Explore the fascinating world of algebraic curves in this comprehensive 35-minute lecture from the MathFoundations series. Delve into the historical development of curves, from ancient Greek concepts to 17th-century Cartesian innovations. Examine the challenges with Jordan's curve definition and discover the polynumber approach to algebraic curves. Learn how to extend elementary calculus beyond functions and investigate various curve types, including Archimedes' spiral, Ptolemy's epicycles, Newtonian cubics, and mechanical curves. Gain insights into the precise definition of a 'curve' and explore algebraic curves using bipolynumbers. Conclude with an examination of the Lemniscate of Bernoulli, setting the stage for generalizing algebraic calculus to algebraic curves.