32. The above neat and concise mode of performing Addition and Subtraction may always be pursued when the amount of the quanties is small, and the number of terms few; but when the quantities are large, and the terms numerous, to add or subtract by the above plan would be very difficult, exceedingly liable to error, and in many casesimpracticable. Therefore another mode has been devised for accomplishing the object, in which neither the amount nor the number of the terms presents any difficulty. 762 343 1105 33. The mode is this. The terms are arranged as in the annexed sum regularly under each other. The units under units, the tens under tens, the hundreds under hundreds, and so on. Then beginning at the right hand figures, that is, at the units, you add them together, and their amount, which is five, you place underneath. You next add the tens, which amount to ten, that is, ten tens, or one hundred. Set down a cypher in the place of tens, and carry 1 to the place of hundreds, and say ONE and THREE are FOUR and SEVEN are ELEVEN. the eleven, the units figure of it in the place of hundreds, and the tens figure advanced a little to the left, and the sum is finished. Observe the eleven is eleven hundreds, or one thousand one hundred, and the sum total of the question one thousand one hundred and five. Set down 762 343 34. In order to show the reason for carrying, let us repeat the above example, but work it another way. Instead of beginning the addition at the units column, let us begin at the hundreds. Then seven and three are ten, that is, ten hundreds, or one thousand, which set down underneath as shown in the annexed example. Then proceed with the tens, and say six and four are ten, that is, ten tens, or one hundred, which set down also underneath the last. Then proceeding with the units, say two and three are five, which set down also underneath the last sum, observing, in all cases, their proper place. Add these three several sums together, and it will be found that their amount is the same as before. 1000 100 5 1105 35. We will repeat the same example, and place it in another point of view, in order to make the matter as plain as possible. 762 700+ 60+2 1105 1000 + 100 +5 36. We will now take the amount of the above example, and separate it into two parts, that is, we will subtract one of the terms from the amount, and the remainder will be the other term. Arranging the question as above, we proceed to subtract it, term from term. Beginning at the extreme right, we say two from five and three, and place the three underneath. Then taking the next terms, we say sixty from one hundred, and forty, which place underneath also; then taking the other two terms, we say seven hundred from ten hundred, and three hundred. Adding the three terms, so found, together, their amount is three hundred and forty-three for the difference. 37. We will take the same question again with a little difference in the division of its parts. From 1105 1000+100+00+5 343 0+300+40+3 Taking the first two terms to the right, we say, two from five and three remain. Then taking the next two terms, we say sixty from nothing we cannot, but borrow one hundred, and we say sixty from one hundred, and forty. We then subtract the other two terms, but first we repay the hundred we borrowed at the preceding terms, by adding it to the seven hundred, which making eight hundred, we then say eight hundred from one hundred we cannot, but borrow one thousand, which, added to the one hundred, will make eleven hundred. then say eight hundred from eleven hundred and three hundred remain. Repaying the thousand we borrowed at the next term; we say a thousand from a thousand and nothing remain. The result, it will be perceived, is the same as before, viz. three hundred and forty-three. We 38. But neither of the preceding methods is that by which Subtraction is usually performed; they are only given to illustrate the principle. The usual method of operation is this: arrange the number to be subtracted under the number it has to be subtracted from, in such a manner that the units of the one number shall be exactly under the units of the other; the same must be observed with regard to the tens, the hundreds, &c. Being so arranged, we draw a line underneath, and then proceed with the Subtraction, thus:-Two from five and three remain; we put down the three under the place of units. Then at the place of tens we say six from nothing we cannot, but borrow C 1105 762 343 en, and say six from ten and four remain. Putting down the four under the place of tens, and to the place of hundreds carrying one for the ten we borrowed at the place of tens, we say one and seven are eight, eight from one we cannot, but borrow ten, which ten, added to the one above, make eleven. Eight from eleven and three. We set down the three in the place of hundreds, and carrying one to the place of thousands say, one from one and nothing. But as cyphers on the left of figures express no value, it is unnecessary to write them down. The result is the same as before, viz. three hundred and forty three. 39. The student, before he proceeds further, ought carefully to read over the foregoing paragraphs several times, and consider their purport. He will perceive that a question in Addition, though worked in three different ways, produces the same result, and that a question in Subtraction, though also worked three different ways, also produces the same result. He should reflect on this, and endeavour to discover the reason for this uniformity of result. (See paragraph 29). He should also endeavour to discover the reason for carrying, both in Addition and Subtraction. But should he, after carefully perusing the foregoing, still be unable to perceive the reason for carrying, let him turn back to paragraphs 9 to 12, and observe what is there said respecting numbers increasing in a tenfold ratioat every step from right to left, and then attend to what he is now about to read. Observe, that ten in the place of units is equal to one in the place of tens,-that ten in the place of tens is equal to one in the place of hundreds-that ten in the place of hundreds is equal to one in the place of thousands, and so on. Hence the reason why, in ADDITION, we carry one to the column of tens for every ten found in the column of units-one to the column of hundreds for every ten found in the column of tensone to the column of thousands for every ten found in the column of hundreds, and so on; and why, in SUBTRACTION, we carry one to the column of tens for every ten we borrow at the column of units-one to the column of hundreds for every ten we borrow at the column of tens-one to the column of thousands for every ten we borrow at the column of hundreds, and so on. 40. We will now elucidatet what is advanced in the foregoing paragraphs by an example. What is the sum of 123+321+234+433? The four numbers to be added are called the terms of the question. If from the sum of all the terms, we subtract any * Purport, design or tendency, meaning or import; from the French pour, for, and porter, to carry. + Elucidate, to enlighten, make clear, plain, apparent, or easy to be understood; from the Latin lux-lucis, light. single term, the remainder will be equal to the sum of the three other terms. 123 First term. 678 - 321 Second do. 234 Third do. 433 Fourth do. 1111 Sum of the four terms. 433 Subtract the fourth term. 678 Remainder equal to the sum of the three first terms. 123+321+234+433–433=123+321+234=678 If from the sum of the three first terms we subtract the third term, the remainder will be equal to the sum of the first and second terms. 678 Sum of three first terms. 444 Remainder, equal the sum of the first and second terms. 123+321+234-234-123+321=444 If from the sum of the two first terms, we subtract the second term, the remainder will be the same as the first term. 444 Sum of the two first terms. 123 Remainder equal to the first term. Let the student apply paragraphs 22, and 25 to 28 to the above operations. EXAMPLES FOR PRACTICE. 41. From the sum of each of the following questions take the sum of the two last terms, and the remainder will be equal to the sum of the two first terms. |