The increasing complexity of computer science and information technology systems requires analytical approaches capable of connecting mathematical reasoning, logical structures, computational models, and systematic problem-solving. This research and review paper develops an integrated analytical framework for understanding information technology systems through mathematical and logical foundations. The study synthesizes concepts derived exclusively from the works of Courant and Robbins, Hardy and Wright, Kac and Ulam, Penrose, and Stewart. The proposed framework considers mathematical abstraction, numerical reasoning, formal logic, structural relationships, and equation-based representation as interconnected analytical dimensions. Rather than treating mathematics as an isolated computational tool, the paper positions it as a foundation for designing, interpreting, validating, and optimizing information technology systems. The methodology develops a conceptual analytical architecture consisting of problem abstraction, mathematical representation, logical validation, computational interpretation, and system-level evaluation. The findings indicate that integrated analytical approaches can improve conceptual clarity, support systematic reasoning, and provide a stronger basis for evaluating complex computational systems. The analysis also identifies limitations associated with abstraction, model assumptions, computational complexity, and the gap between theoretical representations and practical implementations. The study contributes a unified theoretical perspective for researchers examining the relationship between mathematical thinking, logical reasoning, and information technology system development.