The traditional dichotomy between pure and applied mathematics has increasingly dissolved as visual, experimental, and computational methods have become integral to theoretical mathematical research. This chapter examines the historical evolution and contemporary role of these approaches in theoretical mathematics, drawing upon examples from number theory, topology, analysis, and algebra. It is argued that computational experimentation, visualization techniques, and algorithmic methods are not merely auxiliary tools but constitute genuine modes of mathematical reasoning that have fundamentally transformed how theoretical questions are formulated, explored, and resolved.