# 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 $0,1,2,\ldots,9$ and $/$. That is, slash ($/$) is a symbol for a digit. 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 the integers so the cardinality of the rationals is at most the cardinality of the integers. So the rationals are countable.

Title another proof of cardinality of the rationals AnotherProofOfCardinalityOfTheRationals 2013-03-22 16:01:49 2013-03-22 16:01:49 Mathprof (13753) Mathprof (13753) 10 Mathprof (13753) Proof msc 03E10