I was reading this pop math piece on "the different sizes of Infinity." The article explains why the real numbers are uncountably infinite.
Taking a real number, my uneducated mathematical mind intuits that it could be considered as an infinitely-long word made up of letters drawn from an infinitely long alphabet (the rational numbers) in arbitrary combination (hence $+\infty$ to the power of $+\infty$ possible combinations). This would seem to suggest that the real numbers are countably infinite.
Of course, I know my reasoning must be wrong, but I do not have the mathematical background to find out why. Does anyone care to explain?