same manner. The operation consists in taking a less number from a greater, and is called subtraction. Hence, 32. SUBTRACTION is the process of finding the difference between two numbers. The difference or answer to the question is called the remainder. OBS. 1. The number to be subtracted is sometimes called the subtrahend, and the number from which it is subtracted, the minuend. 2. Subtraction, it will be perceived, is the reverse of addition. Addition unites two or more numbers into one single number; subtraction, on the other hand, separates a number into two parts. 3. When the given numbers are of the same denomination, the operation is called Simple Subtraction. (Art. 18. Obs.) 33. Subtraction is often represented by a short horizontal line, which is called minus. When placed between two numbers, this sign shows that the number after it is to be subtracted from the one before it. Thus the expression 8-5, signifies that 5 is to be subtracted from 8; and is read, "8 minus 5," or "8 less 5." OBS. The term minus is a Latin word signifying less. EXERCISES FOR THE SLATE. 34. When we wish to find the difference between two small numbers, it is the most convenient way to perform the subtraction in the mind. But when the numbers are large, it is difficult to retain them in the mind, and carry on the operation at the same time. By setting them down upon a slate or black-board, however, the process of subtracting large numbers is rendered short and simple. (Art. 21.) QUEST.-31. What is the object aimed at in the preceding examples? In what does the operation consist? What is called? 32. What then is subtraction? What is the difference or answer called? Obs. What is the number to be subtracted sometimes called? That from which it is subtracted? Of what is subtraction the reverse? When the given numbers are of the same denomination, what is the operation called? 33. What is the sign of subtraction called? Of what does it consist? What does it show? How is the expression 9-4 read? Obs. What is the meaning of the term minus? 34. What is the most convenient way of finding the difference between two small numbers? What between two large ones? Ex. 1. Suppose a man gave 475 dollars for a span of horses, and 352 dollars for a carriage: how much more did he pay for his horses than for his carriage? Direction. Write the less number under the greater, so that units may stand under units, tens under tens, &c. Now, beginning with the units, proceed thus: 2 units from 5 units leave 3 units; write the 3 in units' place, under the figure subtract Operation. hund. tens Horses, 4 7 5 Dolls. ed. 5 tens from 7 tens leave 2 tens; set the 2 in tens' place, &c. 3 hundreds from 4 hundreds leave 1 hundred; write the 1 in hundreds' place, &c. The remainder is 123 dollars. OBS. It is important for the learner to observe, that we subtract units from units, tens from tens, &c.; that is, we subtract figures of the same order from each other. This is done for the same reason that we add figures of the same order to each other. (Art. 22.) Hence, in writing numbers for subtraction, great care should be taken to set units under units, &c., in order to prevent the mistake of subtracting different orders from each other. 2. A merchant bought 268 barrels of flour. On examination, he found that only 123 barrels were fit for use : how many were damaged? Ans. 145. Suggestion. Write the less number under the greater, &c., and proceed as before. 3. A traveler, having 576 dollars, was robbed of 344 dollars how many dollars had he left? : 4. What is the difference between 648 and 235 ? QUEST.-How do you write numbers for subtraction? Where begin to subtract? Obs. What orders do you subtract from each other? Why not subtract different orders from each other? Why place units under units, &c., in subtraction? 1 35. When the figures in the lower number are each of them smaller than those directly over them, each lower figure, as we have seen in the preceding examples, must be subtracted from that above it, and the remainder must be placed under the figure subtracted. But it often happens that a figure in the lower number is larger than that above it, and consequently cannot be taken from it. 13. It is required to find the difference between 75 and 48. Operation. 75 48 27 Rem. It is plain that we cannot take 8 units from 5 units, for 8 is larger than 5. What then shall we do? Since 75 is composed of 7 tens and 5 units, we can take 1 ten from the 7 tens, and adding it mentally to the 5 units, it will make 15 units. Then subtracting the 8 units from 15 units, will leave 7 units; write the 7 under the units' column. Now, as we took 1 ten from the 7 tens, we have but 6 tens left; and 4 tens from 6 tens leaves 2 tens: write the 2 under the tens' column. The whole remainder, therefore, is 2 tens and 7 units, or 27. 36. The process of taking one from a higher order in the upper number, and adding it to the figure from which the subtraction is to be made, is called borrowing ten, and is the reverse of carrying ten. (Art. 24.) The 1 taken from a higher order is always equal to 10 in the next lower order to which it is added. (Art. 8.) 37. The principle of borrowing may be illustrated by the following analytic solution of the last example. Thus, QUEST.-35. When the figures in the lower number are each smaller than those over them, how proceed? Where do you place the remainder? Is a figure in the lower number ever larger than that above it? 36. What is meant by borrowing 10? 75=60+15 Rem. 20+ 7, or 27. Taking 1 ten from 7 tens, and uniting it with the 5 units, we number. And we simply separate have 60 plus 15 for the upper the lower number into the tens and units of which it is composed. Now subtracting, as in the last article, 8 from 15 leaves 7: 40 from 60 leaves 20. Thus the remainder is 20+7, or 27, the same as before. OBS. It is manifest that this process of borrowing ten does not change the value of the upper number; for it consists simply in transposing a part of one order to another order in the same number, which can no more diminish or increase the number, than it will diminish or increase the amount of money a man has, if he takes a part from one pocket and puts it into another. It is advisable for the pupil to analyze several examples as above, until the process of borrowing becomes familiar. 14. From 6042 Take 2367 Since 7 units cannot be taken from 2 units, we borrow 10, which, added to the 2, will make 12: then 7 (units) Rem. 3675 from 12 (units) leave 5. Now, having borrowed 1 of the 4 tens, it becomes 3 tens; and 6 from 3 is impossible: hence we must borrow again. But the next figure in the upper number, i. e. the figure in the hundreds' place, is a 0, and consequently has nothing to lend. We must therefore borrow 1 from the next order still, i. e. from thousands, and adding it to the 0, it will make 10 hundreds. Now, borrowing 1 of the 10 hundreds and adding it to the 3 tens, it will make 13 tens, and 6 from 13 leaves 7. Now, diminishing the 10 hundreds by 1, (which we borrowed,) it becomes 9, and 3 from 9 leaves 6. Again, diminishing the 6 thousands by 1, (which we borrowed,) it becomes 5, and 2 from 5 leaves 3. The answer is 3675. 37. a. There is another method of borrowing, or rather of paying, which the learner will often find more conven QUEST. What is the 1 taken from the higher order equal to? How illustrate the principle of borrowing upon the black-board? Is the value of the upper number increased by borrowing? Is it diminished? How does this appear? 37. a. When we borrow 10, what other way is there to compensate for it? ient in practice than the preceding, and less liable to lead him into mistakes. When we borrow 10, that is, when we add 10 to the upper figure, instead of considering the next figure in the upper number to be diminished by 1, the result will manifestly be the same, if we simply add 1 to the next figure in the lower number. Thus in the last example, instead of diminishing the 4 tens in the upper number by 1, we may add 1 to the 6 tens in the lower number, which will make 7; and 7 from 14 leaves 7, the same as 6 from 13. Again, adding 1 to the 3 hundreds (to compensate for the 10 we borrowed) makes 4 hundreds; and 4 from 10 leaves 6, the same as 3 from 9. Finally, adding 1 to the 2 (because we borrowed) makes 3; and 3 from 6 leaves 3. The remainder is 3675, the same as before. 15. From 574 Take 326 Rem. 248 6 from 4 is impossible: add 10 to the 4, and it will make 14; then 6 from 14 leaves 8. Adding 1 to the 2 makes 3, and 3 from 7 leaves 4. 3 from 5 leaves 2. Ans. 248. OBS. This method of borrowing depends on the self-evident principle, that if any two numbers are equally increased, their difference will not be altered. That the two given numbers are equally increased by this process, is evident from the fact that the 1 added to the lower number is of the next superior order to the 10 added to the upper number, and will compensate for it; for 1 in a superior order is equal to 10 in an inferior order. (Art. 8.) Hence, 38. When a figure in the lower number is larger than that above it, borrow 10, i. e. add 10 to the upper figure, and from the number thus produced, subtract the lower figure to compensate this, add 1 to the next figure in the lower number; or, diminish the next figure in the upper number by 1, and proceed as before. QUEST. Obs. Upon what does this method of borrowing depend? How does it appear that you increase the given numbers equally? |