What is the sum of the following numbers: 0 + 1 0 +1 +2 0+ 1+ 2+ 3 1+ 2+ 3+ 4 1 + 2 + 3 + 4 + 5 1 + 2 + 3 + 4 + 5 + 6 1 + 2 + 3 + 4 + 5 + 6 + 7 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9. Find the sum of the above numbers, adding from the right. Q. What sign is used to indicate that two numbers are to be added together? What is it called? What does it denote? How would you express that 2 and 3 are to be added together? How do you read 2+3? What is the sign of equality? How used? What does it signify? Give an example. What is 2+3 equal to? 2+4? 2+9? 3+1 3+7? 3+9? 4+3? 4+5? 5+7? 5+9? 8+8? 9+9? 14. Let it now be required to add the numbers 8423 and 1234. 8423 1234 9657 Place the numbers, one under the other, so that units shall be under units, tens under tens, &c. Commencing on the right, we say 4 and 3 are 7, 3 and 2 are 5, 2 and 4 are 6, 1 and 8 are nine, writing down each separate sum, 7, 5, 6, 9, under the numbers which produce it; that is, since 4 units and 3 units make 7 units, 7 must be in the units' place, 3 tens and 2 tens make 5 tens, 5 must be in the tens' place, &c. Hence 9657 is the sum required. 15. Add the numbers 4897, 821, 1634, together. Having set down the numbers as before, 821 1634 7352=sum. we say, 4 and 1 are 5, and 7 are 12; now, 4897 since 12 units contain 1 ten and 2 units, we only set down the 2 units and carry the 1 ten into the column of tens, and say, 1 we carry, and 3 are 4, and 2 are 6, and 9 are 15; but 15 tens contain 1 hundred and 5 tens, we set down 5 in the column of tens and carry 1 to the column of hundreds, and say, 1 we carry and 6 are 7, and 8 are 15, and 8 are 23, and setting down the 3 hundreds, we carry 2 thousands into the next column and proceed, 2 and 1 are 3, and 4 are 7. Hence 7352 is the required sum. 3909 1094 16. Add 3909, 1094, 302, and 20, together. These numbers being set down as before, we have 0 and 2 are 2, and 4 are 6, and 9 are 15, set down 5 and carry 1; 1 we carry and 2 are 3, and 0 are 3, and 9 are 12, and 0 are 12, set down 2 and carry 1; 1 and 3 are 4, and 0 are 4, and 9 are 13, set down 3 and carry 1; 1 and 1 are 2, and 3 are 5. 5325 is the required sum. 302 20 5325=sum. 987641 407180 17. Add 987641 and 407180 together. Here the numbers which are carried in each column are indicated by the small figures placed under the columns; 1 is carried into the column of hundreds and 1394821=sum. 1 into the column of tens of thousands. 1 1 We see by the above examples that if the sum in any one column does not exceed 9, we set down the whole sum under this column; but if it exceed 9, we set down the right hand figure alone, and carry the remaining figure or figures into the next column. From these explanations we deduce the following RULE. I. Set down the numbers, one under the other, so that units fall under units, tens under tens, &c., and draw a line beneath them. II. Beginning at the bottom of the right hand column, add up the figures in each column. If the sum in any one column does not exceed 9, write down the whole sum beneath this column; if it exceed 9, write down the righthand figure and carry the remaining figure or figures into the next column. III. Finally, under the last column write the entire sum of this column. Q. How are numbers set down for addition? Where do you begin to add? If the sum of the figures in any column do not exceed 9, what is set down? If it exceed 9? If the sum in a column be 19, what is set down and what carried? If it be 29? 30? 50? 72? 99? 100? What is set under the last column ? PROOF OF ADDITION. 18. Addition is proved in several ways. 1st Proof. Begin the addition at the top of the right-hand column, and add each column downwards. 2d Proof. Draw a line under the upper number, add the remaining numbers together, and then add their sum to the upper number. 3d Proof. Write the numbers down in a different order from that in which they appear in the example, and then add as the rule directs. If the example be worked correctly, it will have the same result as that given by the proof, Add 3024, 721, and 320 together. In the second proof, the numbers 721 and 320 are first added together, and to their sum, 1041, the first number is added. The total sum, 4065, is the same as shown by the example. By the third proof, the number 721 is placed first, 320 next, and then 3024, and the addition gives the same result. Q. How many ways of proving addition? What is the first proof? Second proof? Third proof? When is the example worked correctly? EXAMPLES. 1. Add 3094, 5211, 6010, and 4372 together. Ans. 18687. 2. Add 1001, 298, 9000, and 32 together. Ans. 10331. 3. Add 11040, 3279, 907, 85, and 3 together. Ans. 15314. 4. Add 3904, 27520, and 1089753 together. Ans. 1121177. APPLICATIONS. 1. There are 12 calendar months in the year. January has 31 days, February 28, March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30, December 31. How many days in the year? Ans. 365 days. Q. How many months in the year? What are they? How many days in each? In the year? 2. How many inhabitants were there in the 15 States by the first census, taken in 1790; and what was the total population of the United States, the number of free persons and slaves in each State being as follows: States. 1. Virginia.... Slaves. Total in each No. of Free No. of Ans. 454,881+293,427=Ans. 2. Massachusetts .... 475,257+000,000 3. Pennsylvania..... 430,636+ 3,737= Ans. 3,199,556 +694,480=3,894,036, Total in the United States. If the total population in each State be correctly determined, the sum of these results will give the total population in the United States. Q. When was the first census taken? How many States at the time? Which was the largest in population? Which the smallest? Which had the most slaves? Next? Which the least? What was the population of the United States in 1790? How many free? Slaves? 3. The United States have had ten Presidents:- -1. General Washington, of Virginia, who served 8 years; 2. John Adams, of Massachusetts, who served 4 years; 3. Thomas Jefferson, of Virginia, 8 years; 4. James Madison, of Virginia, 8 years; 5. James Monroe, of Virginia, 8 years; 6. John Quincy Adams, of Massachusetts, 4 years; 7. Andrew Jackson, of Tennessee, 8 years; 8. Martin Van Buren, of New York, 4 years; 9. William Henry Harrison, of Ohio, who served 1 month, and died in office; and, 10. John Tyler, of Virginia, who succeeded General Harrison, being at the time Vice-President. How many years since the establishment of the United States government to the commencement of General Harrison's term? Ans. 52 years. Q. How many Presidents have the United States had? Who were they? To what States did they belong? How many years did each serve? Who succeeded Van Buren? Did he live out his term? Who succeeded him on his death? What office did Tyler hold before he became President? 4. The public lands of the United States have been obtained by purchase and cession. What is the total amount that has been purchased and ceded?—the number of acres purchased, and the number ceded, being as follows: Ceded by Virginia, New York, Massachu setts, and Connecticut. 169,609,819 Q. How were the public lands of the United States obtained? What States ceded lands? From whom were some of the public lands purchased? 5. What is the total population in the five largest cities in the United States, the population in each being as follows:New York, 312,710; Philadelphia, 258,037; Baltimore, 102,313; New Orleans, 102,193; Boston, 93,383 ? Ans. 868,636. Q. Which are the five most populous cities in the United States? Which is the largest city? How many inhabitants in each, in round numbers? 6. What was the total expense of the United States government for the year 1842, that of each department being as follows:-Civil list, 6,865,451 dollars; Military establish |