I’ve been familiar with the notion of countable and uncountable infinity for a long time and the odd tricks you can do prove trivial stuff like that the number of even numbers is equal to the number of counting numbers (1, 2, 3 …).  An article in Sciencenews.org‘s Math Trek column discusses an alternate proof of new sizes of infinity.  The article mentions Cantor‘s diagonal proof using decimals, I think it’s easier to think about with a binary system.