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Prologue 1 ELEMENTARY ADDITION
THE COUNTING-NUMBERS are 1, 2, 3, 4, and so on. (See Lesson 1, Numeration of the Whole Numbers.) The problem of addition, 6 + 3 = ? -- is the problem of naming the number that results when, starting with the first, you continue counting as many times as there are 1's in the other. Since 3 is three 1's -- 3 = 1 + 1 + 1 -- then, starting at 6, continue counting three times. "Six -- seven, eight, nine." 6 + 3 = 9. We can only add things of the same kind, that is, which must have the same name, and which are the units.
We cannot add 6 girls + 3 boys -- at least not until we call them "children." Skill in adding begins with remembering that 6 + 3 is 9 -- without having to count it. The student should begin by knowing all the sums that are less than 10. 3 + 2 = 5, 4 + 3 = 7, 5 + 3 = 8, and so on. First, though, an elementary fact of addition is that the order in which you add two numbers does not matter. If you know that 6 + 3 = 9, then you would also know that 3 + 6 = 9. But it is simpler to add 3 to 6, the smaller to the larger, than it is to add 6 to 3, the larger to the smaller. In any case, the order does not matter. That will be true for any number of terms. 2 + 3 + 4 = 3 + 2 + 4 = 4 + 3 + 2 = 9. Now practice these sums that are less than 10.
Next, it is useful to know all the ways of adding two numbers to get 10.
9 + 6 = 9 + 1 + 5 = 10 + 5 = 15. Say, "9 + 1 is 10, plus 5 is 15." That is, regroup 1 with 9 to compose 10. It can help to know the sum of a number added to itself. 0 is a number. It can answer the question, "How many?" We will see that when we write a number like 502, the 2 tells us how many ones; the 0 tells us how many tens; and the 5, how many hundreds. We can say that 0 is that number such that when you add it to any number, that number does not change. 5 + 0 = 5. 0 + 6 = 6. 0 means no units of whatever kind. There is a subtle difference between 0 units and nothing. Say that you have an account at the First National Bank and that your balance is $10. If you now withdraw $10, your balance is 0 dollars. But if you do not have an account at that bank, then you do not have a balance of 0 dollars. You have nothing! That is the difference between 0 and nothing. * Practice the following until you remember each one.
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