Monday, February 5, 2018

Are the Digits of Pi Truly Random? - Must Read for Math and Data Geeks

This article covers far more than the title suggests. It is written in simple English and accessible to quantitative professionals from a variety of backgrounds. Deep mathematical and data science research (including a result about the randomness of Pi, which is just a particular case) are presented here, without using arcane terminology or complicated equations.  
The topic discussed here, under a unified framework, is at the intersection of mathematics, probability theory, chaotic systems, stochastic processes, data and computer science. Many exotic objects are investigated, such as an unusual version of the logistic map, nested square roots, and representation of a number in a fractional or irrational base system. 
The article is also useful to anyone interested in learning these topics, whether they have any interest in the randomness or Pi or not, because of the numerous potential applications. I hope the style is refreshing, and I believe that you will find plenty of material rarely if ever discussed in textbooks or in the classroom. The requirements to understand this material are minimal, as I went to great lengths (over a period of years) to make it accessible to a large audience.
The randomness of the digits of Pi is one of the most fascinating, unsolved mathematical problems of all times, having been investigated by many million of people over several hundred years. The scope of this article encompasses this particular problem as part of a far more general framework. More questions are asked than answered, making this document a stepping stone for future research.
This article is structured as follows:
1. General Framework
  • Questions, Properties and Notations about Chaotic Sequences Investigated Here
  • Potential Applications, Including Random Number Generation
2. Examples of Chaotic Sequences Representing Numbers
  • Data Science Step
  • Mathematical Step
  • Numbers in Base 2, 10, 3/2 or Pi 
  • Nested Square Roots
  • Logistic Map
3. About the Randomness of the Digits of Pi
  • The Digits of Pi are Random in the Logistic Map System
  • Paths to Proving Randomness in the Decimal System
  • Connection with Brownian Motions
4. Curious Facts
  • Randomness and The Bad Seeds Paradox
  • Application to Cryptography, Financial Markets, and HPC
  • Exercises
  • Digits of Pi in Base Pi

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