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 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
Canonical name	AnotherProofOfCardinalityOfTheRationals
Date of creation	2013-03-22 16:01:49
Last modified on	2013-03-22 16:01:49
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	10
Author	Mathprof (13753)
Entry type	Proof
Classification	msc 03E10