FOR THE MATHEMATICS TEACHER-TRAINEE: RATIONAL, IRRATIONAL AND FRACTIONAL EXPRESSIONS

Abstract

This study focuses on the concept of the rational, irrational and fractional, and methods of their expressions. In the system of school education, the ideas related to developing the concept of number have an exceptional worldview potential, being the basis for perceiving the dialectical correlation of the rational and irrational, finite and infinite, convergent and divergent, limited and unlimited, as well as other issues. However, when it comes to the expressions with irrationalities, most often these matters are associated with roots of various degrees from non-negative integer or rational numbers, or the number π, which significantly narrows the depth of teacher-trainees’ perceiving and conceptualizing the essence of the concept of irrationality in mathematics.

This article analyzes the ways of representing irrational numbers and expressions in the form of rational expressions, as well as the possibilities that can be exploited in order to expand mathematical horizons of mathematics teacher-trainees. Furthermore, in the article the existing potential connection between irrationalities and chain fractions is illustrated.

A special emphasis in the study is placed on the fact that only quadratic irrationalities (irrational roots of quadratic equations with integer coefficients) get decomposed into periodic chain fractions. Other irrationalities such as π, e, etc. can also break down into chain fractions, but these breakdowns will not be periodic.

It is maintained that based on the procedures mentioned above, mathematics teacher-trainees will be able to conceptualize more deeply the idea of an irrational number, ways of its representation and usage in measuring quantities, as well as in organizing and managing circle work with learners