Remarkably, the Triple Quad Formula (TQF), even though it is purely a one-dimensional result, contains in it the seeds of higher dimensional results. In this video we look at a classical result from ancient Greece, called Heron's formula for the area of a triangle in terms of the side lengths, and reformulate it purely in terms of rational quantities: the quadrances of the sides, not the lengths!We call this reformulation Archimedes' theorem, since it is known from Arab sources that Archimedes had Heron's formula several hundred years before Heron did.
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.