The Deep Structure of the Rational Numbers - Real Numbers and Limits Math Foundations

Explore the intricate structure of rational numbers in this 36-minute mathematics lecture. Delve into the heart of mathematics by examining the crucial role of rational numbers and their potential to reframe concepts traditionally associated with real numbers. Learn about basic definitions, including average and mediant, and focus on the [0,1] interval. Discover the concept of rational number levels and the famous Farey sequences, connecting them to Ford circles. Uncover the layered strata of rational numbers within the [0,1] interval, revealing their complex nature beyond the illusion of uniformity. Gain insights into weighted averages, convex combinations, and the deep connections between different mathematical concepts, setting the stage for future discussions on sequences of rational numbers.

Introduction
Rational numbers
Rational numbers vs Real numbers
Interval [0,1]
Theorem : Average and Mediant
Weighted average
Unit interval [ 0,1 ]
Definition : Rational numbers
Farey sequence
Ford circles