Many people know there are more real numbers than counting numbers, and some don't. Some even find the results contentious or unsettling. Here, rather than getting into the social ramifications of this concept, I'd like to run through the operative part of the most well known proof. I will talk about the diagonal argument.
The diagonal argument, as far as I know, was first introduced by Georg Cantor in his 1891 article "Über eine elementare Frage der Mannigfaltigkeitslehre". I wasn't able to find a pdf copy of the original paper, but Jan scared up this excerpt from a book about Cantor and written by Ernst Zermelo in 1932...I think I can trust the source. The 1891 proof wasn't actually Cantor's first proof that real numbers are more numerous than counting numbers, but it is prettier than his 1874 effort.