Evolving Patterns in Irrational Numbers Using Waiting Times between Digits

There is an increasing interest in determining if there exist observable patterns or structures within the digits of irrational numbers. We extend this search by investigating the interval in position between two consecutive occurrences of the same digit, a kind of waiting time statistics. We characterise these by the burstiness measure which distinguishes if the inter-event times are periodic, bursty, or Poisson processes. Furthermore, the complexity–entropy plane was used to determine if the intervals are stochastic or chaotic. We analyse sequences of the first 1 million digits of the numbers π, e, 2, and ϕ. We find that the intervals between single, double, and triple digits are Poisson processes with a burstiness measure in the range −0.05≤B≤0.05 for the four numbers studied. This result is supported by a complexity–entropy plane analysis, which shows that the time intervals have the same characteristics as Gaussian noise. The four irrational numbers have identical degrees of complexity and burstiness in their inter-event analysis.