Video Description

Real analysis is the foundation of many important areas of maths, with practical applications from the Fourier series to complex and functional analysis. Unfortunately, the typical first reaction to this approach is that it overcomplicates simple ideas when intuitions alone convey them well enough. In this video, with the help of the Cantor set, we will disprove two seemingly obvious examples from analysis: relating the size and measure of a set, and the amount of times a function can cross the axes. Chapters: 00:00 Intro 01:44 Section 1: Cardinality 12:00 Section 2: Measure Zero Sets 17:22 Section 3: The Cantor Set 22:52 Section 4: Crossing the Axis 26:38 Section 5: Distance Function 32:00 Outro Music: Purple Planet Music The picture of the character holding a cube is from the music video Dramaturgy by Eve. https://www.youtube.com/watch?v=jJzw1h5CR-I The animations in this video were created with Manim, an open-source animation library by 3b1b. https://www.manim.community/ This video is a submission to 3b1b’s SoME4 (Summer of Math Exposition).