content simply to admire these new creations. You get into situations in which you want to do arithmetic with them also. If I owe Bart two apples and I owe Teresa four apples, how many apples do I owe all together—that is, what is (–2)+(–4)? If on Monday I get into a situation that leaves me two apples short and this happens again on Tuesday and Wednesday, how many apples short am I then—that is, what is 3×(–2)? Besides enlarging their idea of number, people have had to extend the arithmetic operations to this new larger class of numbers. They have needed to create a new, enlarged number system. The new system, encompassing both positive and negative whole numbers, is called the integers.

How do people decide what arithmetic in this extended system is (or should be)? How do they create recipes for adding and multiplying integers, and what are the properties of these extended operations? They have two guides: (a) intuition and (b) the rules of arithmetic, as described above and in Box 3–1. Fortunately, the guides agree.

Consider first the intuitive approach: Think hard about a lot of different cases and decide what is the right way to add and multiply in each one. To use intuition, you need to think in terms of some concrete interpretation of arithmetic. The yield of financial transactions is a good one for these purposes. Here negative amounts are money you owe, and positive amounts are money that you have or are owed by someone else. If you owe $2 to Joan and $3 to Sammy, then you owe $5 to the two of them together. So (–2)+(–3)=–5. If you owe $2 to three people, then you owe $6, so 3×(–2)=–6. If you have a debt of $2 and someone takes it away, you have gained $2. So –(–2)=2. If someone takes three $2 debts away from you, the amount you owe is then $6 less than before, which means you have $6 more. Therefore (–3)×(–2)=6. Continuing in this way, you can puzzle out what the sum, difference, or product of any two integers should be. The trouble with this approach is that it is somewhat contrived and depends upon making decisions about how to interpret each case in the particular context.5

Another approach6 is to use an exploratory method to reason how the operations should extend from the whole numbers. By extending the patterns in the table below, you find that (–3)×(–2)=6, just as was shown above in context.

3+2=5

3–2=1

3×2=6

(–3)×2=–6

3+1=4

3–1=2

3×1=3

(–3)×1=–3

3+0=3

3–0=3

3×0=0

(–3)×0=0

3+(–1)=

3–(–1)=

3×(–1)=

(–3)×(–2)=

3+(–2)=

3–(–2)=

3×(–2)=

(–3)×(–2)=



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