In this video we introduce Brahmagupta's celebrated formula for the area of a cyclic quadrilateral in terms of the four sides. This is an obvious extension of Heron's formula. We are interested in finding a rational variant of it, that will be independent of a prior theory of `real numbers', `square roots' and `lengths'. For motivation we look at the situation of four affine 1-points. Is there an analog of the Triple Quad Formula? Yes there is: it is the more mysterious and complicated Quadruple Quad Formula. To get it, we will examine some important manipulations for a pair of quadratic equations which are of independent interest.This lecture has some more serious algebra in it: a great place to practice your manipulation and organizational skills. If you are teaching college mathematics, please consider doing yourself and your students a favour: teach them some of the material of this lecture carefully and explicitly!
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.