Cantor says you can't list the real numbers. Here's how you can. Every number is represented by a string of symbols. For example, e is represented by lim n-> [/math]\infty[/math] (1+1/n)^n. Assign a number to each symbol. ( can correspond to 1, + can correspond to 5, and so on. Numbers go after the symbols. If the last symbol is at 1000, then 1 maps to 1001, 2 maps to 1002, etc. If you convert all the symbols representing the number into their mapping, then concatenate them. For example one number can consist of symbols 0010, 0239, 0203, 0200. Then the number is represented by 0010023902030200. Every number has one or more representations using this method. This shows there is an injective mapping from the integers to the reals.