Fractal and statistical analysis on digits of irrational numbers

In this study, for quantifying and comparing the complexity of the digits of irrational numbers, we used the first one million digits to calculate the fractal dimensions of the digits of 10 irrational numbers with long sequence of known digits via box-counting algorithm. The irrational numbers we studied are five algebraic numbers (2,3,5,6and7) and five transcendental numbers (π, e, log(2), ζ(3) and Champernowne’s constant). For statistical analysis, we performed 2000 repeated calculations for each number with segment of digits with length 200, 300, 400 and 500. The distributions of the estimated fractal dimensions seem normal by the histograms. Analysis of variance was used to test the equality of means for the fractal dimensions among the numbers and within the number for different digit lengths. The pattern of complexity of the digits in these numbers based on the estimated fractal dimensions is similar except that of Champernowne’s constant.