  A common question from students on the usual diagonalization proof for the
uncountability of the set of real numbers is: when a representation of real
numbers, such as the decimal expansions of real numbers, allows us to use the
diagonalization argument to prove that the set of real numbers is uncountable,
why can't we similarly apply the diagonalization argument to rational numbers
in the same representation? why doesn't the argument similarly prove that the
set of rational numbers is uncountable too? We consider two answers to this
question. We first discuss an answer that is based on the familiar decimal
expansions. We then present an unconventional answer that is based on continued
fractions.
