Explore the fundamental concept of modeling the continuum through the rational number line in this mathematics lecture. Delve into a geometric presentation of the rational number line, assuming prior knowledge of affine geometry. Discover an alternative algebraic and logical approach to mathematics based on the rational number line as the primary continuum. Examine the projective view of the rational line, including constructions from 19th-century mathematicians. Learn about the basic properties of rationals, the introduction of irrationalities through Euclidean geometry, and attempts at solving polynomial equations. Gain insights into this profound solution to one of mathematics' most important problems.