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MF95: The deep structure of the rational numbers

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. The full playlist is at
Length: 35:42


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