Explore the foundations of complex numbers and their connection to rotations in this 58-minute lecture from the Famous Math Problems series. Delve into W. R. Hamilton's groundbreaking work on quaternions and the rotation problem in 3D space. Learn about the algebraic representation of rotations using complex numbers, the concept of quadrance, and the rational parametrization of a circle. Gain insights into the pure algebraic approach necessary for understanding quaternions, without relying on transcendental notions or real numbers. Discover how turns can substitute for angles and how the rational parametrization of a circle relates to complex numbers. This comprehensive lecture sets the stage for tackling the rotation problem in three dimensions and introduces quaternions in subsequent talks.