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By Turner J., Kautz W.H.

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Evaluate the following sums: (a) 3 + 3; (b) 6 + 3; (c) 9 + 3; (d) 4 + 4; (e) 8 + 4. (f ) What conclusion might you draw? 4. Evaluate the following sums: (a) 2 + 3; (b) 5 + 4; (c) 3 + 4 ; (d) 2 + 7 . (e) What conclusion might you draw? 5. Is the sum of every two natural numbers also a natural number? Why? 6. What meaning would you attach to the indicated sum 4 + 0? To 0 + 4? Explain. 7. On page 34 it is asserted that "the general rule of procedure for solving all problems (in the addition of two natural numbers) can be easily deduced".

Similarly, it could also be shown that 48 NUMBER SYSTEMS AND OPERATIONS OF ARITHMETIC 3+4-4+3 3+5=5+3 1+2=2+1 6 + 1 0 = 10 + 6 0 + 18 = 18 + 0 1,693,432 + 238,984 = 238,984 + 1,693,432, and so on. The general principle involved, that the sum of two numbers is the same in whichever of the two possible orders they may be added, or that the value of a sum of two numbers is not changed by reversing (or commuting) the order in which the two numbers are considered, is called the commutative law for addition.

We say "more-or-less direct" because some people do not do this in a purely direct manner but with the aid of some intermediate set of elements such as a set of fingers, or the numerical representation built into a computing machine, etc. Your set of clams Intermediate set Third leading subset 1 2 3 Any matching of a specified set with a leading subset of the natural numbers, whether direct or indirect (and involving the use of some standard intermediate reference set), is recognized as counting; and since whenever an indirect correspondence is used the intermediate set could be eliminated to produce a direct correspondence, it is such a direct one-to-one correspondence that we shall keep in mind as constituting the essential nature of the counting process.

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