Box 3–3 Consequences of the Basic Properties: Formulas for the Arithmetic of Negation
Subtraction and negation. Subtracting a number is the same as adding its opposite. For example, 5–3=5+(–3) and 5–(–2)=5+2. In general, s–t=s+(–t).
Multiplication and negation. Negation is the same as multiplication by –1. For example, –3=(–1)×3 and 2=(–1)×(–2). In general, –s=(–1)×s.
Opposite of opposite. The opposite of the opposite of a number is the number itself. For example, –(–3)=3. In general, –(–s)=s.
be able to solve subtraction problems. Now, in the integers, subtraction is a true operation in the sense that you can subtract any integer from any other. As described in the rule on additive inverses in Box 3–2, for every integer, there is another integer, called its opposite or additive inverse, that counter-balances it: the two sum to zero. Thus 2+(–2)=0, and –84+84=0. The second equation means that –(–84)=84 and leads to the rule on subtraction and negation in Box 3–3, which says that subtracting an integer gives the same result as adding its additive inverse. Thus 2–3=2+(–3), and 24–(–7)=24+(–(–7)), which is equal to 24+7=31. Thus, at least on a conceptual level, subtraction is merged into addition, and you really only need to have the single operation of addition to capture all the arithmetic of addition and subtraction. As soon as subtraction is made into a true operation by extending the whole numbers to the integers, you also get additive inverses, which allows you to subordinate subtraction to addition. This sort of simplification illustrates a kind of mathematical elegance: Two ideas that seemed different can be subsumed under one bigger idea. As we show below, the analogous thing happens to division when you construct rational numbers. That subordination is the best justification for why mathematicians talk about only the two operations of addition and multiplication when discussing number systems, and not all four operations recognized in school arithmetic.
Forgetting for a moment the triumph with integers, return to the whole numbers and the problem of division. Here the situation is in some sense