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 . Consequently, for every x, the Kronecker’s orbit is defined, namely the set . 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.
Highlights
In 1881, Leopold Kronecker defined what is called the “rationality domain”, which is a body of polynomials in modern terms
Students usually believe that when a computer presents a geometrical space in a geometrical problem or gives a numeric solution to an algebraic problem, this is automatically proven. This must be taken into consideration and we have to explain that the computer is used to help us make a conjecture, but proof is achieved only when carried out based on mathematical principles and procedures
We conclude to the final conjecture: “If we have a fraction in lowest terms k/l, the orbit consists of l elements”
Summary
In 1881, Leopold Kronecker defined what is called the “rationality domain”, which is a body of polynomials in modern terms. Students believe that natural numbers’ properties are applied on rational numbers (Kollias et al, 2004; Voskoglou & Kosyvas, 2012) This student difficulty is reinforced by the fact that they come into contact for the first time with new mathematical symbolisms, which they must understand and learn to use (Ni & Zhou, 2005). Distinction among several types of numbers remains muddy in general, each time depending on their semiotic representations” (Voskoglou & Kosyvas, 2012) This represents a very usual student opinion: fractions are rational numbers and roots are irrational numbers. DGS are programs of geometrical visualisation (Cabri, Sketchpad, EukliDraw, a.o.) Through this software, students become active members of knowledge, understand geometry better, are leaded to conclusions, express questions and conjectures. They will guide the students and will provide the scientific verification of their results
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