We are interested in investigating how to rigorously and carefully extend arithmetic with rational numbers to a wider domain involving the symbol 1/0, represented by a ``sideways 8''. First we have a look at the simpler case of natural number arithmetic, where extending to infinity is relatively simple. For the rational number case, which is quite separate from the natural number case, we look to connect arithmetic with the geometry of the plane, and the projective geometry of lines through the origin. Our strategy is to focus on integral points [a,b] in the plane, and define suitable notions of rational addition and rational multiplication of these. By associating to the integral point [a,b], where b is not zero, the rational number a/b, we can make rational number arithmetic follow from arithmetic with integral points. In this way, the point [1,0] plays a role analogous to infinity. This connects also with earlier videos on the Stern-Brocot tree.
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.