Simplification of calculations for repeating decimal fractions using a direct method

Repeating decimal fractions, both simple and compound, have long posed significant computational challenges in mathematics and education. Traditional methods, which convert these decimals into fractions for algebraic operations, are often time-consuming, error-prone, and complex, particularly when dealing with long, multiple, or nested repeating cycles. This study presents a novel direct method for performing addition, subtraction, and division on repeating decimals without requiring conversion to fractions. By aligning and synchronizing periodic sequences, the method enhances computational speed, maintains high accuracy, and minimizes human error. The approach was systematically developed through detailed analysis of decimal structures, construction of original problems, and rigorous verification across a wide range of cases, including simple cycles, composite cycles, and multiple simultaneous repeating sequences. Comparative evaluation with conventional fraction-based methods demonstrates that the proposed approach significantly reduces computational complexity while achieving identical results. Preliminary observations further indicate promising potential for extending the method to multiplication of repeating decimals. The simplicity, efficiency, and broad applicability of this method make it particularly suitable for educational contexts, theoretical research, and practical applications in engineering, computer science, cryptography, and other related fields. Overall, this study provides a clear, systematic, and reliable framework for handling repeating decimal computations, offering both pedagogical and computational advantages over traditional fraction-based techniques, and opening opportunities for further development and software implementation.