The rational numbers deserve a lot of attention, as they are the heart of mathematics. I am hopeful that modern mathematics will (slowly) swing around to the crucial realization that a lot of things which are currently framed in terms of "real numbers" are more properly understood in terms of the rationals-- in which case richer number theoretical/combinatorial aspects start to become more visible. After a review of basic definitions including the idea of average and mediant of two rational numbers, we focus on the interval [0,1], and discuss how convex combinations allow us to match up any two intervals. We introduce the idea of the level of a rational number, and the famous Farey sequences. These are connected with the notion of Ford circles which we talked about in MF14. A key principle is that even in [0,1], the uniformity of the rational numbers is an illusion; rather they are a layered strata which we can delve into deeper and deeper, yielding more and more complicated numbers.The layered structure of the rationals will play an important role when we start to discuss sequences of rational numbers in a few more videos.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.