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 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.

单词	AnotherProofOfCardinalityOfTheRationals
释义	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 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.
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