The journey of complex numbers from mathematical heresy to fundamental scientific tool represents one of the most remarkable intellectual transformations in history. These numbers of the form a + bi (where i² = -1) have transcended their origins in algebra to become indispensable across physics, engineering, and technology. This comprehensive study examines their historical evolution, deep mathematical properties, and unparalleled applications that continue to shape modern science. Through detailed analysis of pivotal developments, we demonstrate how complex numbers provide the mathematical language for describing phenomena from quantum entanglement to wireless communication.