Analysis of Typical Errors in Algebraic Transformations: From the Domain of Admissible Values to Physical Constraints
Analysis of Typical Errors in Algebraic Transformations: From the Domain of Admissible Values to Physical Constraints
Nikoghosyan Gagik, Manukyan Vardan
Summary
Key words։ identity, extraneous root, discontinuity, modeling, interdisciplinary connections, education The article examines one of the most critical issues in science and mathematics education: the formal assimilation of instructional material and the methods to overcome it. The study emphasizes that disregarding the domain of admissible values (DAV) during algebraic transformations can lead not only to mathematical errors but also to a distorted perception of physical reality.
The scientific and methodological novelty of the work lies in the introduction of the “functional-semantic DAV” concept, which integrates the identity of the formal mathematical apparatus with the constraints of physical modeling. The article provides detailed analysis of typical errors resulting from the neglect of the DAV, using examples of fractional and logarithmic expressions. It is demonstrated that while extraneous roots in equations can be eliminated through verification, the breach of equivalence in inequalities leads to irreversible losses.
In the physics section, a methodological parallel is established between logical “discontinuities” and the “phase transitions” of physical systems. Using examples such as Ohm’s law, thermal phenomena, and fundamental constants (absolute temperature, the speed of light), it is shown how the domain of physical meaning serves as a natural filter for mathematical solutions. The study substantiates that the analysis of the DAV is not a supplementary step but an essential component of the solution algorithm. This approach enables the learner to overcome formulaic “formalism” and construct a comprehensive scientific understanding of the phenomenon.