The sporadic nature of big numbers | Data Structures in Mathematics Math Foundations 176

In this video we derive a fundamental but destabilizing fact about natural numbers: that almost everything we know about arithmetic with natural numbers starts to break down as we proceed to investigate bigger and bigger numbers. By studying complexity and making some estimates using counting, we can precisely support the claim that most of the “numbers“ from 1 to z are in fact completely invisible to us. This raises the reasonable question of whether or not such “numbers“ deserve to be called that after all. And it casts a deep shadow of doubt over many claims of modern number theory. Video Content: 00:00 Destabilizing arithmetic with big numbers 3:58 What is the complexity of an arithmetical expression? 7:50 Observations about complexity 9:47 Mathematical expressions 14:22 Theorem 16:33 Numbers with digit length 19:58 Computing proportions of numbers with a given complexity 22:29 Probability that you can compress complexity to half 24:34 Nature o