Explore the most fundamental and important problem in mathematics in this 44-minute lecture from the "Famous Math Problems" series. Delve into the challenge of modeling the continuum, examining its historical and practical significance across various areas of pure mathematics. Investigate the continuity of curves, segments, planes, and space, and learn how to construct a ruler. Analyze the rational number line geometrically and consider the Pythagorean solution that equates the continuum with the rational number line. Conclude with a thought-provoking challenge on developing an axiomatic framework for this crucial mathematical concept.