Mathematical irrational numbers not so physically irrational

TL;DR

This paper explores the digit distribution in the decimal expansions of π, e, and the golden ratio, revealing universal phase transition behaviors linked to randomness, suggesting these mathematical irrationals are not as physically irrational as previously thought.

Contribution

It uncovers a universal two-phase behavior in digit distributions of key irrational numbers and connects this to phase transition phenomena in physical systems.

Findings

Identifies a universal two-phase behavior in digit distributions

Shows the behavior follows a power law collapse

Links the digit distribution properties to intrinsic randomness

Abstract

We investigate the topological structure of the decimal expansions of the three famous naturally occurring irrational numbers, $\pi$, $e$, and golden ratio, by explicitly calculating the diversity of the pair distributions of the ten digits ranging from 0 to 9. And we find that there is a universal two-phase behavior, which collapses into a single curve with a power law phenomenon. We further reveal that the two-phase behavior is closely related to general aspects of phase transitions in physical systems. It is then numerically shown that such characteristics originate from an intrinsic property of genuine random distribution of the digits in decimal expansions. Thus, mathematical irrational numbers are not so physically irrational as long as they have such an intrinsic property.