Overview

The real numbers can be represented conveniently by a line. Every point on the line corresponds to a real number, but only a few are marked in Figure 2.1. The line stretches infinitely far in both directions, towards ? and ? ?, which are not themselves numbers in the conventional sense but are included among the extended real numbers. The integers are the numbers 0, 1, ?1, 2, ?2, 3, ?3, . We say that there is an infinite but countable number of integers; by this we mean that every integer would eventually appear in the list if we count for long enough, even though we can never count all of them. The rational numbers are those that consist of a ratio of two integers, e.g., 1/2, 2/3, 6/3; some of these, e.g., 6/3, are integers. To see that the number of rational numbers is countable, imagine them listed in an infinite two-dimensional array as in Figure 2.2. Listing the first line and then the second, and so on, does not work, since the first line never terminates. Instead, we generate a list of all rational numbers diagonal by diagonal: first 0, then 1/1; then 2/1, 1/2; then 3/1, 2/2, 1/3; then 4/1, 3/2, 2/3, 1/4; etc. In this way, every rational number (including every integer) is eventually generated. In fact, every rational number is generated many times (e.g., 1/2 and 2/4 are the same number). However,...