Using Commutative Rings and Graphing Harmonic in Solutions and Approximation of Algebraic Equations
DOI:
https://doi.org/10.59992/IJSR.2026.v5n5p2الكلمات المفتاحية:
Complexity، Reciprocal Rings، Harmonic Methods، Merger، Algebraic Equations، Equation Solving، Accuracy، Algorithmsالملخص
Algebraic equations are the cornerstone of many life and scientific applications, whether engineering applications, industrial applications, or environmental applications. When it comes to complex algebraic equations, solutions must be found for this complexity, and through this study, which aims to solve the problems of complex algebraic equations using the methods of reciprocal rings and drawing. Graphics and combinatorial solutions and the combination of these methods, as the two branches of algebra and geometry, or what is called geometric algebra, are combined in a flexible and innovative way through which complex algebraic equations can be solved with high accuracy and through a methodology that relied on description, analysis and comparison methodologies. Three methods were combined using machine learning and neural network techniques and obtaining on the results of solving an algebraic equation related to solving a life problem, which is arranging study courses for a number of students, where the students’ requirements must be fulfilled by attending the largest number of courses without conflicting dates. The results indicated that the combination of these methods has provided many, many effective insights and strategies for solving some of the equations. As indicated by the results, the results of the model were the best by a rate ranging from 5% to 6%, and the accuracy of the model was the highest by a rate ranging between 5% to 11%, and the F1 of the model - the result also obtained the best results compared to the other two methods. At a rate ranging from 2% to 5, there are no significant differences between the expected results and the actual results of the model, which means the model is successful.
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