sion, once you have it, you see that in some sense division is no longer necessary. In enabling division you have created a system in which every (nonzero) number has a *multiplicative inverse* or *reciprocal*. In this system, division by a number (other than zero) is accomplished by multiplying by its reciprocal, which is the source of the “invert and multiply” rule for dividing fractions.

**The Rational Numbers**

You have seen how a desire to solve subtraction problems with no solutions in whole numbers leads to the construction of the integers. In a very similar way, the desire to solve division problems with no solutions in whole numbers leads to the construction of the positive rational numbers (along with zero). But neither of these number systems does it all: There are some integers that will not divide a given integer, and there are some positive rational numbers that cannot be subtracted from a given positive rational number (and still remain within the system). Thus, if you want to be able to always do both operations (except dividing by zero), you have to extend these systems further: You have to annex reciprocals to the integers, and you have to annex negatives to the positive rationals.

That process involves a lot more work. The end result, however, is as elegant as one could wish. It turns out that either procedure produces a system in which all operations are possible, with additive inverses for all numbers and multiplicative inverses for all numbers except zero. In this system, subtraction of a number becomes addition of its additive inverse, and division by a number becomes multiplication by its multiplicative inverse. The