§ 11. John has three apples and Charles has two; how many apples have they between them. Every boy will answer, five. Here a single apple is the unit, and the number five contains as many units as the two numbers three and two. The operation by which this result is obtained is called addition. Hence, ADDITION is the uniting together of several numbers, in such a way, that all the units which they contain may be expressed by a single number. Such single number is called the sum or sum total of the other numbers. Thus, 5 is the sum of the apples possessed by John and Charles. What is the sum of 2 and 4? of 3 and 5? of 6 and 3? of 4, 3 and 1? of 2, 3 and 4? of 1, 2, 3 and 4? of 5 and 7? How many units in 4 and 6? How many units in 9 and 4? Q. What is addition? What is the single number called which expresses all the units of the numbers added? What is the sum of 2 and 4? What is six called? OF THE SIGNS. § 12. The sign+, is called plus, which signifies more. When placed between two numbers it denotes that they are to be added together. Thus, 3+2 denotes that 3 and 2 are to be added together. The sign, is called the sign of equality. When placed between two numbers it denotes that they are equal to each other. Thus, 3+2=5. When the numbers are small we generally read them, by saying, 3 and 2 are 5. Q. What is the sign of addition? What is it called? What does it signify? When placed between two numbers what does it express? Express the sign of equality. When placed between two numbers what does it show? Give an example. § 13. Before adding large numbers the pupil should. be able to add, in his mind, any two of the ten figures. Let him commit to memory the following table, which is read, two and O are two; two and one are three; two and two are four, &c. 6+4=10 7+4=11 8+4=12 9+4=13 6+5=11 7+5=12 8+5=13 9+5=14 6+6=12 7+6=13 8+6=14 9+6=15 6+7=13 7+7=14 8+7=15 9+7=16 6+8=14 7+8=15 8+8=16 9+8=17 6+9=15 7+9=16 8+9=17 9+9=18 Ans. 33 Ans. 89. Ans. 49. Ans. 100. 1. What is the sum of 3 and 3 tens? 2. What is the sum of 8 tens and 9? 3. What is the sum of 4, 5, and 4 tens? 4. What is the sum of 1, 2, 3, 4, and 9 tens? 5. What is the sum of 1, 2, 3, 4, 5, and 6 tens? 6. What is the sum of 1, 4, 9, and 5 tens? 7. What is the sum of 4, 8, 3, and 7 tens? 8. If a top costs 6 cents, a knife 25 cents, a slate pencil 1 cent, and a slate 12 cents, what does the whole amount to? Ans. 85. Ans. 44 cts. 9. John gives 30 cents for a bunch of quills, 18 cents for an inkstand, and 25 cents for a quire of paper, what did they all cost him? 10. Add together the numbers 894 and 637. Ans. 73 cts. OPERATION. 894 637 11 12 14 1531 In this example, the sum of the units is 11, which cannot be expressed by a single figure. But 11 units are equal to 1 ten and 1 unit; therefore, we set down 1 in the place of units, and 1 in the place of tens. The sum of the tens is 12. But 12 tens are equal to 1 hundred, and 2 tens; so that 1 is set down in the hundred's place, and 2 in the ten's place. The sum of the hundreds is 14. The 14 hundreds are equal to 1 thousand, and 4 hundreds ; so that 4 is set down in the place of hundreds, and 1 in the place of thousands. The sum of these numbers, 1531, is the sum sought. thus: OPERATION. 894 637 11 The example may be done in another way, Having set down the numbers, as before, we say, 7 and 4 are 11: we set down 1 in the units place, and write the 1 ten under the 3 in the column of tens. We then say, 1 to 3 is four, and 9 are 13. We set down the three in the tens place, and write the 1 hundred under the 6 in the column of hundreds. then add the 1, 6, and 8 together, for the hundreds, and find the entire sum 1531, as before. 1531 We When the sum in any one of the columns exceeds 10, or an exact number of tens, the excess must be written down, and a number equal to the number of tens, added to the next left hand column. This is called carrying to the next column. The number to be carried may be written under the column or remembered and added in the mind. From these illustrations we deduce the following general RULE. 14. I. Set down the numbers to be added, units under units, tens under tens, hundreds under hundreds, &c., and draw a line beneath them. II. Begin at the foot of the unit's column, and add up the figures of that column. If the sum can be expressed by a single figure, write it beneath the line, in the unit's place. But if it cannot, see how many tens and how many units it contains. Write down the units in the unit's place, and carry as many to the bottom figure of the second column as there were tens in the sum. Add up that column: set down the sum and carry to the third column as before. III. Add each column in the same way, and set down the entire sum of the last column. Q. How do you set down the numbers for addition? Where do you begin to add? If the sum of the first column can be expressed by a single figure, what do you do with it? When it cannot what do you write down? What do you then add to the next column? When you add the tens to the next column, what is it called? What do you set down when you come to the last column? EXAMPLES. 1. What is the sum of the numbers 375, 6321 and 598. In this example, the small figure placed under the 4, shows how many are to be carried from the first column to the second, and the small figure under the 9, how many are to be carried from the second column to the third. OPERATION. 375 6321 598 7294 11 In like manner, in the examples below, the small figure under each column, shows how many are to be carried to the next column at the left. Beginners had better set down the numbers to be carried as in the examples. 15. Begin at the right hand figure of the upper line, and add all the columns downwards, carrying from one column to the other, as before. If the two results agree the work is supposed right. SECOND PROOF. Draw a line under the upper number. Add the lower numbers together, and then add their sum to the upper number. If the last sum is the same as the sum total, first found, the work may be regarded as right. Q. What do the small figures under the columns denote? How do you prove addition by the first method? How do you prove addition by the second method? |