We're looking at extending the natural numbers, so we can do math that the natural numbers can't do. Last time we looked at addition and expanded the natural numbers to the integers. The useful thing about this extension is that it kept all of the established properties of the natural numbers and added new ones.

Last time ended with the question of negative exponents. This is actually an example of a general case of multiplication problems that can't be solved with the integers. From the properties of exponents, it should be true that a^{m}a^{−m}=a^{(m−m)}=1 (as long as a≠0). But it is not generally true that if a and b are integers, there exists an integer x such that ax=b.

The solution is to create the rational numbers as an extension of the integers. The rational numbers keep all the same properties for addition and multiplication as the integers. Multiplication also has the new property that for any rational number x except for 0, there exists a rational number y such that xy=1. (y is the reciprocal or multiplicative inverse of x.) We can define division as multiplication by the reciprocal. Exponentiation is now defined for negative exponents (but not for rational exponents in general) as long as the base is not 0. And the rational numbers are ordered.

The next extension in this sequence is the real numbers, but before I get there, next time I'm going to take a detour through abstract algebra to talk about fields.