Explore algebraic calculus on the unit circle and its imaginary counterpart in this 35-minute mathematics lecture. Delve into polynumbers and polynomials of two variables, examining Taylor polynumbers, subderivatives, and Taylor expansions around points. Investigate tangent planes and lines, and visualize concepts through three-dimensional diagrams of elliptic paraboloids. Compare the algebraic approach to traditional calculus methods, and extend the discussion to the imaginary unit circle. Learn about arranging subderivatives, reversing variable roles, and using tangent planes for approximation. Gain insights into the foundations of mathematics while enjoying a comprehensive exploration of calculus on simple algebraic curves.