We illustrate algebraic calculus on the simplest algebraic curves: the unit circle and its imaginary counterpart. Starting with a polynumber/polynomial of two variables, the derivation of the Taylor polynumber, subderivatives, Taylor expansion around a point [r,s] and various tangents are analogous to the case of a polynumber/polynomial of one variable. We get tangent planes and tangent lines both corresponding to the first tangent.The algebraic derivation is illustrated with three-dimensional diagrams involving the associated elliptic paraboloid to the unit circle.The background noise in parts of the video is due to crickets, common in the Australian summer---sorry about that. I will tell them to keep it down next time.This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.