The integration of mathematics and computer science constitutes a fundamental pillar of modern scientific and technological development. Mathematics provides formal structures, abstract reasoning, and theoretical models, while computer science translates these models into computational processes through algorithms, programming paradigms, and system architectures. This close relationship enables the design, analysis, and optimization of complex computational systems. This article examines the mathematical foundations underlying computer science, including discrete mathematics, linear algebra, probability theory, logic, and numerical analysis. It further explores how these mathematical tools are applied in key computational domains such as artificial intelligence, machine learning, cryptography, data science, scientific computing, and computer graphics. Particular attention is given to algorithmic thinking and optimization, highlighting how mathematical rigor contributes to efficiency, correctness, and scalability in computational solutions. In addition, the paper discusses emerging research directions, including quantum computing, explainable artificial intelligence, and computational mathematics, emphasizing the growing importance of interdisciplinary approaches. By demonstrating the deep interdependence between mathematics and computer science, this study underscores the necessity of integrated education and research to address increasingly complex real-world problems and to drive future innovation.