A number applied to no kind of thing, as 5, 10, 18, 36, is called an abstract number. A number applied to some kind of thing, as 7 horses, 25 dollars, 250 men, is called a denominate number. The unit, or unit value of a number, is one of the kind which the number expresses; thus, the unit of 99 days is 1 day; the unit of 7 dollars is 1 dollar; the unit of 15 acres is 1 acre. In like manner the unit of 9 tens may be said to be 1 ten; the unit of 8 hundred to be 1 hundred; the unit of 6 thousand to be 1 thousand, &c. ADDITION OF SIMPLE NUMBERS. T11. 1. James had 5 peaches, his mother gave him 3 more; how many had he then? Ans. 8. Why? Ans. Because 5 and 3 are 8. 2. Henry, in one week, got 17 merit marks for perfect lessons, and 6 for good behavior; how many merit marks did he get? Ans. Why? 3. Peter bought a wagon for 36 cents, and sold it so as to gain 9 cents; how many cents did he get for it? 4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left; how many walnuts had he at first? 5. A man bought a chaise for 54 dollars; he expended 8 dollars in repairs, and then sold it so as to gain 5 dollars; how many dollars did he get for the chaise ? The putting together of two or more numbers, (as in the foregoing examples,) so as to make one whole number, is called Addition, and the whole number is called the Sum, or Amount. 6. One man owes me 5 dollars, another owes me 6 dollars, another 8 dollars, another 14 dollars, and another 3 dollars; what is the sum due to me? 7. What is the amount of 4, 3, 7, 2, 8, and 9 dollars? 8. In a certain school, 9 study grammar, 15 study arithmetic, 20 attend to writing, and 12 study geography; what is the whole number of scholars? Questions.-11. What is addition? What is the answer, or number sought, called? What is the sign of addition? What does it show? How is it sometimes read? Whence the word plus, and what is its signification? What is the sign of equality, and what does it show? SIGNS.A cross, +, one line horizontal and the other perpendicular, is the sign of Addition. It shows that numbers with this sign between them are to be added together; thus, 4+7+14+16 denote that 4, 7, 14, and 16 are to be added together. It is sometimes read plus, which is a Latin word signifying more. Two parallel, horizontal lines,=, are the sign of Equality. It signifies that the number before it is equal to the number after it; thus, 5+3=8 is read 5 and 3 are 8; or, 5 plus 3 are equal to 8. In this manner let the pupil be instructed to commit the 3+0+9+5= = how many? = 1 2 8 9 9+2+6+4+5: how many? 6 = T12. When the numbers to be added are small, the addition is readily performed in the mind, and this is called mental arithmetic; but it will frequently be more convenient, and even necessary, when the numbers are large, to write them down before adding them, and this is called written arithmetic. 1. Harry had 43 cents, his father gave him 25 cents more; how many cents had he then? SOLUTION.-One of these numbers contains 4 tens and 3 units. The other number contains 2 tens and 5 units. To unite these two numbers together into one, write them down one under the other, placing the units of one number directly under units of the other, and the tens of one number directly under tens of the other, and draw a line underneath. 43 cents. 25 cents. 43 cents. 25 cents. 8 43 cents. 25 cents. Ans. 68 cents. Beginning at the column of units, we add each column separately; thus, 5 units and 3 units are 8 units, which we set down in units' place.. We then proceed to the column of tens, and say, 2 tens and 4 tens are 6 tens, or 60, which we set down directly under the column in tens' place, and the work is done. It now appears that Harry's whole number of cents is 6 tens and 8 units, or 68 cents; that is, 43 +25=68. Units are written under units, tens under tens, &c.; because none but figures of the same unit value can be added to each other; for 5 units and 3 tens will make neither S tens nor 8 units, just as 5 cows and 3 sheep will make neither 8 cows nor 8 sheep. Questions. T 12. What distinction do you make between mental and written arithmetic? How do you write numbers for addition? Where do you begin to add? and where do you set the amount? How do you proceed? Why do you write units under units, tens under tens, &c. ? 2. A farmer bought a chaise for 210 dollars, a horse for 70 dollars, and a saddle for 9 dollars; what was the whole amount? Write the numbers as before directed, with units under units, tens under tens, &c. OPERATION. Chaise, 210 dollars. Horse, 70 dollars. Saddle, 9 dollars. Answer, 289 dollars. Add as before. The units will be 9, the tens 8, and the hundreds 2; that is, 210+ 70+9=289. After the same manner are performed the following examples, in which the amount of no column exceeds nine. 3. A man had 15 sheep in one pasture, 20 in another pasture, and 143 in another; how many sheep had he in the three pastures? 15 +20+143 how many? 4. A man has three farms, one containing 500 acres, another 213 acres, and another 76 acres; how many acres in the three farms? 500+213 +76 how many? 5. Bought a farm for 2316 dollars, and afterwards sold it so as to gain 550 dollars; what did I sell the farm for? 2316 +550 how many? 6. A chair-maker sold, in one week, 30 Windsor chairs, 36 cottage, 102 fancy, and 21 Grecian chairs; how many chairs did he sell? 30+36 +102+21= how many ? 7. A farmer, after selling 500 bushels of wheat to a commission merchant, 320 to a miller, and sowing 117 bushels, found he had 62 bushels left; how many bushels had he at first? 500+320117+62 how many? 8. A dairyman carried to market at one time 231 pounds of butter, at another time 124, at another 302, at another 20, and at another 12; how many pounds did he carry in all? Ans. 689 pounds. 9. A box contains 115 arithmetics, 240 grammars, 311 geographies, 200 reading books, and 133 spelling books; how many books are there in the box? Ans. 999. ¶ 13. Hitherto the amount of any one column, when added up, has not exceeded 9, and consequently has been expressed by a single figure. But it will frequently happen that the amount of a single column will exceed 9, requiring two or more figures to express it. 1. There are three bags of money. The first contains 876 dollars, the second 653 dollars, the third 426 dollars; what is the amount contained in all the bags ? OPERATION. First bag, 876 dollars. 66 SOLUTION. Writing the numbers as already described, we add the units, and find them to be 15, equal to 5 units, which we write in units' place, adding the 1 ten with the tens; which being added together are 15 tens, equal to 5 tens, to be written in tens' place, and 1 hundred, to be added to the hundreds. The hundreds being added are 19, equal to 9 hundreds, to be written in hundreds' place, and 1 thousand, to be written in thousands' place. 1955 66 Ans. 1955 dollars. PROOF.-We may reverse the order, and, beginning at the top, add the figures downwards. If the two results are alike, the work may be supposed to be right, for it is not likely that the same mistake will be made twice, when the figures are added in a different order. NOTE.- Proof by the excess of nines. If the work be right, there will be just as many of any small number, as 9, with the same remainder, in the amount, as in the several numbers taken together. Hence, OPERATION. 876 3 653 5 426 3 In the upper number, 8 (hundreds) is 8 more than a certain number of nines, (5) 7 (tens) is 7 more. Adding the 8 and 7, and the 6 units together, the sum is 212 nines and 3 remainder, which we set down at the right hand, as the excess of nines in this number. In the same manner, 5 is found to be the excess of nines 1955 2 in the second number, and 3 in the third number. These everal excesses being added together, make 1 nine and an excess of 2, which is the same as the excess of nines in the general amount, found in the same manner. This method will detect every mistake, except it be 9, or an exact number of nines. To find what will be the excess after casting the nines out of any number, begin at the left hand, and add together the figures which express the number; thus, to cast the nines out of 892, we say 8 (passing over 9) +2 (dropping 9 from the sum) = 1. From the examples and illustrations now given, we derive the following RULE. I. Write the numbers to be added, one under another, plac Questions. 13. If the amount of the column does not exceed 9, what do you do? What when it exceeds 9? How do you add each column? What do you do with the amount of the left column? For what number do you carry? If the amount of a column be 36, what would you set down, and how many would you carry? On what prin ciple do you do this? How is addition proved? Why? Repeat the rule for addition. |