51, and thence to Hull 41 miles; how many miles are Peterborough, Lincoln, and Hull, from London? SIMPLE SUBTRACTION. Definition. Simple subtraction teaches to deduct, or subtract a less number from a greater of the same denomination, whereby the remainder or difference is found. RULE. Place the less number under the greater, so that units may stand under units, tens under tens, &c. Begin at the unit's place, and subtract each figure in the lower line from the figure above it; if the lower figure be greater than the upper, add ten to the upper figure, from which subtract the lower; set down the remainder, and carry one to the next lower figure. For the proof.-Add the remainder and less number together, and the sum will be the greater. Or, subtract the remainder from the greater number, and the difference will be the less. CLASS II. (9.) From the Creation to the Flood was 1656 years; thence to the building of Solomon's Temple 1336 years; thence to Mahomet, who lived 622 years after Christ, 1630 years. In what year of the world was Christ then born, and how many years is it since the creation? (10.) Sir Isaac Newton was born in the year 1642, and died in 1727, how old was he at the time of his decease, and how many years is it since he died? (11.) A gentleman has two sons, the age of the elder added to his make 126 years, and the age of the younger son is equal to the difference between the age of the father and the elder son. Now, if the father be 80 years of age, how old are each of his sons? (12.) Three boys, A, B, and C, won together 97 marbles at play; now, if the number of marbles B won be added to the number C won, they will make 60; and, if the number A won be added to the number C won, they will make 62. How many marbles did each boy win separately? SIMPLE MULTIPLICATION. Definition 1.-Simple multiplication is a rule by which we increase the greater of two given numbers, of the same denomination, as often as there are units in the less; being a compendious method of performing addition. 2. The number to be multiplied is called the multiplicand; the number you multiply by is called the multiplier; and the number produced by multiplication is called the product. These numbers are sometimes called factors, because they are to constitute a factum or product. The Multiplication Table. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 810121416 18 20 22 24 3 6 912151821 24 27 30 33 36 4 8121620242832 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6121824303642 48 54 60 66 72 71421 2835 42 49 56 63 70 77 84 816243240485664 72 80 88 96 918278645 5463 72 81 90 99108 |10 20 30 40 50 60 70 80 90 100 110 120 1122334455667788 9110121 132 12243648 60/728496108 120132144 Proposition 1. To multiply by a single figure, or any number not exceeding 12. Rule. Begin at the unit's place of the multiplicand, and multiply each figure in it by the multiplier, writing down the whole of such products as are less than 10; but, for such as exceed 10, or a number of tens, write down the excess, and carry an unit, for each ten, to the next product. Prop. 2. When the multiplier is the product of two or more numbers in the table. Rule. Multiply the multiplicand by one of the component parts, and that product by the other, &c. for the whole product. Prop. 3. When the multiplier consists of several figures. Rule. The multiplicand must be multiplied by each figure separately, (beginning with the right-hand figure of the multiplier,) and the first figure of every product must stand exactly under the figure you multiply by. Add these products together for the whole product. Or, begin with the left-hand figure of the multiplier, and multiply every figure in the multiplicand by it; then multiply in a similar manner by the next figure, &c., taking care to place every succeeding product one figure farther out towards the right-hand. Prop. 4. When ciphers are intermixed with the figures in the multiplier. Rule. Omit the ciphers, and let the first figure of each product be placed under its repective multiplier. Prop. 5. When there are ciphers at the end of the multiplicand or multiplier. Rule. Neglect the ciphers, and multiply as before, then to the right-hand of the product annex as many ciphers as were omitted. For the proof. Multiply the multiplier by the multiplicand, and if the product be the same with that of the multiplicand by the multiplier, the work is right. Note 1. If two numbers are to be multiplied together, they will make the same product, whichever number you make the multiplier. 2. If several numbers, as 5, 6, 7, &c. are to be multiplied together, it is the same thing whether 5 be multiplied by the product of 6 and 7, or it be multiplied first by 6 and then by 7, &c. And, if several given numbers are to be multiplied by any number, and the sum of the products taken; it will be the same thing, if you multiply the sum of those given numbers by that multiplier. 3. The product of any two numbers can have at most but as many places of figures as are in both the multiplier and multiplicand, and at least but one less. 4. Multiplication may be proved by casting out the nines as in addition. Thus cast the nines out of the multiplier and multiplicand, and set down the remainders. Multiply the two remainders together; and, if the excess of nines in the product be equal to the excess of nines in the total product, the work is generally right. Examples to Proposition 1. (1.) Multiply 471347325 by 2 Product 942694650 (2.) Multiply 371493407 by 3. |