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Research Article Open access CC BY 4.0

The Set of Rationale Numbers is Countably Infinite-A Simple Proof

Bernd E. Wolfinger

Journal of Advances in Mathematics and Computer Science · pp. 160–166 · Published 19 May 2023

10.9734/jamcs/2023/v38i71781

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.

Cardinality of rationals elementary injective mapping from Q to Z simplification of Cantor’s proof total-ordering of rationals

References (7)

  1. 1 Data and Computer Communications [DOI]
  2. 2 How to Solve It: A New Aspect of Mathematical Method
  3. 3 Recounting the Rationals [DOI]
  4. 4 Proofs of the Cantor-Bernstein Theorem [DOI]
  5. 5 Introduction to Topology [DOI]
  6. 6 Enumerating the Rationals from Left to Right [DOI]
  7. 7 Counting the Positive Rationals: A Brief Survey [DOI]

Cited by 2

Simple Criteria for \(\sqrt[n]{x}\) (n \(\in\) N, n \(\ge\) 2, x \(\in\) R) Being a Rational or an Irrational Number

Bernd E. Wolfinger · Journal of Advances in Mathematics and Computer Science · 2023

Showing 1 of 2 known citations — external sources report more than can currently be individually listed.

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