In arranging the numbers which we want to add, it is obviously needful that the digits of the corresponding columns of each number should be disposed in lines exactly under each other: and the scholar ought to be exercised in the due arrangement of the numbers for himself, and not have them given him arranged by the teacher. The sign, interposed between two numbers, denotes that the numbers are to be added: this sign is technically called plus, from the Latin word signifying more. Thus, 23+5 (read 23 plus 5) denotes the sum of 23 and 5. To add Whole Numbers. 9468 17. RULE.-Place the numbers so that units may stand under units, tens under tens, &c. Find the sum of the column of units, set down the last figure of it below that column, and carry to the next the number expressed by the remaining figure or figures, if there be any. Proceed as before with the remaining columns, and at the last column set down the entire sum. Thus, to add together 9468, 2956, and 79, let them be set as in the margin: then, 9 and 6 are 15 and 8 are 23, set down 3 and carry 2 to the column of tens. Then, 2 and 7 are 9 and 5 are 14 and 6 are 20, set down a cipher and carry 2; again, 2 and 9 are 11 and 4 are 15, set down 5 and carry 1; finally, 1 and 2 are 3 and 9 are 12, set down 12; and the sum or answer required is 12503; that is, twelve thousand five hundred and three. Methods of Proof. 2956 79 12503, sum. 1. Add the several columns, according to the rule, commencing at the top and proceeding downward, and if the result be the same as was obtained by adding them upward, it may be presumed that the work is right. 2. Separate the given numbers into two or more divisions. Find the sum of these divisions severally, and add these partial sums together. if the last result be equal to that found by the common method, the work is right. This will appear obvious from the following example: C This method may also be employed with advantage in finding the sums of large columns, instead of adding the whole at a single operation. The first method is very convenient and useful, when the columns are not very large. 3. Commencing at the left hand, add the several columns, without carrying, and set down the full sum of each column, with the units in their proper place, and the tens below the figure immediately to the left. Add together the two lines thus resulting, and if the last result agree with that obtained by the common method, it may be concluded that both are right. Thus, in the annexed example, the sum of the left hand column is 25, which is set down in full: the sum in the next column is 30; the cipher is set in its proper place, and 3 under the 5; and so with the rest. The sum of the two lines thus obtained is equal to the sum found by the ordinary method. *** 5946 9738 2697 9868 28249, sum. 25029 322, 28249, proof. This method of addition might be used instead of the common method; and as it requires nothing to be carried, it might be employed with advantage when the calculator is liable to interruptions. *Addition may also be proved by casting the nines out of each of the given numbers, as will be explained in multiplication; and by casting the nines out of the sum of the excesses, and out of the sum of the numbers. If these latter excesses be equal, the work is generally right; otherwise, it must be wrong. This method, however, is of little use in proving addition. 16. Required the sum of +756+879109. 37545+23014+756+830 17. Required the sum of 120350+819+365+7569 +1827+10734. 18. Required the sum of 12756+10019+756897+ 13569+8769. 19. Required the sum of 976+3456+897436+ 123456+897030+1978394+7568934. 20. Add together 376454, 1276003, 756000, 376976, 701000134, 37564, and 75603. 21. Add together seven thousand nine hundred and seventy-six, eight thousand six hundred and twenty-four, one million one thousand and thirty, three hundred and sixtyfive, one thousand seven hundred and twenty-seven. 22. The following are the reigns of the Saxon kings of England, from the union of the kingdoms of the Heptarchy under Egbert, to the Danish conquest. 23. Egbert began to reign in 823; when did Alfred's reign commence ? Ans. 872. 24. When did Edwy commence his reign ? Ans. 955. 25. The world was created 4001 years before Christ; how long is it since ? Ans. 5831. 26. Required the aggregate population of the following states: That of Maine, in 1820, 297839 New-Hampshire 244155 Vermont 235764 30. Sir Isaac Newton, the celebrated mathematician, was born in the year 1642, and died in his eighty-fifth year. In what year did he die ? Ans. 31. George Washington was unanimously elected President of the United States in 1789, and presided 8 years; John Adams presided 4 years; Thomas Jefferson 8 years; James Madison 8 years; James Monroe 8 years. year was the last President's time expired? In whate Ans. 32. In 1821, the population of the following towns in |
