another proof of cardinality of the rationals

If we have a rational number $p/q$ with $p$ and $q$ having no common factor,and each expressed in base 10 then we can view $p/q$ as a base 11 integer,where the digits are $0,1,2,\mathrm{\dots },9$ and $/$. That is, slash ($/$) is a symbol for adigit. For example, the rational 3/2 corresponds to the integer $3\cdot {11}^2+10\cdot 11+2$.The rational $-3/2$ corresponds to the integer $-(3\cdot {11}^2+10\cdot 11+2)$.

This gives a one-to-one map into theintegers so the cardinality of the rationals is at most the cardinality ofthe integers. So the rationals are countable.