Mathematical irrational numbers not so physically irrational

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.

💡 Research Summary

The paper investigates whether the decimal expansions of three well‑known irrational numbers—π, e, and the golden ratio φ—exhibit statistical properties akin to those found in physical phase transitions. The authors introduce a novel metric based on the “pair distribution” of the ten decimal digits: for each adjacent two‑digit combination (00, 01, … , 99) they compute the frequency of occurrence, normalize these frequencies, and evaluate a diversity index (essentially Shannon entropy or a related measure). By plotting this diversity index against the number of examined digits (sample size N), they discover a universal two‑phase behavior common to all three numbers.

In the first phase (small N, roughly below 10⁴ digits), the diversity index remains low, indicating that certain digit pairs dominate the sequence—a regime the authors liken to an ordered phase. As N increases, the index rises sharply and reaches a critical point N_c where the curve changes curvature. Beyond N_c the system enters a high‑diversity regime where the index saturates and follows a power‑law scaling D(N) ∝ N^α with an exponent α≈0.5–0.6. This scaling appears as a straight line on a log‑log plot, reminiscent of critical phenomena in statistical physics.

To test whether this pattern is intrinsic to the irrational numbers or merely a consequence of random digit sequences, the same analysis is performed on synthetic data consisting of truly random, uniformly distributed decimal digits. Remarkably, the random sequences reproduce the same two‑phase curve and the same scaling exponent, demonstrating that the observed behavior stems from the genuine randomness embedded in the decimal expansions of π, e, and φ, rather than any special arithmetic property of these constants.

The authors draw a parallel between the two phases and the ordered/disordered states of a physical system undergoing a continuous phase transition. The low‑diversity phase corresponds to an ordered state, the high‑diversity phase to a disordered state, and the critical point N_c to a critical temperature. This analogy provides a fresh perspective on the longstanding question of normality: a normal number would display each digit (and each digit pair) with equal asymptotic frequency, which would manifest as the high‑diversity, power‑law regime observed here. Although normality of π, e, and φ has not been rigorously proven, the empirical evidence presented supports the conjecture that they behave as normal numbers at the scales examined.

Methodologically, the study offers a quantitative tool for assessing the randomness of number sequences that goes beyond simple frequency counts. By focusing on pairwise correlations and their scaling with sample size, the approach captures both short‑range structure (the ordered phase) and long‑range statistical uniformity (the disordered phase). The paper suggests several extensions: applying the technique to other bases (binary, hexadecimal), higher‑order tuples (triplets, quadruplets), and directly mapping the statistical model onto known physical systems such as Ising or Potts models.

In conclusion, the research demonstrates that the decimal expansions of π, e, and φ are not “physically irrational” in the sense of lacking statistical regularity. Instead, they exhibit a universal two‑phase behavior that mirrors phase transitions in physical systems, rooted in an intrinsic property of genuine random digit distribution. This insight bridges number theory and statistical physics, providing a new framework for evaluating the randomness of mathematical constants.