Characterization of Rational Numbers Using Kronecker’s Orbit: A Didactic Approach

Abstract

For every real number x, we define as integer part the biggest integer k so that k ≤ x and is expressed [x]. The difference of the number from its integral part is defined as decimal part of x and expressed with () [) 0,1 x ∈. Consequently, for every x, the Kronecker's orbit is defined, namely the set () { } nx n ∈ . Through Kronecker's orbit, rational numbers are characterized as the numbers whose orbit is a bounded set, while irrational numbers are characterized as the numbers whose orbit is a dense set. Using this fundamental theoretical result and utilizing a computer, a didactic approach was established, initially referring to the definition of rational numbers as fraction equivalence classes and basically to the segregation of rational and irrational numbers. This didactic approach also incorporates elements of ancient Greek mathematics history. The proposition was applied to students and was evaluated.