The Set of Rationale Numbers is Countably Infinite-A Simple Proof
Bernd E. Wolfinger *
Computer Science Department, University of Hamburg Vogt-Koelln-Str. 30 D-22527, Hamburg, Germany.
*Author to whom correspondence should be addressed.
Abstract
This research note presents a very simple proof of the interesting fact that the set Q of rationale numbers is still countably infinite as is the set of natural and integer numbers. The proof is based on several innovative ideas and neither relies on Cantor’s well-known diagonalization approach nor on the non-trivial Cantor- Schroeder-Bernstein Theorem.
In addition, we present a new proposal for a simple injective function f: Q\(\to\)Z, which allows one to encode rationals in a highly efficient manner and at the same time it can be understood much more easily (even by non-mathematicians). Moreover, also the inverse function f -1 can be derived in an extremely simple manner. Nevertheless, the growth of length is only logarithmic if we compare the resulting length of f(r=p/q) with the value of p, while the length of q has no impact at all on the length of f (r). Our approach also allows us to introduce a total ordering for the set of rationale numbers in a straight-forward manner.
Keywords: Cardinality of rationals, elementary injective mapping from Q to Z, simplification of Cantor’s proof, total-ordering of rationals