ADDITION. 23. Addition is uniting two or more numbers in one. The Sum or Amount is the number found by addition. Thus, 5 added to 7 are 12; twelve, the number obtained, is the sum or amount. NOTES.-I. The sum or amount contains as many units as the numbers added. For, the numbers added are composed of units; and the whole is equal to the sum or all its parts. (Art. 3.) 2. When the numbers added are the same denomination, the operation is called Simple Addition. SIGNS. 24. Signs are characters used to indicate the relation of numbers, and operations to be performed. 25. The Sign of Addition is a perpendicular cross called plus (+), placed before the number to be added. Thus 7 +5, means that 5 is to be added to 7, and is read "7 plus 5." NOTE. The term plus, Latin, signifies more, or added to. 26. The Sign of Equality is two short parallel lines (=), placed between the numbers compared. Thus 7+5=12, means that 7 and 5 are equal to 12, and is read, "7 plus 5 equal 12," or the sum of 7 plus 5 equals 12. Read the following numbers: 66 23. What is addition? The result called? Note. When the numbers adde are the same denomination, what is the operation called? 24. What are signs? 25. The sign of addition? Note. The meaning of plus? 26. Sign of equality? How is 7+5=12 read? 27. The Sign of Dollars is a capital S with two perpendicular marks across it ($), prefixed to the number of dollars to be expressed. Thus, $245 means 245 dollars. NOTE.-The term prefix, from the Latin prefigo, signifies to place More mistakes are made in adding than in any other arithmetical operation. The first five digits are easily combined; the results of adding 9, being 1 less than if 10 were added, are also easy. The others. 6, 7, 8, are more difficult, and therefore should receive special attention. 27. What is the sign of dollars? CASE I. 28. To find the Amount of two or more numbers, when the Sum of each column is Less than 10. Ex. 1. A man owns 3 farms; one contains 223 acres, another 51 acres, and the other 312 acres: how many acres has he? OPERATION. hunds. teng. units. 223 ANALYSIS.-Let the numbers be set down as in the margin. Beginning at the right, we proceed thus: 2 units and I unit are 3 units, and 3 are 6 units; the sum being less than ten units, we set it under the column of units, because it is units. Next, I ten and 5 tens are 6 tens, and 2 are 8 tens; the sum being less than 10 tens, we set it under the column of tens, because it is tens. Finally, 3 hundreds and 2 hundreds are 5 hundreds; the sum being less than 10 hundreds, we set it under the columu of hundreds, for the same reason. Therefore, he has 586 acres. All similar examples are solved in like manner. 51 312 Ans. 586 By inspecting the preceding illustration, the learner will discover the following principle: Units of the same order are added together, and the sum is placed under the column added. (Art. 9.) NOTES.-I. The same orders are placed under each other for the sake of convenience and rapidity in adding. 2. We add the same orders together, units to units, tens to tens, etc., because different orders express units of different values, and therefore cannot be added to each other. Thus, 5 units and 5 tens neither make 10 units nor 10 tens, any more than 5 cents and 5 dimes will make 10 cents or 10 dimes. 3. We add the columns separately, because it is easier to add one order at a time than several. 4. The sum of each column is set under the column added, because being less than 10, it is the same order as that column. 7. What is the sum of $2321 + $123+$3245? 8. What is the sum of 3210 pounds + 2023 pounds + 4601 pounds? 9. What is the sum of 130230 +201321+402126 ? CASE II. 29. To find the Amount of two or more numbers, when the Sum of any column is 10, or more. OPERATION $436 324 645 1. What is the sum of $436, $324, and $645? ANALYSIS.-Let the numbers be set down as in the margin. Adding as before, the sum of the first column is 15 units, or I ten and 5 units. We set the 5 units under the column added, and add the I ten to the next column because it is the same order as that column. Now, I added to 4 tens makes 5 tens, and 2 are 7 tens, and 3 are 10 tens, or 1 hundred and o tens. We set the o, or right hand figure, under the column added, and add the I hundred to the next column, as before. The sum of the next column, with the I added, is 14 hundreds; or 1 thousand and 4 hundreds. This being the last column, we set down the whole sum. The answer is $1405. All similar examples may be solved in liko manner. By inspecting this illustration, it will be seen, $1405 When the sum of a column is 10 or more, we write the units' figure under the column, and add the tens' figure to the next column. NOTES.-I. We set the units' figure under the column added, and add the tens to the next column, because they are the same orders as these columns. 2. We begin to add at the right hand, in order to carry the tens as we proceed. We set down the whole sum of the last column, because there are no figures of the same order to which its left hand figure can be added. 30. Adding the tens or left hand figure to the next column, is called carrying the tens. The process of carrying the tens, it will be observed, is simply taking a certain number of units from a lower order, and adding their equal to the next higher; therefore, it can neither increase por diminish the amount. NOTE. We carry for tent instead of seven, nine, eleven, etc., because in the Arabic notation the orders increase by the scale of ten. If they increased by the scale of eight, twelve, etc., we should carry for that number. (Art. 13.) 31. The preceding principles may be summed up in the following GENERAL RULE. I. Place the numbers one under another, units under units, etc.; and beginning at the right, add each column separately. II. If the sum of a column does not exceed NINE, write it under the column added. If the sum exceeds NINE, write the units' figure under the column, and add the tens to the next higher order. Finally, set down the whole sum of the last column. NOTES.-I. As soon as the pupil understands the principle of adding, he should learn to abbreviate the process by simply pronouncing the successive results, as he points to each figure added. Thus, instead of saying 7 units and 9 units are 16 units, and 8 are 24 units, and 7 are 31 units, he should say, nine, sixteen, twenty-four, thirty-one, etc. Again, if two or more numbers together make 10, as 6 and 4, 7 and 3; or 2, 3, and 5, etc., it is shorter, and therefore better, to add IO at once. 31. How write numbers to be added? The next step? If the sum of a column does not exceed nine, what do you do with it? If it exceeds nine? The sum of the last column? 28. Note. Why write units under units, etc.? Why add the columns separately? Why not add different orders together promiscuously? Is the sum of 3 units and 4 tens, 7 units or 7 tens? When the sum of a column does not exceed 9, why set it under the column? 29. Note. If the sum of a column is 10 or more, why set the units' figure under the column added, and carry the tens to the next column? 30. What is meant by carrying the tens? Why does not carrying change the amount? Why carry for 10 instead of 6, 8, 12, etc. |