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[parent] another proof of cardinality of the rationals (Proof)

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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Cross-references: countable, cardinality, cardinality of the rationals, map, one-to-one, rational, digits, integer, base, factor, rational number
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This is version 7 of another proof of cardinality of the rationals, born on 2006-06-23, modified 2007-04-26.
Object id is 8073, canonical name is AnotherProofOfCardinalityOfTheRationals.
Accessed 1383 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
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