Rescaled range and detrended fluctuation analysis for decimal digits of irrational numbers

Irrational numbers appear in various research areas, from science to high-level technologies, and even in human life, from living organisms to music. This research aims to quantify autocorrelation for decimal digits of four famous irrational numbers, including [Formula: see text], [Formula: see text], e and [Formula: see text] using rescaled range analysis and detrended fluctuation analysis, up to their one billion digits. In this way, the Hurst exponent and fluctuation exponent are extracted by the mentioned methods. The estimated Hurst and fluctuation exponents, as statistical metrics, indicate short-range memory in the sequence of decimal digits of the studied irrational numbers. Both exponents show finite-size scaling behavior by increasing the number of decimal digits, from [Formula: see text] up to [Formula: see text], and they generally tend to reach their marginal value of 0.5, which confirms the lack of long-range correlation. Consequently, the decimal digits of irrational numbers can be considered as truly random sequences, with uncorrelated randomness.