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# 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.

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03E10*no label found*

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## http://planetmath.org/?op=getobj&id=8073&from=objects

I removed my comments.