SECTION IV. SIMPLE SUBTRACTION. Simple Subtraction teaches to take one number from another of the same kind, that is greater; or, which amounts to the same thing, to separate this last into two parts, one of which shall be the given number. If, for instance, we have the number 9 and wish to take 4 from it, we shall, in so doing, separate it into two parts, which, by addition, would be the same again. To take one number from another when they are small, it is necessary to pursue a course opposite to that pursued in the addition of small numbers; that is, in the series of names of numbers, we ought to begin from the greater of the numbers in question, and descend as many places as there are units in the smaller, and we shall come to the name given to the difference required. Thus in descending four places below 9, we come to 5, which expresses the number that must be added to 4 to make 9; or which shows how much 9 is greater than 4. When numbers are large, the subtraction is performed part at a time, by taking successively from the units, tens, hundreds, &c. of the greater number, the corresponding units, tens, hundreds, &c. of the smaller number, as in this example :— (Lacroix.) From 587 take 345. 587 345 242 Ans. Five units taken from seven units leave 2; 4 tens from 8 tens leave 4; and 3 hundreds So the remain from 5 hundreds leave 2. der is 2 hundreds, 4 tens, and 2 units, or 242, which shows how much 587 is greater than 345. This process evidently gives a true result, because in taking from the greater of the two numbers all the parts of the smaller, we certainly take from it the whole of the smaller. When the units, tens, hundreds, &c. in the larger number are smaller than the corresponding figures in the smaller number, this process requires some modification. If, for instance, 397 is to be taken from 524 : 524 397 127 In performing this operation, we cannot at first take the units in the lower, or smaller number from the units in the upper or larger number; but we may do this, by taking one of the two tens in upper number and joining it with the 4 units; the we have taken 14 units, which is the same as 4 units and 1 ten. Then 7 units taken from 14 leave 7. We have now left in the upper number but 1 ten, from which we cannot subtract 9, but we may take 1 hundred from the 5 and join it with the 1 ten; we shall then have 4 hundreds and 11 tens; taking from these tens the 9 tens in the lower number, and 2 remain. Remaining or remainder is a term frequently used, and it requires some explanation. Take for illustration a sum and separate its parts, or decompose it. From 125 Take 69 56 We cannot take 9 units from 5 units, and of course it becomes necessary to take one of the tens from the place of tens and reduce it to units, and add to the 5 which will increase it to 15, from which 9 can be taken and 6 remain. The 1 hundred in the next place borrowed and reduced to tens, will make, with the one that remains, eleven tens, from which 6 tens can be taken, and 5 tens remain. Thus 125 is equal to 11 tens and 15 units; 56 There remain 5 tens and 6 units. When there are ciphers between the figures of the larger number, it is necessary to go to the first figure on the left to borrow the ten that is wanted. See an example : 7002 3495 As we cannot take the 5 units of the lower number from the 2 of the upper, we borrow 10 units from the 7000, denoted by the figure 7, which leaves 6990; joining the 10 we borrowed to the figure 2, the upper number is now decomposed into 6990 and 12; as may be seen in this manner. In like manner decompose the lower number. 6000+900+90+12=7002 3000+400+90+ 5=3495 3000+500+00+7=3507 Taking the 5 (units) of the lower number, we obtain 7 for the units of the remainder. Then take 9 (tens) from 9 (tens) and nothing remains; 4 (hundreds) from 9, (hundreds) and 5 (hundreds) remain; 3 (thousands) from 6, (thousands) and 3 (thousands) remain. Then the whole remainder is 3 thousand 5 hundred, 0 tens, and 7 units, or 3507. (Lacroix.) EXAMPLES. 1. What is the difference between the numbers 7005 and 467? 7005 6538 Ans. 2. America was discovered by Columbus in 1492; the first permanent settlement was made at Jamestown in 1607: how many years between these two events? 1607 115 Ans. 3. From 1832 take 1699. What remains? Ans. 133. 4. What number must be added to 425 to make 793 ? 793 425 368 Ans. 5. If 368 be taken from 793, how many will remain ? Ans. 425. 6. How much greater is the number 2741 than 1417? Ans. 1324, 7. From 1000000 take 999999. What remains? Ans. 1. Note. For greater convenience, when it is necessary to decrease the upper figure by unity, we can suffer it to retain its value, and add this unit to the corresponding lower figure, which thus increased gives as is wanted, a result one less than would arise from the written figures. (Lacroix.) Adding one to the lower figure is the same in effect as taking one from the upper. The foregoing explanations prepare the way for the following RULE. 1. Place the less number under the greater, so that units may stand under units, tens under tens, &c. and draw a line under them. 2. Beginning at the right, take each figure in the subtrahend from the figure over it, and set the remainder under the line. 7 3. If the lower figure be greater than the one over it, add ten to the upper figure, from which figure so increased, take the lower, and write the remainder, carrying one to the next figure in the lower line, and thus proceed till the whole is finished. METHOD OF PROOF. 1. Take the excess of 9's in the less from the excess of 9's in the greater; their difference will be equal to the excess of 9's in the remainder. But if the excess of 9's in the smaller number exceed that in the greater, add 9 to the excess in the greater, and proceed as before. Let A represent the number of nines in the minuend, and B the number of nines in the subtrahend; C the excess of nines in the minuend, and D the excess of nines in the subtrahend. A and B then being an even number of nines, the difference between them must of course be an even number of nines. All that remain of the given numbers are the two excesses C and D; and it is evident that the difference between these excesses will be equal to the excess of nines in the difference of the two given numbers; for, in fact, it is the same thing, as may be shown : A-C E F Let A, B, C, and D be the same as before; let E represent the number of nines in the remainder, and F the excess of nines in the remainder. Since A, B, and E, each represent an even number of nines, they may be rejected; then C will represent the remainder, D the subtrahend, and F the remainder, and F must be equal to itself. RULE. 2 Reject the 9's from the minuend, subtrahend, and re *There are other methods of proving subtraction. 1. The remainder added to the least number, exactly gives the greatest. (Lacroix.) 2. Add the remainder to the subtrahend, and if their sum be equal to the minuend, the work is right. (Thompson.) mainder; then subtract the excess of 9's in the remainder from the excess of 9's in the minuend, and the remainder will be equal to the excess of 9's in the subtrahend. If the excess of 9's in the remainder be greater than the excess of 9's in the minuend, add 9 to the latter and subtract as before. The constant application of this rule in the transactions of business, sufficiently exhibits its usefulness. SECTION V. SIMPLE MULTIPLICATION. Simple Multiplication teaches how to find the amount of any given number of one denomination, repcated a certain number of times. It is a compendious method of performing Addition. The number to be multiplied is called the Multiplicand. The number by which we multiply is called the Multiplier. The number found after the work is finished is called the Product. Both the Multiplicand and Multiplier are commonly called terms or factors. "When numbers to be added are equal to each other, addition takes the name of multiplication, because in this case the sum is composed of one of the numbers repeated as many times as there are numbers to be added." How much will 5 books come to at 25 cents for each? 25 25 25 25 25 125 This example is easily performed by addition: but if it had been required to find the price of 30, 50, or 100 books, the operation must have been very tedious, on account of the number of times, which the price of one book, 25 cents, must have been written down. In performing this example by adding the number 25, written 5 times, we find that the units |