to the 2 of the second, making 9; then the 2 tens of the first with the 3 of the second, making 5 tens. The two results, taken together, form a total of 5 tens and 9 units, or 59, which is the sum of the numbers proposed. What is here said applies to all numbers, however large, that are to be added together; but it is necessary to observe, that the partial sums, resulting from the addition of two numbers, each expressed by a single figure, often contain tens, or units of the next higher collection, and these ought consequently to be joined to their proper collection. In the addition of the numbers 49 and 78, the sum of the units 9 and 8 is 17, of which we should reserve 10, or ten, to be added to the sum of the tens in the given numbers; next we say that 4 and 7 make 11, and joining to this the ten we reserved, we have 12 for the number of tens contained in the sum of the given numbers; which sum, therefore, contains 1 hundred, 2 tens and 7 units, that is, 127. 11. By proceeding on these principles, a method has been devised of placing numbers, that are to be added, which facilitates the uniting of their collections of units, and a rule has been formed, which the following example will illustrate. Let the numbers be 527, 2519, 9812, 73 and 8; in order to add them together, we begin by writing them under each other, placing the units of the same order in the same column; then we draw a line to separate them from the result, which is to be written underneath it. 527 2519 9812 73 8 Sum 12939 We at first find the sum of the numbers contained in the column of units to be 29, we write down only the nine units, and reserve the 2 tens, to be joined to those which are contained in the next column, which, thus increased, contains 13 units of its own order; we write down here only the three units, and carry the ten to the next column. Proceeding with this column as with the others, we find its sum to be 19; we write down the 9 units and carry the ten to the next column, the sum of which we then find to be 12; we write down the 2 units under this column and place the ten on the left of it; that is, we write down the sum of this column, as it is found. By this means we obtain 12939 for the sum of the given numbers. 12. The rule for performing this operation may be given thus, Write the numbers to be added under each other, so that all the units of the same kind may stand in the same column, and draw a line under them. Beginning at the right, add up successively the numbers in each column; if the sum does not exceed 9, write it beneath its column, as it is found; if it contains one or more tens, carry them to the next column ; lastly, under the last column write the whole of its sumt. Examples for practice. Add together 8635, 2194, 7421, 5063, 2196 and 1225. Add together 84371, 6250, 10, 3842 and 631. Add together 66947, 46742 and 132684. SUBTRACTION. 13. AFTER having learned to compose a number by the addition of several others, the first question, that presents itself, is, how to take one number from another 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 + The best method of proving addition is by means of subtraction. The learner may, however, in general, satisfy himself of the correctness of his work by beginning at the top of each column and adding down, or by separating the upper line of figures and adding up the rest and then adding this sum to the upper line. number 9, and we wish to take 4 from it, we should, by doing this, separate it into two parts, which by addition would be the same again. To take one number from another, when they are not large, it is necessary to pursue a course opposite to that prescribed in the beginning of article 10, for finding their sum; that is, in the series of the names of numbers, we ought to begin from the greatest of the numbers in question, and descend as many places as there are units in the smallest, and we shall come to the name given to the difference required. Thus, in descending four places below the number nine we come to five, which expresses the number that must be added to 4 to make 9, or which shows how much 9 is greater than 4. In this last point of view, 5 is the excess of 9 above 4. If we only wished to show the inequality of the numbers 9 and 4, without fixing our attention on the order of their values, we should say that their difference was 5. Lastly, if we were to go through the operation of taking 4 from 9, we should say that the remainder is 5. Thus we see that, although the words, excess, remainder, and difference, are synonymous, each answers to a particular manner of considering the separation of the number 9 into the parts 4 and 5, which operation is always designated by the name subtraction. 14. When the numbers are large, the subtraction is performed, part at a time, by taking successively from the units of each order in the greatest number, the corresponding units in the least. That this may be done conveniently, the numbers are placed as 9587 and 345 in the following example; and under each column is placed the excess of the upper number, in that column, over the lower, thus ; 5, taken from 7, leaves 2, 4, taken from 8, leaves 4, 3, taken from 5, leaves 2, and writing afterwards the figure 9, from which there is noth ing to be taken; the remainder, 9242, shows how much 9587 is greater than 345. That the process here pursued gives a true result is indisputable, because in taking from the greatest of the two numbers all the parts of the least, we evidently take from it the whole of the least. 15. The application of this process requires particular attention, when some of the orders of units in the upper number are greater than the corresponding orders in the lower. If, for instance, 397 is to be taken from 524. In performing this question we cannot at first take the units in the lower number from those in the upper; but the number 524, here represented by 4 units, 2 tens and 5 hundreds, can be expressed in a different manner by decomposing some of its collections of units, and uniting a part with the units of a lower order. Instead of the 2 tens and 4 units which terminate it, we can substitute in our minds 1 ten and 14 units, then taking from these units the 7 of the lower number, we get the remainder 7. By this decomposition, the upper number now has but one ten, from which we cannot take the 9 of the lower number, but from the 5 hundred of the upper number we can take 1, to join with the ten that is left, and we shall then have 4 hundreds and 11 tens, taking from these tens the tens of the lower number, 2 will remain. Lastly, taking from the 4 hundreds, that are left in the upper number, the three hundreds of the lower, we obtain the remainder 1, and thus get 127 as the result of the operation. This manner of working consists, as we see, in borrowing, from the next higher order, an unit, and joining it according to its value to those of the order, on which we are employed, observing to count the upper figure of the order from which it was borrowed one unit less, when we shall have come to it. 16. When any orders of units are wanting in the upper number, that is, when there are ciphers between its figures, it is necessary to go to the first figure on the left, to borrow the 10 that is wanted. See an example 7002 3495 Remainder 3507. 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 decompounded into 6990 and 12; taking from 12 the 5 units of the lower number, we obtain 7 for the units of the remainder. This first operation has left in the upper number 6990 units or 699 tens instead of the 700, expressed by the three last figures on the left; thus the places of the two ciphers are occupied by 9s, and the significant figure on the left is diminished by unity. Continuing the subtraction in the other columns in the same manner, no difficulty occurs, and we find the remainder, as put down in the example. 17. Recapitulating the remarks made in the two preceding articles, the rule to be observed in performing subtraction may be given thus. Place the less number under the greater, so that their units of the same order may be in the same column, and draw a line under them; beginning at the right, take successively each figure of the lower number from the one in the same column of the upper; if this cannot be done, increase the upper figure by ten units, counting the next significant figure, in the upper number, less by unity, and if ciphers come between, regard them as 9s. 18. 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. In the first of the following examples, after having taken 6 units from 14, we count the next figure of the lower number 8, as 9, and so in the others. |